Study on the Influence of Multiple Factors on the CH₄/CO₂ Adsorption Selective Prediction Model in Coal

Abstract

More accurate prediction of CO₂/CH₄ adsorption selectivity coefficients in the CO₂ Enhanced Coal Bed CH₄ Recovery (CO₂-ECBM) project can help to judge the CO₂ adsorption concentration and the desorption purity of CH₄ during the CO₂ injection process, and to achieve the maximization of CO₂ sequestration as well as the optimization of the CH₄ recovery rate. To this end, a coal molecular slit model with 16 sizes including micro-, meso-, and macropores was constructed in this study, and the competitive adsorption characteristics of CO₂ and CH₄ gas mixtures in bituminous coal molecules were investigated using molecular dynamics and giant canonical Monte Carlo simulations. The CO₂/CH₄ adsorption selectivity coefficients (S_c) as a function of gas ratio, gas pressure, pore size, and temperature were analyzed using a large amount of adsorption isotherm data. Based on the simulation results, considering the neglect of pressure and component changes when calculating the adsorption selectivity coefficient using the traditional extended Langmuir (E-L) model, a correction term regarding the pressure of the mixed gas and the mole fraction of CO₂ is set, and a modified equation is proposed. The results show that the adsorption potential energy of CO₂ is significantly higher than that of CH₄, giving it an absolute advantage in the competition. Through multiple regression analysis, the ranking of the influence weights of the four factors on S_c is as follows: pore size > mixed gas pressure > molar fraction of CO₂ > temperature. The negative exponential function can describe the variation of S_c with four factors. The fitting degree between the modified prediction model and the S_c data obtained through simulation reaches 0.84, and the model effect is good. The research results provide theoretical guidance for the optimization of gas injection parameters in the CO₂-ECBM project.

1. Introduction

The technology of injecting CO₂ into coal seams to enhance CH₄ recovery (CO₂-ECBM) has emerged against the backdrop of CO₂ geological sequestration and carbon emission reduction [1,2]. This technology not only improves the economic viability of CO₂ storage in coal seams but also strengthens mine gas control. Due to its multifaceted benefits, it has garnered widespread attention and research worldwide [3,4].

To predict the changes in adsorption selectivity, many scholars have conducted in-depth explorations on the competitive adsorption mechanism of multiple gases in coal and the improvement of the competitive adsorption equilibrium model; during this process, the giant canonical Monte Carlo simulation (GCMC) has been widely applied in the study of coal adsorption due to its accuracy [5,6]. Studies show that under the same conditions, the adsorption capacity of coal for CO₂ is approximately twice that of CH₄, and it varies with specific environmental factors and the internal structure of coal [7,8,9]. Huang et al. [10] explained from a thermodynamic perspective that CO₂ has a stronger adsorption capacity compared to CH₄. Zhao [11] and Zhang et al. [12] studied the influence of different influencing factors on the competitive adsorption of CO₂ and CH₄ through numerical simulation. The results show that an increase in temperature, gas concentration, pressure and coal pore size will reduce the adsorption advantage of CO₂. The adsorption selectivity coefficient [13,14] is commonly used to evaluate the strength or weakness of competitive adsorption of binary mixed gases in adsorbents. Many scholars have analyzed the research results through this coefficient. Liao et al. [15] experimentally inferred that the adsorption selectivity coefficient of CO₂ relative to CH₄ S_c (CO₂/CH₄ adsorption selectivity coefficient) first increased and then decreased with the increase in CO₂ fraction. However, the simulation results of Xie et al. [16] indicated that S_c continuously decreased with the increase in CO₂ fraction. Jia et al. [17] found through molecular simulation methods that with the increase in the volume fraction of CO₂ and the equilibrium pressure of the system, the competitive adsorption strength of CO₂ compared with CH₄ showed a decreasing trend. Dutta et al. [18] obtained the same view through experimental means. Unlike the slit model, Wang et al. [19] constructed an organic nanopore model to adsorb CO₂/CH₄ mixed gas and found that the adsorption selectivity coefficient of CO₂ decreased with the increase in temperature. The E-L model and the Ideal Adsorption Solution Theory (IAST) are the most commonly used methods for calculating the adsorption capacity of multi-component gases [20,21]. However, the calculation results are insufficient to reflect the variation of competitive adsorption characteristics with different factors discovered in the existing studies. Therefore, on this basis, many scholars have proposed improved models by adding correction factors to the adsorption constant and equilibrium pressure according to the variation mechanism of competitive adsorption strength [22,23,24]. However, due to the complex form of the equation, it is not convenient to calculate the adsorption selectivity coefficient.

Although existing studies have systematically explored the influences of temperature, pressure, gas ratio, and pore size on the competitive adsorption of CO₂/CH₄, different scholars still have differences regarding the variation law of S_c with key factors. In addition, the S_c prediction model based on the traditional E-L model is solely determined by the Langmuir adsorption constant, and it is difficult to reflect the dynamic influence of multiple factors on S_c. In order to establish an S_c prediction model considering factors such as pore size, pressure, gas ratio, and temperature based on the E-L model, so as to obtain more accurate S_c in the CO₂-ECBM and CO₂ sequestration processes, this paper simulates the competitive adsorption of CO₂ and CH₄ in the pores of bituminous coal under different conditions, and obtains the influence laws of various factors such as coal quality characteristics and environmental conditions on the competitive adsorption. The functional relationships of each factor on the adsorption selectivity coefficient are analyzed through a large amount of mixed gas adsorption data. Combined with the simulation results, the correction parameters are set to quantify these influences and optimize the calculation method of S_c based on the E-L model, and provide a theoretical reference for the optimization of gas injection parameters for CO₂ injection into coal seams.

2. Simulation Method of Competitive Adsorption of CO₂/CH₄ in Coal

2.1. Model Construction

Bituminous coal has abundant reserves worldwide and is often used in coalbed methane adsorption experiments [25]. In this paper, the Wiser bituminous coal molecular model was selected to construct the adsorption system in Materials Studio 2020 (MS) software. This model is widely used due to its relatively comprehensive structure and high stability [26]. Each single unit cell created was composed of two coal molecules, with the molecular formula C₁₈₄H₁₅₅N₃O₂₀S₃ [27], and the cycle boundary for it was set. The Compass force field was used to describe the molecular inter- and intramolecular interactions, the charge distribution was calculated using the Q_eq method, and the final density of the coal molecules was 1.30 g/cm³, which is in the range of bituminous coal densities (1.25 to 1.5 g/cm³) [28]. The final cell geometry parameters were a = b = c = 19.98 Å, α = β = γ = 90°, and the pore model construction process is shown in Figure 1. Geometric and energy optimization were carried out using the forcite module, the annealing cycle was 20, the initial temperature was 293 K, the maximum temperature was 600 K, the NVT ensemble was selected, the Nose temperature control method was adopted, the time step was 1 fs, and the simulation time was 200 ps. Then, the unit cell units were expanded along the x, y, and z directions to obtain a super unit cell with a 1 × 1 × 2 structure. And, based on this, a slit model with a vacuum layer was constructed to simulate coal pores of different sizes. The equilibrium step size and total step size of the adsorption process were 2.5 × 10⁵ and 5 × 10⁶ steps, respectively, and the truncation radius was set to 12.5Å.

The molecular models of adsorbent CO₂ and CH₄ were directly called from the software model library, and the same method as that of coal molecular model was used for structure optimization and energy optimization. After inspection, the established coal slit pore model and adsorbate molecules were electrically neutral.

2.2. Simulation Parameter

In this study, the control variable method was adopted to design the test conditions. Considering the breakthrough pressure during the actual storage process [29,30], the test temperature range of 20~50 °C and the pressure range of 0~4 MPa were determined. The adsorption of the mixed gas of CO₂ and CH₄ in three types of coal pore diameters (microporous < 2 nm ≤ mesoporous < 50 nm ≤ macroporous [31]) within the range of 0.6~100 nm was simulated. Micropores in bituminous coal occupy a sizable specific surface area and contribute the most to the total adsorption, so the effects of total pressure, gas ratio, and temperature on the adsorption selectivity were explored in 2 nm micropores. The parameter settings for each simulation group are shown in

Table 1. Simulation parameter setting table.

No.	Pore Size (nm)	Gas Ratio n1:n2	Temp (°C)	Total Pressure (MPa)	Pressure Gradient (MPa)	No.	Pore Size (nm)	Gas Ratio n1:n2	Temp (°C)	Total Pressure (MPa)	Pressure Gradient (MPa)
1	0.6	1:1	30	0~4	0.05&0.5	17	2	0:1	30	0~4	1.0
2	0.8	1:1	30	0~4	0.05&0.5	18	2	0:1	30	0~4	1.0
3	1.0	1:1	30	0~4	0.05&0.5	19	2	1:9	30	0~4	1.0
4	1.6	1:1	30	0~4	0.05&0.5	20	2	2:8	30	0~4	1.0
5	2.0	1:1	30	0~4	0.05&0.5	21	2	3:7	30	0~4	1.0
6	6.0	1:1	30	0~4	0.05&0.5	22	2	4:6	30	0~4	1.0
7	10	1:1	30	0~4	0.05&0.5	23	2	6:4	30	0~4	1.0
8	20	1:1	30	0~4	0.05&0.5	24	2	7:3	30	0~4	1.0
9	30	1:1	30	0~4	0.05&0.5	25	2	8:2	30	0~4	1.0
10	40	1:1	30	0~4	0.05&0.5	26	2	9:1	30	0~4	1.0
11	50	1:1	30	0~4	0.05&0.5	27	2	1:1	20	0.1~0.5	0.1
12	60	1:1	30	0~4	0.05&0.5	28	2	1:1	25	0.1~0.5	0.1
13	70	1:1	30	0~4	0.05&0.5	29	2	1:1	35	0.1~0.5	0.1
14	80	1:1	30	0~4	0.05&0.5	30	2	1:1	40	0.1~0.5	0.1
15	90	1:1	30	0~4	0.05&0.5	31	2	1:1	45	0.1~0.5	0.1
16	100	1:1	30	0~4	0.05&0.5	32	2	1:1	50	0.1~0.5	0.1

.

In the table, groups 1~16 study the effect of pore size on competitive adsorption, setting 0.5 MPa as the pressure gradient in the range of 0~4 MPa. Different pressure points on the same isotherm are used to study the effect of different total pressures, and are supplemented with the pressure points of 0.05 MPa, which is close to the adsorption group under the pressure of 0 MPa. Groups 5 and 19~26 explore the effect of the proportion of the gas, which is the ratio of the amount of gas substance, where Groups 17 and 18 were set as pure gas adsorption control groups. Group 5 and groups 27~32 were investigated for the effect of temperature.

Under high pressure conditions, the compressibility of gases varies. To control the amount of the two gases to be consistent under different total pressures, the partial pressure of the two gases at each pressure point is calculated using the following system of equations.

$$\left\{\begin{array}{l}P=P_1+P_2\\ {\displaystyle \frac{P_1}{P_2}}={\displaystyle \frac{Z_1{n}_1}{Z_2{n}_2}}\end{array}\right.$$

(1)

where P is the total pressure of mixed gas, P₁ and P₂ are the pressures of CO₂ and CH₄, respectively, and n₁ and n₂ are amount of substance of CO₂ and CH₄, respectively. Z₁ and Z₂ are the compression factors of CO₂ and CH₄, respectively, and the compression factor, which is also a function of the gas pressures, is calculated using the virial equation expanded to the second term [32,33,34]:

$$\left\{\begin{array}{l}Z=1+{\displaystyle \frac{B_{V}P}{RT}}\\ B_V={\displaystyle \frac{R{T}_c}{P_c}}\left[0.083-{\displaystyle \frac{0.422}{{T_{\mathrm{r}}}^{1.6}}}+\omega (0.139-{\displaystyle \frac{0.172}{{T_{\mathrm{r}}}^{4.2}}})\right]\\ T_{\mathrm{r}}={\displaystyle \frac{T}{T_c}}\end{array}\right.$$

(2)

where B_V is the second virial coefficient, p is the gas pressure, MPa; T is the gas temperature, K, R is the gas constant, J/(kg·K); T_c is the gas critical temperature, K; P_c is the critical pressure, MPa; T_r is the reduction temperature, dimensionless; and ω is the eccentricity factor, dimensionless. The physical parameters of CO₂ and CH₄ gases are known by querying the NIST database, and the values are taken as shown in

Table 2. Physical properties parameters of CO₂ and CH₄.

Gas Species	Tc (K)	Pc (MPa)	ω (Dimensionless)	R (J/kg/k)
CH4	190.56	4.60	0.0114	0.520
CO2	304.13	7.38	0.2239	0.189

below. The compression factors at different pressure points calculated by the virial equation are compared with the data in the NIST database as shown in Figure 2, and the two curves overlap well, which verifies the reliability of the equation.

It is also worth noting that fugacity is used in MS to represent gas pressure, and for real gases, fugacity f and pressure p can be converted by the following equation:

$$\phi =f/p$$

(3)

$$\ln \phi ={\displaystyle {\int }_{{\mathrm{p}}^{\mathsf{\theta }}}^p(\frac{Z-1}{p})d_p}$$

(4)

where φ is the fugacity coefficient, p^θ is a constant infinitely close to 0, and p is the gas pressure, MPa. From this, the fugacity of CO₂ and CH₄ gases at different pressures in specific substance amount ratios can be obtained.

Combining the above equation with the gas data, the formula for calculating the fugacity from pressure at different temperatures was obtained, as shown in

Table 3. Calculation formulas for the fugacity of CO₂ and CH₄ gases at different temperatures.

Temperature T (K)	CO2 Fugacity Calculation Equation	CH4 Fugacity Calculation Equation
293.15	f = p·e−0.0532p	f = p·e−0.0179p
298.15	f = p·e−0.0502p	f = p·e−0.0168p
303.15	f = p·e−0.0475p	f = p·e−0.0157p
308.15	f = p·e−0.0449p	f = p·e−0.0148p
313.15	f = p·e−0.0425p	f = p·e−0.0139p
318.15	f = p·e−0.0402p	f = p·e−0.0130p
323.15	f = p·e−0.0381p	f = p·e−0.0123p

, and these can be input into the simulation software as pressure points on the adsorption isotherm to calculate the adsorption amount.

In molecular simulations, the adsorption of various gases by coal can be calculated from the average number of gas molecules adsorbed per cell of the coal slit model in the simulated system:

$$V=1000\times {\displaystyle \frac{N}{N_{\mathrm{a}}{M}_{\mathrm{c}}}}$$

(5)

where V is the gas adsorption amount, mmol/g; N is the number of adsorbed gas molecules, N_a is the number of molecules in the cell, here 4; M_c is the molecular molar mass of waser bituminous coal molecules, g/mol, with a specific value of 2821.

3. The Influence Law of Multiple Factors on the Competitive Adsorption of CO₂ and CH₄

3.1. Adsorption Isotherm of CO₂/CH₄ Mixed Gas

Before the simulation began, six simulation groups in micro-, meso-, and macropores were selected in this paper, and repeated simulations were conducted 3 to 4 times. The simulation results showed that the range of the number of adsorbed molecules under the same conditions was much smaller than the average value, proving that the adsorption results were stable under the simulation parameters set in this study. After the adsorption system in the model reached equilibrium, adsorption isotherms for an equimolar CO₂/CH₄ gas mixture in coal pores of different sizes were obtained. Figure 3a,b show the adsorption isotherms of the component gases and the mixture of gases, respectively, and all the adsorption isotherms are typical of the Langmuir-type adsorption curves. It can be seen that the adsorption amount of CO₂ is larger than that of CH₄ in each pore size, which reflects that CO₂ has an advantage in the adsorption process of competing with CH₄.

The distribution of gas adsorption probability at each adsorption potential energy point during the adsorption process can be obtained through Monte Carlo simulation. The larger the absolute value of the adsorption potential energy, the more stable the adsorption and the higher the priority of the adsorption point. If a gas has a greater adsorption capacity at the preferred adsorption point, it indicates that the adsorption affinity of the gas is stronger. Figure 4 shows the adsorption probability distribution of pure CO₂ and pure CH₄ in a 2 nm pore at 30 °C. The results reveal that the adsorption probability of coal molecules for gases follows a normal distribution with respect to adsorption potential energy. The optimal adsorption sites for CO₂ and CH₄ exhibit adsorption potentials of −6.85 kcal/mol and −3.95 kcal/mol, respectively. The adsorption potential energy of CO₂ is generally higher than that of CH₄.

Furthermore, the distribution range of the adsorption probability of CO₂ is relatively wide. There is a certain adsorption probability at the points with lower adsorption potential energy, indicating that CO₂ also has a certain adsorption at the weak adsorption points on the coal surface. However, CH₄ is concentrated around the points with an adsorption potential energy of −3.95 kcal/mol, suggesting that CH₄ tends to attach to specific adsorption sites. Moreover, it has a relatively high competitive adsorption capacity on this type of adsorption site, which explains the preferred adsorption of CO₂ in the adsorption system. It can be predicted that the adsorption affinity of CH₄ gradually increases after the priority adsorption sites are occupied.

The isothermal adsorption curves of CO₂/CH₄ mixed gases with different molar ratios in 2 nm pores are shown in Figure 5. Figure 5a presents the component gas adsorption isotherms, demonstrating that coal exhibits significantly stronger adsorption affinity for CO₂ than CH₄. Even when CO₂ only accounts for 20% of the mixed gas, its adsorption quantity still exceeds that of CH₄. This phenomenon occurs because CO₂ is a polar molecule while CH₄ is non-polar, making CO₂ more easily attracted by polar functional groups (such as hydroxyl and carboxyl groups) abundantly distributed on coal surfaces. Additionally, the smaller molecular diameter of CO₂ facilitates its access to coal pores. Figure 5b displays the overall adsorption curves of the mixed gas, revealing that higher CO₂ proportions result in greater mixed gas adsorption capacity.

The adsorption isotherms of CO₂/CH₄ mixed gases in equal substance ratios at different temperature environments are shown in Figure 6. The adsorption amounts of both gases decreased with increasing temperature, and, in terms of the adsorbed molecular weights, the change of temperature had a more obvious effect on CO₂ adsorption, which was caused by the larger adsorption amount of CO₂ molecules.

3.2. The Variation in the Adsorption Selectivity Coefficient Under the Influence of Pore Size

In the adsorption model, the CO₂/CH₄ adsorption selectivity coefficient can be calculated according to Equation (6):

$$S_{\mathrm{c}}={\displaystyle \frac{N_{{\mathrm{CO}}_2}/N_{{\mathrm{CH}}_4}}{n_{{\mathrm{CO}}_2}/n_{{\mathrm{CH}}_4}}}$$

(6)

where N_CO2 and N_CH4 are the number of molecules of adsorbed CO₂ and CH₄, respectively, and n_CO2 and n_CH4 are the amounts of substance of CO₂ and CH₄ in the free phase, respectively. The ratio of the amounts of substance of the two is a certain value under a specific partial pressure, and this constant value is the ratio of the gases in the different simulation groups; the values are shown in Table 1. When S_c is greater than 1, it indicates that CO₂ adsorption is preferred over CH₄. A high CO₂/CH₄ adsorption selectivity coefficient indicates that CO₂ is more enriched in the adsorbed phase than CH₄ compared to the free phase ratio.

To explore the influence of pore size on the competitive adsorption of CO₂ and CH₄, in experimental groups 1 to 16, four pressure points, 0.5 MPa, 1.0 MPa, 2.0 MPa, and 4.0 MPa, were selected among the measured pressure points for side-by-side comparisons. The change in S_c with pore size is shown in Figure 7; it can be seen that among the three types of pores (microporous < 2 nm ≤ mesoporous < 50 nm ≤ macroporous), S_c shows a negative exponential decreasing trend with the increase of pore size: in the range of micropore sizes, S_c is the most sensitive to changes in pore size, and its value decreases rapidly with the increase in pore size; in pore sizes from 0.6 nm to 2.0 nm, the S_c value of the various pressure points generally decreases by more than 1; in the medium and large pore sizes, the trend of the decrease in the S_c slows down and gradually tends to stabilize with the increase in pore size, but always stabilizes above 1. The change rule of S_c with pore size at different pressure points is consistent, and it shows the phenomenon of decreasing the advantage of adsorption of CO₂ with the increase in pressure.

The density distribution of gas molecules adsorbed in different pore sizes at 2 MPa in the slit model is shown in Figure 8. The resolution of the scattered points represents the density of the gas in space. Red represents the density of CO₂ molecules, and green represents the density of CH₄ molecules. It can be seen that the gas molecules are concentrated on the surface of coal molecules. As the pore size increases from 1.0 nm to 60.0 nm, the density of CO₂ gradually decreases while that of CH₄ increases. This is because in the micropores, the specific surface area is relatively high and the free space volume is small, the intermolecular forces are strong, and CO₂ and CH₄ undergo intense competitive adsorption in the narrow space. CO₂ has a significant adsorption advantage. When the pore size increases, the free space increases, the competitive intensity between gas molecules weakens, and the inhibitory effect of CO₂ on CH₄ adsorption decreases. Therefore, the larger pore size weakens the adsorption affinity of coal molecules for CO₂.

In order to quantitatively analyze the effect of pore size (d) on S_c, the change curves were fitted exponentially, and the results are shown in

Table 4. Fitting equation for the relationship between S_c and pore size.

Pressure (MPa)	Fitted Equation	Coefficient of Determination (R2)
0.5	Sc = 2.67 + 3.51e−0.050d	0.944
1.0	Sc = 2.06 + 4.03e−0.046d	0.961
2.0	Sc = 1.81 + 4.21e−0.059d	0.967
4.0	Sc = 1.55 + 4.11e−0.064d	0.982

. The coefficients of determination of the fitted curves, R², are all above 0.94, and the form of the equation expressed in Equation (7) basically explains the law of the effect of pore size on S_c:

$$S_{\mathrm{c}}=a+b{\mathrm{e}}^{cx}$$

(7)

where a, b, and c are the fitting parameters, and x is the independent variable representing each of the influencing factors. e is the base of the natural logarithm.

The results show that the micropores of coal have a stronger adsorption capacity for CO₂, and the proportion of micropores and mesopores in high-rank coal is relatively high. It can be predicted that it has a greater adsorption capacity for CO₂. Within the range of mesopores and macropores, especially when it is greater than 50 nm, S_c enters a stage of gradual decline. At this time, the pore size has little influence on the adsorption selectivity. It can be considered to sacrifice the competitive adsorption advantage and select coal seams or coal bodies with a large distribution of macropores for injection to improve the injection efficiency.

3.3. The Variation in the Adsorption Selectivity Coefficient Under the Influence of Gas Proportion

To explore the influence of the proportion of gas components on the competitive adsorption of CO₂ and CH₄, the simulation set nine different gas mixture ratios, with the CO₂ mole fraction increasing in 10% increments from 10% to 90%. The variation in S_c with the percentage of CO₂ in the mixed gas obtained from the simulation and the fitted curves is shown in Figure 9.

The selective adsorption effect of the coal on CO₂ gradually weakens with the increase in the CO₂ molar fraction in the mixed gas, under different pressures, as the molar proportion of CO₂ increases from 10% to 90%, the reduction rate of S_c is generally above 30%; the decreasing tendency is also exponential, and the overall pattern is consistent at different pressures. This indicates that coal exhibits stronger CO₂ adsorption capacity when the CO₂ concentration is lower; this is similar to the research results of Jia et al. [17], and is because at low CO₂ fractions, CO₂-surface interactions dominate, allowing CO₂ molecules to fully occupy the active sites on coal surfaces, resulting in high adsorption selectivity. As CO₂ concentration increases, intermolecular repulsion between CO₂ molecules weakens their interaction with coal surfaces. The accumulated CO₂ molecules near the surface create a shielding effect, increasing CH₄’s access to adsorption sites. Consequently, CH₄ adsorption rises.

Denoting the CO₂ molar fraction, %, by r, the relationship between CO₂ molar fraction and S_c was also fitted by Equation (7), and the results are shown in

Table 5. Fitting equation for the relationship between S_c and CO₂ molar fraction.

Pressure (MPa)	Fitted Equation	Coefficient of Determination (R2)
1.0	Sc = 4.56 + 3.58e−0.035r	0.980
2.0	Sc = 4.45 + 3.15e−0.038r	0.984
3.0	Sc = 4.41 + 3.51e−0.047r	0.996
4.0	Sc = 4.46 + 4.11e−0.073r	0.947

. From the fitted coefficient of determination, Equation (7) still explains the relationship.

3.4. The Variation in the Adsorption Selectivity Coefficient Under the Influence of Mixed Gas Pressure

To explore the influence of the total pressure of the mixed gas on the competitive adsorption of CO₂ and CH₄, comparing the adsorption isotherms of adsorbing equal amounts of mixed CO₂ and CH₄ gases in four pore sizes at 30 °C, S_c was calculated at each pressure point, and Figure 10 shows the variation in S_c with the total pressure of the mixed gases. As the total pressure increases, the selectivity of coal adsorption on CO₂ gradually decreases, and it can be seen that the action law still conforms to the negative exponential function. Under the same pressure, the smaller the pores, the larger the S_c. As the pressure increases from 0 to 4 MPa, the S_c of the pores at 1.0 nm, 2.0 nm, 30 nm, and 100 nm decreases by approximately 34%, 36%, 60%, and 57% respectively. The decreasing trend of S_c exhibits a rapid decrease, a slow decrease, and a gradual steady decrease when the pressure is 0~0.5 MPa, 0.5~2 MPa, and more than 2 MPa, respectively. This is consistent with the findings of Xie et al. [16]. In addition, Liao et al. [15] conducted an experiment on the adsorption of 50%CH₄ + 50%CO₂ mixed gas by shale. The experimental results also showed that S_c changed negatively exponentially with pressure, which confirmed the accuracy of the simulation results.

This is because at lower pressure, the adsorption sites on the coal surface are relatively more numerous, and CO₂ molecules will preferentially occupy the dominant adsorption sites on the coal surface due to their stronger adsorption capacity during the adsorption process of the coal, and at this time, the adsorption selectivity of coal on CO₂ is stronger. With the increase in gas pressure, the advantageous adsorption sites on the coal surface are gradually occupied, and the remaining adsorption sites are weak energy sites; at this time, the competitive ability of CO₂ relative to CH₄ is weakened, and the S_c is also reduced. Curve fitting was performed on the scatter points, and the fitted equations for the relationship between the mixed gas pressure and S_c are shown in

Table 6. Fitting equation for the relationship between S_c and total pressure.

Pore Size (nm)	Fitted Equation	Coefficient of Determination (R2)
1.0	Sc = 5.34 + 3.33e−2.37p	0.967
2.0	Sc = 4.75 + 3.07e−1.82p	0.946
30	Sc = 2.25 + 3.59e−1.61p	0.952
100	Sc = 1.44 + 2.34e−1.82p	0.961

, with p denoting the total mixed gas pressure, MPa; the fitted equations in each aperture have a high degree of fit, with the coefficients of determination above 0.94.

3.5. The Variation in the Adsorption Selectivity Coefficient Under the Influence of Temperature

To explore the influence of temperature on the competitive adsorption of CO₂ and CH₄, the simulation of adsorption of CO₂ and CH₄ gases mixed in equal amounts at different temperatures in 2 nm coal pores was carried out, and the S_c at the same pressure point on different isotherms was taken for comparison, and the results are shown in Figure 11. S_c shows a slow decreasing trend with increasing temperature. Within the temperature difference range of 20~50 °C, the variation range of S_c in all pressure groups was only about 1, and the change was not obvious. Under the buried condition, in the horizontal direction of the same coal seam, if it is not affected by local geologic structures such as magma intrusion and strong groundwater activity, the temperature difference is smaller. In the vertical direction, the buried depth span of the same coal seam tends to be small, and the temperature variation is also small, so the unevenness of the temperature of the coal seam does not have much effect on the selectivity of CO₂ and CH₄ adsorption.

In terms of the nature of adsorption, the adsorption process is an exothermic process, and an increase in temperature at the same pressure will desorb the gas and reduce the amount of adsorption. Compared with CH₄, CO₂ is more affected by temperature. Because CO₂ has a stronger force with the surface of the coal, the heat of adsorption is greater, and its desorption tendency is more obvious when the temperature increases, making the S_c decrease. Still trying to fit the effect of temperature on S_c with an exponential regression model, this paper denotes the temperature by T, °C. The fitted equation is shown in

Table 7. Fitted equations for the relationship between S_c and temperature.

Pressure (MPa)	Fitted Equation	Coefficient of Determination (R2)
0.1	Sc = 9.91 − 1.63e0.015T	0.885
0.2	Sc = 11.31 − 3.11e0.009T	0.861
0.3	Sc = 8.52−0.75e0.023T	0.876
0.4	Sc = 5.61 + 9.27e−0.057T	0.95
0.5	Sc = 4.83 − 3.09e0.02T	0.476

, which shows a good fit in terms of the coefficient of determination.

3.6. Ranking of the Importance of Factors Affecting Competitive Adsorption

From the above analysis, it can be seen that the pore width, CO₂ molar fraction, gas pressure, and temperature are all negatively correlated with S_c, but with different strengths of influence on the S_c value. In order to compare the intensity of the influence of the four factors on the adsorption selectivity of coal adsorption of CO₂/CH₄ mixed gas, the relationship between the four factors and S_c was analyzed by multiple regression. In the multiple regression model, the size of the absolute value of the regression coefficients of the influencing factors indicated their importance to the dependent variable S_c.

The four influencing factors have different units and orders of magnitude, which may lead to misjudgment of the importance of other variables. In order to eliminate the influence of the order of magnitude and scale of the variables, this study applied range normalization to the independent variables, mapping the range of the values of these data with different scales into a specific interval (set as 0~1 in this paper), so that each variable is comparable. The variable standardization method is:

$$X^{\prime }={\displaystyle \frac{(X-X_{\min })}{(X_{\max }-X_{\min })}}$$

(8)

wher, X′ represents the range-normalized variable value, X denotes the original variable value before normalization, and X_max and X_min correspond to the maximum and minimum values of the variable X, respectively.

In order to simplify the calculation and facilitate the conversion to a multiple linear regression model, this paper takes the natural logarithm on both sides of the equation at the same time, and has the following equations for a single influencing factor:

$$\ln S_{\mathrm{c}}=\ln b+cx+m$$

(9)

where m is a fitting constant. Considering the combined effects of four factors on S_c, regression analysis of the normalized independent variables yields the following model equation:

$$\ln S_{\mathrm{c}}={\beta }_0+c_1{d}^{\prime }+c_2{r}^{\prime }+c_3{p}^{\prime }+c_4{T}^{\prime }+\omega$$

(10)

where β₀ is the intercept, ω is the error term, and c₁, c₂, c₃, and c₄ are the normalization coefficients of the pore width normalized quantity d′, CO₂ molar fraction normalized quantity r′, mixed gas pressure normalized quantity p′, and temperature normalized quantity T′, respectively, and the magnitude of its absolute value reflects the importance of the corresponding influencing factors on S_c. Using the existing simulation results as data samples, Equation (10) was used as the model for multivariate linear fitting, and the results are shown in

Table 8. Results of multivariate linear fitting of the effect of multiple factors on lnS_c.

Influencing Factors	Corresponding Coefficients	Value of the Fitting Coefficient	Coefficient of Determination (R2)
—	β0	2.139	0.904
Pore size	c1	−1.279
CO2 molar fraction	c2	−0.369
Mixed gas pressure	c3	−0.451
Temperature	c4	−0.120

.

The coefficients of the regression equation are all negative, and the absolute values are |c₁| > |c₃| > |c₂| > |c₄|, which shows that the values of the four influencing factors are negatively correlated with lnS_c, and the importance of the four influencing factors is ranked as pore size > mixed gas pressure > CO₂ molar fraction > temperature. This indicates that the size of the coal pore size has the most critical influence on the adsorption selectivity of CO₂ and CH₄, and the smaller the pores in the coal, the stronger the adsorption selectivity of CO₂, followed by the gas pressure, and the CO₂ molar fraction and the ambient temperature have a weaker influence on the adsorption selectivity.

4. Optimization of Adsorption Selectivity Coefficient Model

In the CO₂-ECBM work, an accurate adsorption selectivity coefficient can better predict the adsorption concentration of CO₂ and the desorption purity of CH₄ in the coal after CO₂ is injected into the coal seam and the adsorption equilibrium is reached, which can help to determine the pressure and quantity during the CO₂ injection process, and to evaluate the CH₄ production at different development stages. The expression of the adsorption selectivity coefficient for the competitive adsorption of CO₂ and CH₄ obtained from E-L model is [13]

$$S_{\mathrm{c}}={\displaystyle \frac{V_{\mathrm{L}1}{P}_{\mathrm{L}2}}{V_{\mathrm{L}2}{P}_{\mathrm{L}1}}}$$

(11)

where V_L1 and V_L2 denote the Langmuir volume of CO₂ and CH₄, respectively, cm³/g; P_L1 and P_L2 denote the Langmuir pressure of CO₂ and CH₄, respectively, MPa. Equation (11) shows that the adsorption selectivity coefficient in the process of competitive adsorption of two gases is completely determined by the adsorption constants of a single gas at the time of adsorption. Temperature and pore structure have a large influence on the adsorption constants [35], and changes in both will change the adsorption constants; therefore, Equation (11) can reflect the influence of temperature and pore structure on adsorption selectivity, but it cannot accurately reflect the changes in the competitive adsorption characteristics with the gas pressure, the ratio of component gases, and other factors.

To investigate the response patterns of S_c to these factors, we modified the existing E-L model accordingly. Based on the above analysis and considering that pore size distribution and temperature variations remain relatively constant in homogeneous coal seam [36], the two are not used as correction factors. According to the molecular simulation results, S_c shows exponential changes with both gas pressure and CO₂ molar fraction and an exponential correction factor is set to Equation (11), and the correction model is shown in Equation (12):

$$S_{\mathrm{c}}={\mathrm{e}}^{{\alpha }_{1}p+{\alpha }_{2}r+{\alpha }_3}{\displaystyle \frac{V_{\mathrm{L}1}{P}_{\mathrm{L}2}}{V_{\mathrm{L}2}{P}_{\mathrm{L}1}}}$$

(12)

where α₁, α₂, and α₃ are correction parameters.

The simulation primarily used 2 nm pores as the basis for testing adsorption isotherms. Therefore, to validate the modified model’s applicability in predicting adsorption selectivity, we substituted the single-component gas adsorption parameters (CO₂ and CH₄) from 2 nm pores at 30 °C into Equation (12) and used this equation to fit the S_c values for mixed gas adsorption under identical temperature and pore size conditions. Figure 12 presents both the single-component adsorption isotherms and their corresponding Langmuir adsorption functions.

Figure 13 demonstrates the S_c values corresponding to different total pressures and CO₂ molar fractions at 30 °C in the 2 nm pores from the previous simulation results. Based on the adsorption constants obtained, surface fitting of the data points using Equation (12) achieved a goodness-of-fit of 0.84, suggesting that the modified model can provide a more accurate prediction of adsorption selectivity coefficients for the competitive adsorption of CO₂/CH₄. The modified computational model for the CO₂/CH₄ adsorption selectivity coefficient considering pressure and CO₂ molar fraction was obtained as Equation (13).

$$S_{\mathrm{c}}=1.6{\mathrm{e}}^{-0.1p-0.5r}{\displaystyle \frac{V_{\mathrm{L}1}{P}_{\mathrm{L}2}}{V_{\mathrm{L}2}{P}_{\mathrm{L}1}}}$$

(13)

In homogeneous coal seams, Equation (13) can represent the calculation method of the CO₂/CH₄ adsorption selectivity coefficient in coal with pore diameters mostly concentrated at 2 nm at a coal seam temperature of 30 °C. It is known that the proposed model Equation (12) still has certain limitations. This is because in this paper, the correction parameters are mainly obtained through simulation to obtain Equation (13). In fact, when calculating with Equation (12), conducting a large number of experiments in advance to obtain empirical parameters in order to obtain the accurate values of the correction parameters α₁, α₂, and α₃ is a more rigorous and accurate calculation method. However, the correctness of the improved model in the form of equations has been verified by the simulation results. The comparison of the advantages and disadvantages between the improved model and the traditional model can be seen in

Table 9. The comparison between the traditional E-L model and the improved model.

Sc Calculation Model	Model Equation	Limitations	Advantage
E-L model	$S_{\mathrm{c}}={\displaystyle \frac{V_{\mathrm{L}1}{P}_{\mathrm{L}2}}{V_{\mathrm{L}2}{P}_{\mathrm{L}1}}}$	Lack of consideration for the interaction between gases.	Simple calculation process
proposed model	$S_{\mathrm{c}}=1.6{\mathrm{e}}^{-0.1p-0.5r}{\displaystyle \frac{V_{\mathrm{L}1}{P}_{\mathrm{L}2}}{V_{\mathrm{L}2}{P}_{\mathrm{L}1}}}$	A large number of experimental results are required to obtain the values of corrected parameters α1, α2, and α3.	More accurate calculation results

.

To verify the rationality of the improved model in practical applications, the results of single gas adsorption and competitive adsorption of CO₂ and CH₄ in shale obtained by Liao et al. [15] through experiments were used for testing. Based on the absolute isothermal adsorption amounts of CO₂ and CH₄ single gases adsorbed by marine shale of the Ordovician Wufeng Formation in the Sichuan Basin at 323.15 K temperature and different pressure points measured in the experiment, the adsorption constants of the two gases can be obtained by fitting the adsorption data using the Langmuir model, as shown in

Table 10. Adsorption constants of pure gases such as CO₂ and CH₄ in shale.

Type of Gas	VL (mmol/g)	PL (MPa)	Coefficient of Determination (R2)
CH4	0.083	2.41	0.992
CO2	0.26	3.06	0.994

. The S_c of the 25%CH₄ + 75%CO₂ mixed gas under different pressures was calculated by Equation (13). The real experimental results, the calculated values of the traditional E-L model, and the calculated results of the improved model were compared, as shown in Figure 14. It can be seen that the improved model can very well display the variation of S_c with gas pressure. When the gas pressure is above 1 MPa, the predicted results are closer to the true results, but there are still errors. This is due to the errors in the values of the calculation parameters α₁, α₂, and α₃. Obtaining the parameter values applicable to this shale sample will lead to more accurate results.

It can be seen from the verification results of real experiments that although the correction parameter values in Equation (13) were only obtained from the adsorption data in 2 nm pores at 30 °C, Equation (13) may have wider applicability in the S_c prediction of actual coal and rock masses, which awaits further testing.

In summary, the modified model incorporates the two key influencing factors of total gas pressure and CO₂ molar fraction into the calculation of the adsorption selectivity coefficient, which supplements the defects of the traditional model that ignores the influence of pressure and gas composition on the adsorption selectivity in the actual working conditions, and can make the model more closely match the real mixed gas adsorption results.

5. Conclusions

In this paper, through the methods of molecular dynamics and massive canonical Monte Carlo simulation, the competitive adsorption behavior of CO₂/CH₄ in the pores of bituminous coal and the dynamic variation law of its adsorption selectivity coefficient with the characteristics of the coal and environmental conditions were systematically studied. Based on a large amount of adsorption data, the negative exponential influence mechanism of pore size, gas pressure, gas ratio, and temperature on the adsorption selectivity coefficient was quantified. Correction factors were set for the gas proportion and the pressure of the mixed gas. Based on the S_c calculation formula of the extended Langmuir model, a correction equation of the adsorption selectivity coefficient suitable for homogeneous coal seams was proposed, and a more accurate and reliable S_c calculation method was obtained. However, its applicability still needs further study. The core conclusions are as follows:

(1)

The adsorption isotherm of coal on CO₂, CH₄ single-component and mixed gases under different conditions were obtained by molecular simulation, and found that the main adsorption potential energy was −6.85 kcal/mol for CO₂ and −3.95 kcal/mol for CH₄, which explains the absolute dominance of CO₂ in the competition, from the perspective of energy.

(2)

Based on the adsorption isotherm results from molecular simulations, the effects of four factors, namely, pore size, gas pressure, CO₂ molar fraction, and temperature, on the CO₂/CH₄ adsorption selectivity coefficient were analyzed and found to have negative exponential relationships.

(3)

The influence weights of four factors on CO₂/CH₄ adsorption selectivity coefficient were investigated by multiple linear regression analysis as pore size > mixed gas pressure > CO₂ molar fraction > temperature, which verified the superiority of using the high-ranking coal seam as a target layer for CO₂ sequestration.

(4)

Aiming at the shortcomings of the traditional extended Langmuir (E-L) model, which ignores the dynamic changes of pressure and components, a correction equation is proposed to introduce an exponential correction term for the CO₂ mole fraction and total gas pressure; after the verification of the simulation results and the experimental results of shale adsorption in the existing studies, the prediction error of the newly built model is lower than that of the traditional model.

(5)

The newly constructed model has achieved success in terms of variance, but the corrected parameters are derived from the simulation data of bituminous coal adsorption, and it assumes that the coal seam is homogeneous. Its applicability in other coal types (such as anthracite, lignite), heterogeneous coal seams, and broader temperature scenarios awaits further study.