The Distribution Characteristics of Adsorbed CH₄ in Various-Sized Pore Structures of Coal Seams

Abstract

The distribution characteristics of adsorbed CH₄ across pores of various sizes underpin coal mine gas disaster prevention, resource assessment, and efficient coalbed methane (CBM) extraction. Utilizing Grand Canonical Monte Carlo (GCMC) simulations as a theoretical framework, this study establishes a mathematical model linking microscopic pore structure to macroscopic CH₄ adsorption thermodynamics in coal. Results reveal that micropores (0.38–1.5 nm) dominate pore structures in coal. For micropores (0.419–1.466 nm), CH₄ adsorption follows the Dubinin-Astakhov (DA) equation. The adsorption parameters change significantly as pore diameter increases, indicating that micropore size distribution predominantly governs CH₄ adsorption in coal. For larger pores (1.619–4.040 nm), Langmuir equation analysis reveals no significant changes in CH₄ adsorption parameters with increasing pore size, suggesting that the CH₄ adsorption behavior in pore structures larger than 1.5 nm is relatively consistent and does not vary substantially with respect to pore size. The accuracy of the mathematical model improves with coal rank, reducing prediction errors from 35.371% to 11.044%. Decomposed CH₄ adsorption isotherms reveal that while CH₄ adsorption capacity increases with equilibrium pressure for all pores, smaller pores achieve saturation at lower pressures. The proportion of total adsorption attributed to smaller pores peaks before declining with further pressure increases.

1. Introduction

Coalbed methane (CBM), a significant co-product of coal mining, is primarily composed of methane (CH₄). In the environmental context, CH₄ acts as a potent greenhouse gas with a global warming potential 25 times greater than that of CO₂ [1]. From an energy perspective, CH₄ represents an efficient and clean fuel. The calorific value of one cubic meter of CH₄ is 35.9 MJ, equivalent to the energy released by the combustion of 1.2 kg of standard coal [2]. CH₄ poses a significant hazard during coal mining, capable of triggering coal and gas outbursts, gas explosions, and other catastrophic accidents [3]. Its extraction from coal reservoirs not only mitigates environmental pollution but also enables CH₄ to be repurposed for energy use. This fundamentally reduces the risk of such accidents. Meanwhile, the extraction and development of CBM are conducive to the healthy development of the energy industry. They also contribute to the realization of carbon emission reduction goals [4,5,6,7]. Efficient CBM recovery depends on accurately characterizing its distribution across coal’s complex pore networks, offering critical guidance for extraction strategies and resource assessment [8,9,10,11,12,13].

As is widely known, coal inherently contains a vast network of pores with diverse sizes, shapes, and origins, which provide storage space for CH₄ and govern its adsorption characteristics [14,15]. It is widely accepted that CH₄ exists in coal in both adsorbed and free states within pores of varying sizes. The adsorbed state predominantly resides in micropores and accounts for over 80% of the total gas. However, the detailed distribution of adsorbed CH₄ across different pore sizes remains unclear, and adsorption mechanisms and theoretical models for CH₄ in coal are still debated [16,17]. Previous studies primarily inferred CH₄ adsorption behavior in coal by correlating adsorption characteristics with pore structural parameters such as pore volume and specific surface area. Moore et al. [18] found that the CH₄ adsorption capacity of coal reservoirs is related to the pore surface area rather than pore volume. This suggests that CH₄ is predominantly stored on the coal pore surface via adsorption. Bustin et al. [19] revealed that the CH₄ adsorption capacity of coal is closely related to the micropore structure, particularly for micropores with a diameter less than 2 nm. They concluded that CH₄ is mainly stored on the surface of micropores smaller than 2 nm in coal through adsorption. An et al. [20] noticed similar experimental phenomena and emphasized that the specific surface area of micropores determined the adsorption characteristics of CH₄ in coal. However, Tao et al. [21,22] compared the CH₄ Langmuir volume (V_L) and the Brunauer–Emmett–Teller (BET) specific surface area of coal samples with different metamorphic degrees to evaluate their correlation. Using the BET specific surface area derived from low-pressure N₂ adsorption (LPGA-N₂), they found a poor correlation with the CH₄ adsorption capacity. Thus, they concluded that the specific surface area of pores is not the dominant factor controlling the CH₄ adsorption capacity of coal. Given that CH₄ adsorption in coal is a physical process [23,24] and its pore structures are predominantly composed of micropores, Wang et al. [25] employed mercury intrusion porosimetry (MIP) and LPGA-N₂/CO₂ adsorption to demonstrate that micropores (<2 nm) account for the vast majority of the total specific surface area (SSA_to). They also found that the CH₄ adsorption capacity increases with micropore volume (PV_mi) and micropore specific surface area (SSA_mi). These findings confirm that the micropore structure is the dominant factor governing CH₄ adsorption capacity. In addition, Lozano-Castello et al. [26] conducted further research by comparing the relationship between the CH₄ adsorption capacity of the samples and their pore characteristic parameters. They demonstrated that the pore volume and pore size distribution characteristics of the micropores jointly affect the CH₄ adsorption capacity of coal. Jin et al. [27,28] noticed discrepancies between micropore structure and high-pressure CH₄ adsorption characteristics by comparing the relationship between BET surface area (SSA_BET), pore volume, and V_L. They speculated that high-pressure adsorption capacity is influenced by a combination of <2 nm micropores, SSA_BET, and pore volume. This interpretation is further corroborated by our recent findings [29,30], which experimentally show that the density of adsorbed-phase CH₄ remains below that of the liquid-phase across all pressures. This implies that CH₄ adsorption in coal is not solely governed by available pore volume. Thus, pore volume is ruled out as the sole determinant of adsorption capacity.

In our previous work [2,31], adsorption science and theoretical quantum mechanical approaches were combined to study CH₄ adsorption behavior in pores of varying sizes. We analyzed the ultimate CH₄ adsorption capacity within the multi-scale pore structure of coal and developed a quantitative characterization model based on microscopic pore structure. The model reveals that at infinite CH₄ pressure, adsorbed CH₄ primarily exists in a micropore-filling state. Only a small portion of CH₄ is adsorbed in the form of a monolayer on the surface of pores larger than 1.5 nm [32]. The distribution of CH₄ molecules across different pore sizes is illustrated in Figure 1 [33]. Although the thermodynamic distribution of adsorbed CH₄ at saturation is partly understood, its variation across pore sizes—especially under low-pressure conditions—remains unclear. Therefore, there is a need to develop a mathematical model that correlates the microscopic pore structure with the thermodynamic parameters of CH₄ adsorption. This model will elucidate how adsorbed CH₄ is distributed under varying equilibrium conditions.

In this study, the complex pore network of coal was discretized into uniform adsorption pore units of fixed size. Each unit corresponds to a specific pore size interval, for which CH₄ adsorption isotherms were obtained using Grand Canonical Monte Carlo (GCMC) simulations. Micropores (0.38–1.5 nm) were divided into 12 units based on the DFT-derived pore size distribution, while pores larger than 1.5 nm were simplified into a single unit to represent monolayer adsorption. GCMC simulations were employed to obtain CH₄ adsorption isotherms for these pore structures at various scales. This approach allows the development of a mathematical model that connects the microscopic pore structure of coal to the thermodynamic characteristic parameters of CH₄ adsorption. By integrating the pore size distribution results from LPGA-N₂/CO₂ experiments, the CH₄ isothermal adsorption behavior within the complex pore system can be quantitatively characterized. Specifically, micropore size distributions (0.38–1.5 nm) were determined from LPGA-CO₂ (273 K) using the DFT method, and mesopore/macropore distributions (2–300 nm) were obtained from LPGA-N₂ (77 K) using the BJH method. Experimentally derived pore volumes and surface areas serve as input parameters for the adsorption model, directly linking experimental characterization with theoretical modeling. Furthermore, it allows us to determine the thermodynamic distribution of adsorbed CH₄ across different pore sizes under varying equilibrium pressures. This study is the first to systematically explain the distribution patterns of adsorbed CH₄ in different-sized coal pores under different pressures. It offers theoretical guidance for efficient CH₄ extraction and accurate resource assessment.

2. Adsorption Model Development

2.1. Quantitative Characterization Model for CH₄ Isothermal Adsorption Characteristics in Coal

Coal, as a typical porous medium, exhibits pore structures spanning from molecular to micrometer scales [33,34]. This multi-scale nature precludes the comprehensive characterization of CH₄ adsorption behavior using a single isothermal adsorption model. In this study, the complex pore network of coal was divided into adsorption units of uniform size. The isothermal adsorption characteristics of CH₄ within these pore structures were then quantitatively characterized. Combining the test results of pore structures with different sizes in coal enabled the quantitative characterization of CH₄ isothermal adsorption in complex pore structures. As demonstrated by Hu et al. [31], CH₄ adsorption in pores with diameters of 0.38–1.5 nm occurs through micropore filling, whereas pores exceeding 1.5 nm exhibit monolayer adsorption. Therefore, the quantitative model for CH₄ isothermal adsorption in the complex pore structure of coal is:

$$Q=Q_{\mathrm{mi}}+Q_{\mathrm{mo}}$$

(1)

where Q is the CH₄ adsorption capacity of all pore sizes in coal, cm³/g; Q_mi is the CH₄ adsorption capacity of coal adsorbed in the form of micropore filling, cm³/g; Q_mo is the CH₄ adsorption capacity of coal adsorbed in the form of monolayer adsorption, cm³/g.

Due to the distinct adsorption mechanisms of CH₄ in different-sized pores, the theoretical models used to quantitatively characterize CH₄ isothermal adsorption behavior also vary. Based on the classical isothermal adsorption theories [35,36,37,38,39,40,41,42], the DA equation and the Langmuir equation were applied to quantify the isothermal adsorption characteristics of CH₄ in coal pore structures. The DA equation was used for pores sized 0.38–1.5 nm, while the Langmuir equation was employed for pores larger than 1.5 nm.

$$Q_{\mathrm{mi}}={\displaystyle \underset{r=0.38}{\overset{r=1.5}{\int }}{V}_{\mathrm{mi},r}\exp \left\{-K_r{\left[\ln \left(\frac{p_0}{p}\right)\right]}^{n_r}\right\}}$$

(2)

$$Q_{\mathrm{mo}}={\displaystyle \underset{r=1.5}{\overset{r=\infty }{\int }}\frac{V_{\mathrm{mo},r}p}{P_{\mathrm{L},r}+p}}$$

(3)

where r is the pore diameter, nm; p is the gas pressure, MPa; p₀ is the CH₄ saturated gas pressure, MPa; V_mi,r is the ultimate filling adsorption capacity of the pore structure with a pore size of r, cm³/g; K_r is the adsorption proportion constant of the pore structure with a pore size of r; n_r is the adsorption non-uniformity parameter of the pore structure with a pore size of r; V_mo,r is the CH₄ Langmuir volume of the pore structure with a pore size of r, cm³/g; P_L,r is the CH₄ Langmuir pressure of the pore structure with a pore size of r, MPa.

Although pore sizes vary inconsistently among different coal samples, the CH₄ adsorption characteristic parameters remain consistent for pores of the same size. This consistency allows the parameters in the formulas to be separated. However, since CH₄ adsorption occurs in different forms which depend on pore size, the factors controlling its ultimate adsorption capacity also vary. As shown in Figure 2, the CH₄ adsorption capacity of micropore and macropore structures with the same surface area can differ by as much as 3.6 times [2]. Therefore, Equations (2) and (3) can be transformed into the following form:

$$Q_{\mathrm{mi}}={\displaystyle \underset{r=0.38}{\overset{r=1.5}{\int }}{V}_r\cdot v_{\mathrm{mi},r}\cdot \exp \left\{-K_r{\left[\ln \left(\frac{p_0}{p}\right)\right]}^{n_r}\right\}}$$

(4)

$$Q_{\mathrm{mo}}={\displaystyle \underset{r=1.5}{\overset{r=\infty }{\int }}{S}_r\cdot \frac{v_{\mathrm{mo},r}\cdot p}{P_{\mathrm{L},r}+p}}$$

(5)

where V_r is the volume of the pore structure with a pore size of r, cm³/g; ν_mi,r is the CH₄ adsorption capacity per unit pore volume, cm³/cm³; S_r is the surface area of the pore structure with a pore size of r, m²/g; ν_mo,r is the CH₄ adsorption capacity per unit pore surface area, cm³/m².

Combining Equations (4) and (5), the CH₄ adsorption isotherm in coal with complex pore structures can be obtained:

$$Q={\displaystyle \underset{r=0.38}{\overset{r=1.5}{\int }}{V}_r\cdot v_{\mathrm{mi},r}\cdot \exp \left\{-K_r{\left[\ln \left(\frac{p_0}{p}\right)\right]}^{n_r}\right\}}+{\displaystyle \underset{r=1.5}{\overset{r=\infty }{\int }}{S}_r\cdot \frac{v_{\mathrm{mo},r}\cdot p}{P_{\mathrm{L},r}+p}}$$

(6)

2.2. Simplification of the Quantitative Characterization Model

Based on our previous work [31], micropores smaller than 1.5 nm dominate the pore structure in coal. Under saturation conditions, CH₄ adsorbed in a monolayer form constituted a small fraction (<10%). Furthermore, molecular simulations [43,44,45] indicate that mesopore size has little effect on CH₄ adsorption characteristics. Micropores between 0.38 and 1.50 nm were thus represented as 12 adsorption units of similar sizes. Pores larger than 1.50 nm were simplified into a single unit. This simplification reduces Equations (4) and (5) to:

$$Q_{\mathrm{mi}}={\displaystyle {\sum }_{i=1}^{i=12}{V}_i\cdot v_{\mathrm{mi},i}\cdot \exp \left\{-K_i{\left[\ln \left(\frac{p_0}{p}\right)\right]}^{n_i}\right\}}$$

(7)

$$Q_{\mathrm{mo}}=S_{\mathrm{BET}}\cdot {\displaystyle \frac{v_{\mathrm{mo},\overline{\mathrm{r}}}\cdot p}{P_{\mathrm{L},\overline{\mathrm{r}}}+p}}$$

(8)

where $V_i$ is the volume of the pore structure with a pore size range of i, cm³/g; $v_{\mathrm{mo},\overline{\mathrm{r}}}$ is the CH₄ adsorption capacity per unit equivalent cylindrical pore structure specific surface area, cm³/m²; $S_{\mathrm{BET}}$ is the external specific surface area of all mesopores and macropores, m²/g; $P_{\mathrm{L},\overline{\mathrm{r}}}$ is the CH₄ Langmuir pressure of equivalent cylindrical pore structure, MPa.

Therefore, Equation (6) for the CH₄ adsorption isotherm with a complex pore structure in coal can be simplified as:

$$Q={\displaystyle {\sum }_{i=1}^{i=12}{V}_i\cdot v_{\mathrm{mi},i}\cdot \exp \left\{-K_i{\left[\ln \left(\frac{p_0}{p}\right)\right]}^{n_i}\right\}}+S_{\mathrm{BET}}\cdot {\displaystyle \frac{v_{\mathrm{mo},\overline{\mathrm{r}}}\cdot p}{P_{\mathrm{L},\overline{\mathrm{r}}}+p}}$$

(9)

To elucidate how the micropore structure of coal governs CH₄ adsorption behavior, the pore structure was quantitatively characterized. Combining the CH₄ adsorption characteristic parameters for different pore sizes yields the theoretical adsorption isotherm for the complex coal pore structure. The theoretical adsorption isotherm (Equation (9)) consists of 13 distinct CH₄ adsorption isotherms. These isotherms correspond to different pore sizes. Therefore, the CH₄ occurrence region can be partitioned into 13 adsorption zones. Consequently, partitioning Equation (9) enables the derivation of the CH₄ adsorption behavior in different-sized coal pores under varying pressure conditions.

2.3. CH₄ Adsorption Parameters of Quantitative Characterization Model

2.3.1. Construction of Pore Structures of Different Sizes and Simulation of CH₄ Adsorption

Previous studies have shown [43,44,45,46] that the CH₄ adsorption characteristics in pores of different sizes can be predicted by MS. Additionally, slit pores more closely resemble the actual pores in coal samples. Slit pores of varying sizes were modeled with graphite structures (Figure 3). The volume and surface area of different pore sizes were obtained using the Atom Volumes and Surfaces tool, where Grid resolution was set to Fine and Connolly radius to 0.19 nm. The equivalent pore diameter of each slit pore structure could be derived based on its pore volume and surface area of different pore sizes.

The selection of a molecular force field is critical for predicting gas adsorption isotherms in pores of different sizes. This study employs the PCFF force field, which is built upon CFF91 and includes additional computational parameters for metals, polymers, and zeolites. This makes it suitable for simulating organic polymers and zeolite systems [43]. It should be noted that in GCMC simulations, chemical potential is input as fugacity rather than absolute pressure. Fugacity relates the chemical potential of a real gas to that of an ideal gas at the same temperature. It represents the pressure that an ideal gas would need to exert to achieve an equivalent chemical potential. The Peng-Robinson state equation was used to convert between fugacity and pressure in this study [47]. GCMC simulations provided the lowest-energy configurations of CH₄ in slit pores across a range of pressure. As shown in Figure 4, we take 0.519 nm and 3.518 nm slit pore structures as examples.

By comparing the lowest energy configurations in the 0.519 nm and 3.518 nm slit pore structures, we can find that CH₄ molecules almost completely filled the 0.519 nm slit pore at 0.421 MPa. As the pressure increased, the concentration of CH₄ molecules in the pores remained relatively stable. For the 3.518 nm slit pore, when the CH₄ pressure was raised to 5.810 MPa, there was still a considerable gap between the CH₄ molecular concentration in the center of the pore and that on the walls. This indicates that no filling phenomenon occurred. This result supports the view of Ortiz et al. [2,48] that CH₄ exhibits micropore filling behavior in pore structures below 1.5 nm, but no similar volume filling phenomenon occurs in larger pore structures. By comparing the lowest energy configurations of slit pore structures of other sizes, we observed that the smaller the size of the slit pore, the more sensitive the CH₄ isothermal adsorption data is to gas pressure. For slit pore structures below 0.6 nm, only extremely low gas pressure (<0.421 MPa) is needed to achieve half of their ultimate adsorption capacity. This indicates that the smaller the pore size, the greater its surface energy [44,49].

2.3.2. CH₄ Adsorption Parameters of Pore Structures of Different Sizes

To accurately characterize CH₄ adsorption behavior in slit pores of different sizes and obtain the corresponding adsorption characteristic parameters, it was essential to define the boundaries between adsorbed and free state CH₄ in the simulated crystal cell structure. This study adopted the approach of Song et al. [50] to separately define pore structures that exhibit micropore filling and surface coverage adsorption behaviors. It was assumed that in micropores smaller than 1.50 nm, all gas was in the adsorbed state. In pores larger than 1.50 nm, the adsorption space was defined as the region with a monolayer thickness of adsorbate molecules on the pore space area. Void space beyond the adsorption region was considered free gas. Consequently, the absolute adsorption capacity in micropores (0.38–1.5 nm) was calculated using Equation (10), while Equation (11) was used for larger pores (>1.50 nm):

$$N_{\mathrm{a},\mathrm{mi}}^{\mathrm{ab}}=N$$

(10)

$$N_{\mathrm{a},\mathrm{mo}}^{\mathrm{ab}}=N-N_{\mathrm{A}}{\displaystyle \frac{{\rho }_{\mathrm{g}}\left(V_{\mathrm{pore}}-V_{\mathrm{a},\mathrm{mo}}\right)}{M}}$$

(11)

where $N_{\mathrm{a},\mathrm{mi}}^{\mathrm{ab}}$ is the absolute adsorption capacity of gas molecules in the pore structure ranging from 0.38 to 1.5 nm, individual; $N_{\mathrm{a},\mathrm{mo}}^{\mathrm{ab}}$ is the absolute adsorption capacity of gas molecules in the pore structure larger than 1.5 nm, individual; $V_{\mathrm{a},\mathrm{mo}}$ is the volume of gas adsorption space in the pore structure greater than 1.5 nm, cm³; $N$ is the total number of gas molecules in the simulated unit cell pore structure, individual.

The absolute adsorption amount calculated via Equations (10) and (11) uses a unit that is inconsistent with the commonly used unit for gas adsorption amount. To perform unit conversion on the above simulation results, we first convert the number of adsorbed CH₄ molecules to the adsorption volume using the ideal gas state equation, whose expression is shown in Equation (12):

$${V_{\mathrm{a}}}^{\prime }={\displaystyle \frac{N_{\mathrm{a}}^{\mathrm{ab}}}{N_{\mathrm{A}}}}\times 22400$$

(12)

The gas adsorption volume ${V_{\mathrm{a}}}^{\prime }$ in a single simulated unit cell obtained via the above calculation has a unit of “cm³”, which cannot be used directly. Given the research focus of this study, we do not use the commonly used unit “cm³/g” (typically employed to measure the gas adsorption capacity of pore structures with different sizes). For microporous structures (0.38–1.5 nm), we use the “gas adsorption amount per unit pore volume” (unit: cm³/cm³) to measure the gas adsorption capacity of micropores of different sizes.

$$V_{\mathrm{a}}={\displaystyle \frac{{V_{\mathrm{a}}}^{\prime }}{V_{\mathrm{a},\mathrm{mi}}}}$$

(13)

For pore structures larger than 1.5 nm, the amount of adsorbed gas per unit pore surface area is used as the unit (cm³/m²) to measure the gas adsorption capacity of pore structures of different sizes.

$$V_{\mathrm{a}}={\displaystyle \frac{{V_{\mathrm{a}}}^{\prime }}{S_{\mathrm{a},\mathrm{mo}}}}$$

(14)

where N_A is Avogadro’s constant; V_a,mi is the volume within a single simulated unit cell, cm³; S_a,mo is the pore surface area within a single simulated unit cell, m².

The number of CH₄ molecules within a single simulated unit cell at varying pressures was converted. This conversion was performed to determine the corresponding quantity of adsorbed molecules per unit mass (or volume). Unit conversion was applied through Equations (12)–(14) with integration of the results. This process yielded the gas adsorption isotherms for pore structures of different sizes, as shown in Figure 5.

The CH₄ adsorption isotherm of slit pore structures of different sizes showed that the smaller the pore size, the lower the pressure required for the pore to reach its ultimate adsorption capacity. Moreover, the smaller the pore size, the higher the adsorption potential energy [44,49]. Under the same gas pressure conditions, as the pore size increased from 0.419 nm to 1.466 nm, the ultimate CH₄ adsorption capacity that the pore structure could accommodate did not decrease monotonically with increasing pore size. According to geometric theory, the maximum packing density of gas molecules within pores varies irregularly with pore size. This indicates that the ultimate density of adsorbed CH₄ varied among pores of different sizes [44]. For slit pore structures ranging from 1.619 to 4.040 nm, the CH₄ adsorption isotherms of different pore sizes were essentially consistent, reflecting similar adsorption behavior of CH₄ within this size range. The Dubinin-Astakhov (DA) equation was applied to fit the CH₄ isothermal adsorption data for pores of 0.38–1.5 nm, yielding the total micropore volume V₀, proportion coefficient K, and heterogeneity parameter n, as shown in

Table 1. CH₄ adsorption characteristic parameters of slit pore structures with different sizes.

r (nm)	νmi,r (cm3/cm3)	Kr	nr	R2	r (nm)	νmo,r (cm3/m2)	PL,r (MPa)	R2
0.419	500.7	0.019	2.044	0.9971	1.619	0.211	2.107	0.9997
0.469	447	0.018	2.124	0.9977	1.819	0.209	2.275	0.9999
0.519	400.2	0.018	2.281	0.998	2.04	0.207	2.252	0.9997
0.574	368.4	0.026	2.366	0.9982	2.219	0.205	2.216	0.9998
0.719	393.6	0.061	2.238	0.9994	2.418	0.202	1.977	0.9998
0.825	396.6	0.066	2.309	0.9997	2.618	0.201	1.989	0.9999
0.905	373.3	0.088	2.182	0.9996	2.819	0.202	2.176	0.9997
0.994	348.8	0.128	1.984	0.9996	3.04	0.2	2.139	0.9997
1.101	339.9	0.184	1.785	0.9997	3.518	0.198	1.933	0.9998
1.24	321.1	0.23	1.665	0.9998	4.04	0.198	2.137	0.9984
1.353	297.6	0.242	1.637	0.9998	—	—	—	—
1.466	278.5	0.255	1.607	0.9998	—	—	—	—

. The Langmuir equation was used to fit the CH₄ isothermal adsorption data for pores larger than 1.5 nm, obtaining the characteristic parameters V_L and P_L of CH₄ adsorption in pore structures above 1.5 nm, as shown in Table 1.

For CH₄ adsorption in slit pores ranging from 0.419 to 1.466 nm, the correlation coefficient between the CH₄ isothermal adsorption data and the DA equation rose sharply with pore size. This was mainly due to larger relative errors in adsorption data from fewer CH₄ molecules in smaller pores and insufficient GCMC simulation iterations, which resulted in less smooth data. For CH₄ adsorption in slit pores ranging from 1.619 to 4.040 nm, the CH₄ isothermal adsorption data generally aligned well with the Langmuir equation. The correlation coefficient exceeded 0.9984, indicating that the monolayer adsorption theory effectively describes CH₄ adsorption within this pore size range. Within slit pores ranging from 0.419 nm to 1.466 nm, the CH₄ adsorption characteristic parameter ν_mi,r exhibited irregular variations but overall decreased as pore size increased. The adsorption characteristic parameter K_r increased monotonically with pore size, rising from 0.018 to 0.255 (representing an approximately 14 times increase). The parameter n_r initially increased and subsequently decreased with pore size, changing from 2.044 to 2.366 before falling to 1.607. The adsorption characteristic parameter, Langmuir volume (V_L), decreased monotonically with the increase in pore size for slit pores ranging from 1.619 to 4.040 nm. In contrast, the Langmuir pressure (P_L) remained nearly constant. This suggests that the adsorption potential energies across these pore sizes are closely related, and that variations in ultimate adsorption capacity are potentially linked to the division of the adsorption region.

3. Experimental

3.1. Sample Preparation

To evaluate the relationship between the micropore structure and macroscopic CH₄ adsorption characteristics in coal, and to validate the proposed theoretical model against experimental results, four coal samples were selected from different mining areas in China following GB/T 19222-2003 [51]. The metamorphic degree of coal samples was characterized based on proximate analysis parameters (volatile matter, V_daf, and fixed carbon, FC_ad). As summarized in

Table 2. Basic physical parameters of coal samples with different metamorphic degrees.

Sample Number	Sampling Location	Industrial Analysis
		Mad (%)	Ad (%)	Vdaf (%)	FCad (%)
XTR	Xintian coal mine 4 coal seam	0.84	8.33	4.96	86.39
11110Y	Pingmei 13th mine 15–17 coal seams	0.51	10.77	16.31	74.29
QNY	Qinan coal mine 10 coal seam	0.55	7.64	33.61	60.98
XQY	Qingqing coal mine 7 coal seam	1.49	6.29	37.26	57.92

, the volatile matter decreases from 37.26% to 4.96%, while fixed carbon increases from 57.92% to 86.39%, indicating a clear progression from low-rank to high-rank coal. The basic physical property parameters of the coal samples were determined according to GB/T 212-2008 [52] using a 5E-MAG6600 fully automatic industrial analyzer (Changsha Kaiyuan Instrument Co., Ltd., Changsha, China). The test results of the basic physical property parameters are shown in Table 2.

3.2. Testing Methods

To quantitatively evaluate the distribution characteristics of adsorbed CH₄ within pore structures of varying sizes in coal, this study employed the LPGA-N₂ (77 K) and LPGA-CO₂ (273 K) methods for the segmented quantitative characterization of different pore size ranges within coal samples. The LPGA-N₂ (77 K) and LPGA-CO₂ (273 K) experiments were conducted using an Autosorb iQ2 automated gas sorption analyzer (Quantachrome Instruments, Boynton Beach, FL, USA). The meso-macropore structures of the coal samples were quantitatively characterized in accordance with GB/T 21650.2-2008 [53], and the detailed experimental procedures followed previously published literature [31]. The micropore structures of the coal samples were quantitatively characterized in accordance with GB/T 21650.3-2011 [54], and the detailed experimental procedures followed previously published literature [55].

This study employed the volumetric method to examine the adsorption characteristics of coal samples with varying metamorphic degrees. The tests were conducted under different gas conditions at the National Engineering Research Center of Coal Mine Gas Control, China University of Mining and Technology. The experiments utilized a KDXJ-II coal gas adsorption/desorption dynamics analyzer, manufactured by Jiangsu Kedi Petroleum Instrument Co., Ltd. (Haian, China) [56] (See Figure 6 for the experimental instrument). The gas adsorption characteristics of coal samples with different metamorphic degrees were tested according to GB/T 19560-2008 [57]. The adsorption test steps mainly include the following: hermeticity testing, calibration of the reference and sample cell volumes, coal sample pretreatment, free space calibration of the sample cell, gas adsorption testing, and experimental data processing. Specific experimental details are available in previously published literature [58,59]. The adsorption experiments were carried out using the volumetric method under specified conditions. To ensure reliability, all experiments—including LPGA-CO₂ (273 K), LPGA-N₂ (77 K), and high-pressure CH₄ adsorption—were conducted in triplicate for each coal sample. The reported results are averaged values, with deviations among parallel tests within ±3%, confirming good repeatability and a negligible influence on the conclusions.

4. Results and Discussion

4.1. Pore Size Distribution in Coal Samples

To quantitatively characterize the pore structures of varying sizes in coal, LPGA-N₂ (77 K) adsorption/desorption isotherms and LPGA-CO₂ (273 K) adsorption isotherms were measured for four coal samples of varying ranks. These isotherms are shown in Figure 7 and Figure 8, respectively.

The IUPAC classification now includes eight types of physical adsorption isotherms [60], an update from the original six. Thus, the LPGA-N₂ (77 K) adsorption isotherms (Figure 7) of coal samples reflect a combination of Type I, Type IV (a), and Type II isotherms. At low relative pressures, the isotherm rises sharply, consistent with the low-pressure region of Type I isotherms. This behavior is attributed to N₂ micropore filling within the coal matrix [60]. After micropore filling, the heterogeneity of the coal surface potential energy decreases significantly. This allows probe molecules to undergo monolayer adsorption, multilayer adsorption, and capillary condensation in mesopores and macropores. Capillary condensation typically causes irreversible adsorption/desorption, observable as hysteresis loops. This corresponds to the adsorption behavior near P/P₀ ≈ 0.5 in Type IV (a) isotherms, suggesting the presence of mesopores around 4 nm in the coal [60,61,62]. As relative pressure approaches 1, the adsorption isotherm of coal showed varying degrees of rapid rise without the extreme adsorption plateau observed in category IV (a) adsorption isotherms [60,63]. This is consistent with the high-pressure stage of category II adsorption isotherms, indicating the existence of abundant macropores larger than 300 nm in coal [64]. Conversely, the LPGA-CO₂ (273 K) adsorption isotherms (Figure 8) of coal samples with different metamorphic degrees displayed a convex shape increasing with relative pressure, corresponding to the adsorption isotherm of class I. This indicates the presence of a large number of micropore structures in coal [65,66].

Considering the applicability and analysis ranges of various analytical methods [55,67], the DFT method was selected to process LPGA-CO₂ (273 K) isothermal adsorption data to obtain pore size distribution characteristics of micropores (0.38 to 1.50 nm). The micropore size range was then divided into 12 intervals (0.33 to 0.38 nm, 0.38 to 0.45 nm, 0.45 to 0.52 nm, 0.52 to 0.59 nm, 0.59 to 0.66 nm, 0.66 to 0.76 nm, 0.76 to 0.82 nm, 0.82 to 0.92 nm, 0.92 to 1.03 nm, 1.03 to 1.14 nm, 1.14 to 1.26 nm, 1.26 to 1.37 nm, and 1.37 to 1.50 nm). Segmental pore volumes for pores of different sizes were calculated using interpolation methods [31]. Based on LPGA-N₂ (77 K) adsorption data, the BJH method was applied to determine the pore volume and specific surface area of pores ranging from 2 to 300 nm in different coal samples [68,69]. The pore volume parameters for different pore size ranges in coal are summarized in

Table 3. Pore volume parameters in different size ranges of different coal samples.

Coal Sample Serial Number	Segmental Pore Volume (×10−3 cm3/g)
	XTR	11110Y	QNY	XQY
0.38–0.45	12.433	1.319	0.993	7.650
0.45–0.52	4.164	4.650	2.377	2.794
0.52–0.59	18.837	10.874	9.274	11.860
0.59–0.66	15.948	10.361	8.973	10.910
0.66–0.76	6.689	4.817	4.222	5.292
0.76–0.82	3.279	2.958	1.749	2.956
0.82~0.92	3.528	2.745	2.193	3.715
0.92–1.03	4.339	3.815	2.655	3.913
1.03–1.14	3.880	2.978	2.165	3.026
1.14–1.26	3.668	2.724	1.998	2.853
1.26–1.37	3.566	2.567	1.920	2.899
1.37–1.50	3.504	2.487	1.880	2.679
>1.50	3.717	2.692	1.994	3.102

.

Analysis of the pore volume parameters for different size ranges of various coal samples (Table 3) shows that the total pore volume for four samples (0.38–300 nm) is between 0.042 and 0.088 cm³/g. Among the 13 artificially defined pore size intervals, the 0.52–0.59 nm interval accounts for the highest proportion of the total pore volume (18.63–21.88%). Pores within the ultra-micropore range (0.38–0.76 nm) constitute a substantial 58.23% to 66.33% of the total pore volume, while micropores (0.38–1.5 nm) represent 95.10% to 95.75%. These findings demonstrate that micropores (0.38–1.5 nm) dominate the pore structure of these coals. Within this micropore fraction, ultra-micropores (0.38–0.76 nm) contribute over 50% of the total pore volume, providing substantial adsorption sites for CH₄ storage.

4.2. Validation of the Theoretical CH₄ Adsorption Isotherms

To accurately characterize the CH₄ adsorption features of pore structures within specific size ranges, each pore size range was treated as an adsorption unit containing only single-sized pores. The equivalent pore diameter was defined as the average pore diameter of its segment. The equivalent diameters for the 12 segments are 0.415 nm, 0.485 nm, 0.555 nm, 0.625 nm, 0.710 nm, 0.789 nm, 0.868 nm, 0.972 nm, 1.084 nm, 1.198 nm, 1.315 nm, and 1.437 nm. Based on the CH₄ adsorption characteristic parameters for different pore structures obtained from GCMC simulation (see Table 1), the CH₄ adsorption characteristic parameters (V₀, K, n, V_L, and P_L) of slit pores under different conditions were determined using interpolation methods, as shown in

Table 4. CH₄ adsorption characteristic parameters of slit pores with different sizes at 30 °C.

Pore Size (nm)	DA Adsorption Model			Pore Size (nm)	Langmuir Adsorption Model
	V0 (cm3/cm3)	K	n		VL (cm3/m2)	PL (MPa)
0.415	505.110	0.019	2.038	1.619	0.211	2.107
0.485	432.081	0.018	2.174	1.819	0.209	2.275
0.555	379.544	0.023	2.336	2.040	0.207	2.252
0.625	377.234	0.038	2.321	2.219	0.205	2.216
0.710	392.030	0.058	2.246	2.418	0.202	1.977
0.789	395.564	0.064	2.285	2.618	0.201	1.989
0.868	383.998	0.078	2.240	2.819	0.202	2.176
0.972	354.831	0.118	2.032	3.040	0.200	2.139
1.084	341.351	0.175	1.817	3.518	0.198	1.933
1.198	326.751	0.216	1.701	4.040	0.198	2.137
1.315	305.512	0.238	1.646	—	—	—
1.437	283.404	0.251	1.615	—	—	—
1.500	216.615	0.242	1.634	—	—	—

.

To validate the theoretical CH₄ adsorption model for coal’s complex pore structure, the pore size distribution parameters (see Table 3) and their corresponding CH₄ adsorption parameters (see Table 4) were substituted into Equation (9). This generated theoretical adsorption isotherms under the slit-pore assumption. The measured isothermal adsorption data and theoretical CH₄ adsorption isotherms were then compared, as shown in Figure 9. To quantify the discrepancy between them, we calculated the average relative error (E_rela) at each adsorption pressure. The E_rela values were used to assess model accuracy:

$$E_{\mathrm{rela}}={\displaystyle \frac{100}{m}}{\displaystyle \sum _{i=1}^m\left|\frac{\left(V_i^{\mathrm{meas}}-V_i^{\mathrm{theo}}\right)}{V_i^{\mathrm{theo}}}\right|}$$

(15)

where m is the number of measured isothermal adsorption data, individual; $V_i^{\mathrm{meas}}$ is measured values of coal sample adsorption CH₄ under different pressure conditions, cm³/g; $V_i^{\mathrm{theo}}$ is theoretical values of coal sample adsorption of CH₄ under different pressure conditions, cm³/g.

By comparing the theoretical CH₄ adsorption isotherms with the actual isotherm data of different coal samples under 30 °C conditions (see Figure 9), it is evident that with the intensification of coal chemical metamorphism, the average relative error between the theoretical CH₄ adsorption isotherms and the measured isothermal adsorption data decreases rapidly from 35.371% to 11.044%. This demonstrates that the thermodynamic characterization model for CH₄ adsorption in coal—established based on CH₄ adsorption isotherm characteristics derived from a graphitic molecular structure model—is feasible. Although significant discrepancies exist when characterizing CH₄ adsorption in medium-rank and low-rank coals, the model still yields the thermodynamic distribution characteristics of adsorbed CH₄ within pore structures of varying sizes under different equilibrium pressures. These characteristics hold significant reference value for anthracite, which exhibits a graphite-like structure. The reason for this is that the macromolecular structure of coal gradually transforms into a graphite-like structure as the degree of metamorphism increases [70,71,72]. The graphite molecular structure model can reflect the CH₄ adsorption characteristics of high-rank coal, which differ significantly from those of medium-rank and low-rank coal. In addition, Mosher et al. [44,46] observed significant deviations between theoretical predictions and experimental measurements when predicting CH₄ adsorption isotherms for high-volatile bituminous coal based on the adsorption characteristics of individual pore sizes. Incomplete pore characterization data and oversimplified theoretical adsorption models were likely the primary factors contributing to these discrepancies.

4.3. Distribution Characteristics of Adsorbed CH₄

As outlined in Section 2.2, the theoretical CH₄ adsorption isotherm for a coal sample is derived by combining the adsorption isotherms of its individual pore units. Consequently, by deconstructing the theoretical adsorption isotherm, we can reveal the CH₄ adsorption characteristics of different pore structures under various pressures. Using the XTR coal sample as an example, Figure 10 displays the adsorption capacity and its proportion relative to the total capacity across different pore size ranges under increasing pressures. The results show that the adsorption capacity for all pore sizes increases as the equilibrium pressure rises. At an equilibrium pressure of 6 MPa, the adsorbed amounts for the pore size interval reached 2.091, 8.112, 6.028, 2.504, 1.369, and 0.026 cm³/g, respectively. These results indicate that the adsorption amount first increases and then decreases as the pore size grows. Specifically, pores in the 0.45–0.52 nm range exhibited the highest adsorption, while pores larger than 1.5 nm had the lowest-with an adsorbed volume significantly less than that of all other pore sizes. A comparison of adsorption trends across different pore sizes reveals distinct behaviors. For pores smaller than 0.52 nm (see Figure 10a,b), the adsorption amount increases rapidly at very low pressures and quickly approaches saturation. For pores between 0.52 and 1.5 nm (see Figure 10c–e), the adsorption amount rises rapidly initially and then gradually approaches saturation. For pores larger than 1.5 nm (see Figure 10f), the adsorbed amount increases continuously across the pressure range. Although the rate of increase slows, saturation is not achieved. This suggests that CH₄ molecules preferentially fill smaller micropores as pressure increases.

Analysis of the adsorption capacity ratio of different pore structures of XTR coal samples under different pressure conditions at 30 °C (see Figure 10) reveals that the proportion of adsorption capacity attributed to pores smaller than 0.52 nm decreases gradually with increasing pressure. By combining this analysis with the lowest energy configurations of slit pores of different sizes at various CH₄ pressures (see Figure 4), it is observed that CH₄ molecules almost fully saturate the 0.519 nm pores at 0.1 MPa. Subsequent increases in pressure cannot accommodate more CH₄ in these pores, while larger pores continue to adsorb CH₄. This causes the proportion of CH₄ adsorption capacity from sub-0.52 nm pores to decline slowly as pressure increases. The proportion of CH₄ adsorption capacity for pores between 0.52 and 0.76 nm first increases and then decreases with pressure. This indicates that at lower pressures, the adsorption capacity of CH₄ molecules in these pores increases gradually. As the CH₄ pressure increases, the total amount of CH₄ adsorbed by the complex pore structure in coal also increases. However, the increase in adsorption capacity provided by this pore size range mainly contributes to the total increase in adsorption capacity. This leads to an upward trend in the proportion of CH₄ adsorption capacity from this pore size range as pressure increases. When the gas pressure further increases, the increase in adsorption capacity provided by this pore size range accounts for only a small proportion of the total adsorption capacity increase. This leads to a downward trend in the proportion of CH₄ adsorption capacity from this pore size range as pressure increases. The peak pressure at which the proportion of adsorption capacity of different pore sizes increases with CH₄ pressure varies; the smaller the pore size, the lower the CH₄ pressure required to reach the peak. This indicates that smaller pore structures require less gas pressure to be filled with CH₄ molecules. The proportion of CH₄ adsorption capacity from pore structures larger than 0.92 nm in coal increases continuously with increasing pressure. This indicates that when the CH₄ pressure is below 5.9 MPa, pore structures larger than 0.92 nm are not fully filled with CH₄ molecules. Based on the changes in the proportion of CH₄ adsorption capacity from pore structures larger than 1.5 nm in XTR coal samples under varying pressures, it is observed that under extreme conditions, CH₄ molecules adsorbed on the surface of pores larger than 1.5 nm in a monomolecular form account for less than 1% of the ultimate adsorption capacity. As the pressure on CH₄ decreases, the proportion of adsorption capacity from pores larger than 1.5 nm further decreases.

As stated in the introduction, a key application for this work is to provide a more accurate basis for resource assessment. Our thermodynamic model improves upon traditional methods in several critical ways. Firstly, conventional assessments often rely on a single set of ‘lumped’ parameters (e.g., from the Langmuir model) to characterize the entire coal matrix, which oversimplifies the heterogeneous pore system. Our model, by contrast, adopts a ‘distributed’ approach, quantifying the contribution of each pore size class to total gas storage. This provides a more physically grounded estimate of the gas in place (GIP). Secondly, and perhaps more importantly, our model differentiates between total GIP and recoverable reserves. The adsorption potential is much stronger in smaller pores, meaning gas is held more tightly. As demonstrated by the dynamic distribution analysis (Figure 10), our model can predict which portion of the gas is released at various stages of pressure depletion. This is crucial for forecasting production behavior and estimating ultimate recovery, which offers a significant advantage over methods that cannot distinguish the varying energy states of adsorbed gas molecules.

4.4. Limitations and Future Perspectives

While this study provides a robust framework for analyzing adsorbed gas distribution, several limitations are acknowledged, which also represent important avenues for future research.

(1)

The analysis in this study was conducted isothermally at 30 °C. However, in situ reservoir temperatures can vary significantly with depth. To extend the model’s applicability, future work should involve performing GCMC simulations across a range of geologically relevant temperatures. This would enable the establishment of temperature-dependent thermodynamic parameters, which in turn allows the model to predict gas distribution under diverse geothermal gradients—a capability crucial for deep coalbed methane (CBM) exploration.

(2)

The experimental validation and simulations in this study were limited to a maximum pressure of 6 MPa, constrained by our experimental setup. While this pressure range is sufficient for many conventional CBM scenarios, deep coal seams typically exhibit much higher pressures. Future studies should aim to acquire high-pressure experimental data to validate and potentially refine the model’s predictive behavior in the supercritical region—this will ensure the model’s robustness for deep reservoir conditions.

(3)

It should be noted that the GCMC simulations in this study employed a simplified slit-pore geometry. Although coal contains various pore morphologies, such as ink-bottle, wedge-shaped, and irregular pores, previous research has shown that slit pores are the most frequent type in the coal matrix [44,73]. They provide a large specific surface area and significantly influence methane adsorption capacity. Furthermore, GCMC studies have demonstrated that experimentally measured isotherms align more closely with slit-pore model predictions, whereas circular and square pore models tend to underestimate actual adsorption [46]. Therefore, the slit-pore model was adopted to simulate methane adsorption behavior, ensuring comparability and reducing computational complexity. However, it may overlook the effects of complex pore connectivity and geometric features. Future research will incorporate more realistic pore geometries to improve predictive accuracy.

Addressing these aspects will enhance the model’s fidelity and broaden its application scope, building upon the foundational understanding of dynamic gas distribution established in this work.

5. Conclusions

This study presents a quantitative thermodynamic framework to characterize CH₄ adsorption behavior in coal with complex pore structures of various sizes. By integrating GCMC simulations with pore structure characterization and experimental validation, we systematically revealed the spatial distribution and governing mechanisms of CH₄ adsorption under different pressure conditions. The main conclusions are as follows:

(1)

Micropores (0.38–1.5 nm) dominate the pore structure of the studied coals. They account for 95.10–95.75% of the total pore volume. Within this range, ultra-micropores (0.38–0.76 nm) contribute 58.23–66.33%, serving as the principal adsorption sites for CH₄ storage.

(2)

CH₄ adsorption behavior strongly depends on pore size. Smaller pores require lower pressure to reach CH₄ saturation and exhibit higher adsorption potential energy. In slit pores from 0.419 to 1.466 nm, the characteristic parameter ν_mi,r decreases irregularly overall (from 500.7 to 278.5 cm³/cm³), Kr increases significantly (from 0.018 to 0.255, 14 times), and n_r initially rises (from 2.044 to 2.366) and then declines (to 1.607). For larger slit pores (1.619–4.040 nm), V_L gradually decreases (from 0.211 to 0.198 cm³/m²), while P_L remains almost constant. This suggests similar CH₄ adsorption behavior in these pores.

(3)

The proposed thermodynamic model is based on GCMC-derived adsorption behavior. It shows good agreement with experimental CH₄ adsorption isotherms, especially for high-rank coals. As coal rank increases, the mean relative deviation between theoretical and measured CH₄ adsorption isotherms drops sharply from 35.371% to 11.044%. This supports the model’s applicability to coals with graphite-like molecular structures.

(4)

Adsorbed CH₄ distribution analysis reveals dynamic pressure-dependent pore contributions: pores <0.52 nm dominate at low pressures but saturate rapidly, reducing their relative share. Pores in the range of 0.52–0.76 nm show a peak contribution at intermediate pressures. Pores larger than 0.92 nm continue to increase their contribution as pressure rises, highlighting their importance for high-pressure CH₄ storage.