Comparative Study of Variable-Flow Gas Injection Patterns on CH₄ Diffusion Dynamics: Experimental Insights into Enhanced Coalbed Methane Recovery

Abstract

Variable-flow displacement has been effectively used to enhance oil recovery; however, it has rarely been investigated for coalbed methane production, and the CH₄ diffusion laws in this process are not clear. In this paper, we carried out a study on the CH₄ diffusion law in the bidirectional diffusion process displaced by variable-flow gas injection. The emission and desorption quantity of CH₄ under variable-flow gas injection, the displacement effect under the principle of equal time and quantity, and the applicability of the CH₄ diffusion model for the bidirectional diffusion process were analyzed. The results indicate that the variable-flow injection modes emit more CH₄ compared to constant flow injection. The CH₄ emission and desorption quantities for each injection mode are as follows: step-changed > sinusoidal-changed > constant flow. Secondly, the order of CH₄ emission and desorption quantity in each gas injection mode is as follows: step-changed > sinusoidal-changed > constant flow. When CO₂ is the injection gas source, the outlet CH₄ emission and desorption quantity are larger than N₂ injection. Thirdly, through the analysis of the principle of equal time and equal quantity, the variable-flow injection modes consume less gas for each volume of emitted CH₄, resulting in a more effective displacement. Finally, the diffusion fitting effect of the bidisperse model for CH₄ in the bidirectional diffusion process is better than that of the unipore model, and the bidisperse diffusion model can better fit the m_t/m_∞ curve of CH₄ diffusion during the variable-flow gas injection replacement process.

1. Introduction

Gas injection displacement has emerged as a globally recognized effective technology for enhancing coalbed methane (CBM) production, demonstrating significant improvements in CBM recovery rates [1,2,3,4,5]. More recently, researchers have successfully applied this technology to mine gas disaster mitigation, with field applications showing promising results [6,7]. The predominant gas injection agents include N₂ [8,9,10], CO₂ [11,12,13], and their binary gas mixtures [14,15]. CO₂ demonstrates superior adsorption affinity in coal compared to CH₄ [16,17,18,19], owing to its stronger molecular interactions with coal matrices. When injected, CO₂ preferentially occupies CH₄ adsorption sites through competitive adsorption mechanisms, thereby displacing and promoting CH₄ desorption. Conversely, N₂ primarily functions through physical displacement in coal fractures. As an inert gas with lower adsorption capacity, N₂ reduces CH₄ partial pressure within the pore system while establishing concentration gradients that enhance CH₄ diffusion and liberation [10,20,21]. Therefore, the whole displacement process can be regarded as a comprehensive process for co-adsorption of injection gas and CH₄ in the matrix of coal pores, in which the injected gas and CH₄ are mixed in the fracture network of the coal seam, the injected gas diffuses from the fracture into the matrix of coal, and the desorbed gas diffuses from the matrix to the fracture. Because the diffusion of injected gas and CH₄ is a migration process along the same cleats but in opposite directions, it is referred to in this study as a “bidirectional diffusion process”, which is different from the “unidirectional diffusion process” of single gas in coal.

In the previous research, most scholars have done a lot of work on the displacement process under the condition of constant flow gas injection by controlling the gas injection type [22,23], gas injection pressure [24,25,26], gas injection time [27], degree of coal metamorphism [28,29], coal sample gas pre-adsorption pressure [30,31], and stress conditions [32,33]. A variety of displacement conditions are simulated, and the displacement mechanism and the permeability evolution law are analyzed under each of the conditions. For the final adsorption state of injected gas and gas, that is, the law of “competitive adsorption”, scholars have also done a lot of research through quasi-static adsorption experiments, considering gas types [34,35], coal rank [36,37], mixed gas ratio [38], coal sample humidity [36,39], temperature [40], and so on. In addition, the previous research on the gas diffusion law connecting adsorption in pores and seepage in fractures mainly focused on the single gas diffusion mechanism and also formed widely accepted characterization models such as the unipore diffusion model [41,42], bidisperse diffusion model [43,44,45], and dispersion diffusion model [46,47]. In conclusion, scholars have done a lot of research on the change of seepage law and competitive adsorption characteristics in the process of gas injection displacement, and achievements in the research on the diffusion characteristics of single gas in coal were made, but there are few experimental studies on the law of the bidirectional diffusion process in gas injection displacement.

N₂ and CO₂ injection displace coal seam gas, and water injection [48] and gas injection [49,50] in oil production are technically “injection-production” displacement types. Many similar research results indicate that in the “injection-production” displacement, the variable-flow technology can achieve better gas and liquid utilization efficiency and displacement performance. The research on variable-flow injection-production [51,52,53] is widely used in the oil industry, and its displacement effect and injection source utilization rate are better than those of constant flow injection, but it is rare to apply the variable-flow gas injection method to the research of coalbed methane production increase or gas disaster prevention. Note that the previous studies are based on the boundary conditions of constant flow gas injection/quasi-static adsorption, and the special flow field environment in the coal body caused by the variable-flow gas injection displacement also poses some key problems that cannot be solved by the previous research findings, such as (1) whether the bidirectional diffusion process of CH₄ can be described by the unidirectional diffusion law under the condition of variable-flow gas injection displacement, and whether the diffusion coefficient of each gas is the same as that of its unidirectional diffusion; (2) the displacement effect of gas injection under the condition of variable-flow lacks quantitative description compared with that of constant flow gas injection.

In this study, a custom-designed experimental system was developed to investigate bidirectional gas diffusion under variable-flow injection conditions. The dynamic characteristics of CH₄ diffusion during bidirectional displacement processes under variable-flow conditions were systematically investigated. A bidirectional diffusion model was developed, with subsequent quantitative analysis of CH₄ diffusion coefficient variations under transient flow conditions. Comparative evaluation of displacement efficiencies between variable-flow and constant-flow regimes was performed through parametric analysis. The results provide theoretical support for optimizing in situ CBM drainage operations and enhancing gas recovery efficiency.

2. Experimental Samples, Apparatus, and Procedures

2.1. Preparation of Coal Sample

Coal samples were collected from the 13^# coal seam at Xiaojiawa Coal Mine (located in Lvliang City, China). The average thickness of the seam is 12.02 m, which is a relatively stable and mineable coal seam across the entire area. The roof was dominated by sandy mudstone, followed by mudstone, and the bottom was dominated by mudstone, followed by sandy mudstone. The coal samples are gas coals with a low degree of metamorphism. The gas content of the No. 13 coal seam ranges from 1.65 to 2.65 m³/t. Despite the relatively low gas content per tonne of coal, the high-intensity mining operations have resulted in significant gas emissions. Simultaneously, due to the low porosity (only 4.82%), conventional drainage methods cannot effectively reduce the gas content. The location of coal samples is shown in Figure 1. Fresh coal samples were wrapped in plastic, sealed, and transported to the laboratory, where they were crushed and sieved to obtain coal particles with a diameter of 1–3 mm, dried, and used for experiments. The basic parameters of the coal samples are tabulated in

Table 1. Basic parameters of coal samples [54].

Coal Sample	Moisture (%)	Ash (%)	Volatile Matter (%)	Porosity (%)
Xiao Jiawa 13#	2.09	31.39	29.87	4.82

.

2.2. Experimental Methods

2.2.1. Experimental Apparatus

A bidirectional diffusion apparatus was designed to study the diffusion behavior of CH₄ under variable-flow gas injection conditions, as illustrated in Figure 2. The apparatus includes a gas injection system, a gas injection flow control and monitoring system, a thermostat box, a coal sample tank, a back-pressure valve, a gas concentration sensor, a vacuum pump, and data acquisition computer, and a gas injection steel pipe. The gas injection system consists of high-pressure cylinders (He, CH₄, N₂, and CO₂), a six-way valve, and a relief valve to control the pressure and composition of the injected gas. The flow control and monitoring system consists of two flow-meters, installed at both the inlet and outlet ends of the coal sample tank. The inlet flow-meter controlled the gas injection flow rate, while the outlet flow-meter monitored the flow rate of the gas leaving the tank. The thermostat box was used to maintain a constant temperature in which equipment such as inlet and outlet flow-meters, coal sample tank, back-pressure valve, gas pressure sensor, stop valve, etc., was placed. The internal volume of the coal sample tank was approximately 800 mL. Quartz chips were positioned at both ends of the canister to facilitate uniform dispersion of the injected gas. The data acquisition system collected experimental data on parameters including temperature, inlet and outlet gas pressure, gas flow rate, gas concentration, and other variables. Additionally, the back-pressure valve regulated the gas pressure within the tank. Following the completion of each experimental set, the apparatus was evacuated using a vacuum pump.

2.2.2. Experimental Procedures

Unidirectional and bidirectional diffusion experiments were conducted to investigate the diffusion behavior of CH₄ under variable-flow gas injection conditions.

1. The coal sample tank was filled with the coal sample, totaling 842.2 g of coal, and the free space volume of the experimental apparatus was calculated. The void space, consisting of the sample tank and connecting pipes, was calculated as 232.58 mL using Equation (1), following the methodology established in a previous study [6].

$${\mathrm{V}}_{\mathrm{void}}\cdot ({\mathrm{P}}_1-{\mathrm{P}}_2)={\mathrm{P}}_{\mathrm{a}}\cdot {\mathrm{V}}_{\mathrm{de}}$$

(1)

where ${\mathrm{V}}_{\mathrm{void}}$ is the free space volume of the apparatus; P_a is the atmospheric pressure in the laboratory; ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ denote the gas pressures in the pipeline after He injection and partial discharge, respectively; and ${\mathrm{V}}_{\mathrm{de}}$ is the discharged He volume.

2. Unidirectional diffusion experiments of CH₄, N₂, and CO₂ were conducted based on the experimental design parameters outlined in

Table 2. Experimental parameters for unidirectional diffusion experiments.

Number	Adsorbed Gas	Adsorption Equilibrium Pressure/MPa
1	CH4	0.5, 1.0, 1.5
2	N2
3	CO2

.

① Sample Preparation and Evacuation

The coal sample was charged into the experimental tank, followed by system evacuation for ≥12 h under vacuum conditions.

② Gas Injection and Equilibrium Criteria

To achieve the target adsorption equilibrium pressure, gas injection commenced after closing the back-pressure valve and opening the inlet stop valve. Adsorption equilibrium was confirmed when gas pressure stability persisted for 24 consecutive hours.

③ Unidirectional diffusion and Parameter Monitoring

Opening the back-pressure valve initiated gas release, while pressure transducers flanking the coal tank and outlet flow rates were continuously monitored.

④ Experiment Completion Criteria

Experimental termination occurred when all pressure sensors and the flow meter registered null values.

⑤ Replication Procedure

Unidirectional diffusion experiments were finalized by iterating Steps ①–④.

3. Bidirectional diffusion experiments under variable-flow gas injection for CH₄ replacement conditions with flow parameters designed as shown in

Table 3. Experimental parameters for bidirectional diffusion experiments.

Number	Injected Gas	Adsorbed Gas	Gas Injection Method	Gas Injection Flow Rate/(mL/min)	Adsorption Equilibrium Pressure/MPa	Replacement Pressure/MPa
1	N2	CH4	Constant	$y=$ 200	0.5, 1.0, 1.5	0.5, 1.0, 1.5
			Sinusoidal-changed	$y=100\mathit{sin}{\displaystyle \frac{\pi }{30}}t+200$
			Step-changed	$y=\left\{\begin{array}{c}300, 0+60n\le t\le 30+60n\\ 100, 30+60n\le t\le 60+60n\end{array}(n=0,1,2,3\dots )\right.$
2	CO2	CH4	Constant	$y=$ 200	0.5, 1.0, 1.5	0.5, 1.0, 1.5
			Sinusoidal-changed	$y=100\mathit{sin}{\displaystyle \frac{\pi }{30}}t+200$
			Step-changed	$y=\left\{\begin{array}{c}300, 0+60n\le t\le 30+60n\\ 100, 30+60n\le t\le 60+60n\end{array}(n=0,1,2,3\dots )\right.$

Note: t is the injection time, min; y is the gas injection flow rate at moment t, mL/min.

.

Two cycles of variable-flow injection, sinusoidal-changed and step-changed, were implemented to study their effects on CH₄ diffusion behavior. The injection flow rate waveforms for the first cycle are illustrated in Figure 3. The right Y-axis of Figure 3 presents the programmed flowmeter control values for gas injection rate, while the left Y-axis shows the corresponding measured values. In each injection cycle, the injection time of the two injection methods was 3600 s, the injection volume was 12,000 mL, and the injected gas types were selected as N₂ and CO₂, respectively.

The experimental procedure was conducted as follows:

① Initial System Configuration

Consistent with unidirectional diffusion protocols, the experimental system underwent CH₄ pre-adsorption following standard evacuation procedures.

② Gas Injection Phase

The back pressure valve was adjusted to set the gas discharge pressure. Internal gas outflow from the coal sample canister occurs only when the pressure exceeds the back pressure valve setting, which more closely matches actual gas injection displacement engineering scenarios. Simultaneously, N₂/CO₂ was introduced by opening the inlet valve, with injection rates dynamically controlled via the inlet flowmeter program to maintain predetermined variable flow rates into the coal sample tank.

③ Experimental Data Acquisition

Gas flow sensors were installed at the inlet and outlet ends of the coal sample tank, respectively, to monitor the instantaneous flow rates of injected gas and discharged gas mixture in real time. Simultaneously, a gas concentration detection device at the outlet end monitors the gas composition and volume fractions of the discharged gas. Through the integration of these devices, we achieve real-time and precise monitoring of the composition, volume fractions, and instantaneous flow rates of the outlet gas mixture.

④ Data Acquisition and Termination Criteria

Gas pressure, effluent flow rate, and concentration data were tracked in real-time by the automated acquisition system. Experimental termination was triggered when the CH₄ concentration persistently remained below 1% for ≤30 consecutive minutes.

⑤ Experimental Replication

Bidirectional diffusion experiments were concluded by repeating the sequence (Steps ①–④).

3. Experimental Results

3.1. Unidirectional Diffusion Experiments with N₂, CO₂, CH₄

The results of unidirectional diffusion experiments under different adsorption equilibrium pressures are demonstrated in Figure 4. There are two stages. In the initial diffusion stage (0–6 min), equilibrium is disrupted as the pressure gradient decreases. Free gas rapidly flushes out, and small amounts of adsorbed gas desorb. In the second stage (6 min to the end), the gas pressure in the coal samples tank gradually decreases to 0. In this stage, the gas emission is mainly contributed by the desorption of the adsorbed state gas. Furthermore, at identical adsorption equilibrium pressures, the emission amount by unidirectional diffusion followed CO₂ > CH₄ > N₂. This relationship is determined by the coal’s adsorption capacity for each gas. At identical adsorption equilibrium pressures, more gas is adsorbed when the adsorption strength is greater. Consequently, compared to CH₄ and N₂, more CO₂ was ultimately released from the coal.

Table 4. The unidirectional diffusion experimental results.

Gas Type	Adsorption Equilibrium Pressure (MPa)	Gas Emission (mL/g)	Gas Desorption (mL/g)
CH4	0.5	2.45	1.35
	1.0	4.73	1.94
	1.5	7.44	3.47
CO2	0.5	2.85	/
	1.0	5.79
	1.5	8.84
N2	0.5	2.27	/
	1.0	4.43
	1.5	6.37

presents the unidirectional diffusion experimental results. It is evident that for a given gas, increasing the adsorption equilibrium pressure led to greater gas emission and desorption.

3.2. Bidirectional Diffusion Experiments with Constant Flow Gas Injection

3.2.1. Effect of Pre-Adsorbed Gas Equilibrium Pressure on CH₄ Diffusion During Constant Flow Injection

Using constant flow N₂ injection for CH₄ replacement as an example, Figure 5 demonstrates the trend of CH₄ emission and desorption with time under different adsorption equilibrium pressure conditions. At adsorption equilibrium pressures of 0.5 MPa, 1.0 MPa, and 1.5 MPa, the cumulative CH₄ emissions were 3.56 mL/g, 5.61 mL/g, and 9.06 mL/g, respectively, and the cumulative CH₄ desorption was 2.20 mL/g, 2.87 mL/g, and 4.98 mL/g, respectively. It is evident that higher adsorption equilibrium pressures lead to increased CH₄ emissions and desorption over the same time period. In comparison to unidirectional CH₄ diffusion, bidirectional diffusion processes resulted in a respective increase of 45.96%, 18.53%, and 21.71% in CH₄ emissions at each pre-adsorption gas pressure level. Similarly, CH₄ desorption exhibited increases of 163.69%, 188.82%, and 161.12% in the same order. The injection of N₂ not only enhances the removal of free CH₄ but also effectively promotes the desorption of CH₄ in the adsorbed state.

3.2.2. Effect of Different Injection Gas Types on the Diffusion of CH₄

Figure 6 demonstrates the trend of CH₄ emission with time for N₂/CO₂ injection under different adsorption equilibrium pressure conditions. It can be found that the CH₄ diffusion pattern is the same when CO₂ is injected at constant flow and N₂ is injected at constant flow, but the cumulative CH₄ emission and desorption are significantly different. In order to further reflect the differences in the bi-directional diffusion patterns of the different gas injection types to drive CH₄, we compared the final CH₄ emissions and desorption, as shown in Figure 7. Under varying adsorption equilibrium pressures, the cumulative emissions of CH₄ during CO₂ injection were as follows: 4.09 mL/g (0.5 MPa), 6.72 mL/g (1.0 MPa), and 9.67 mL/g (1.5 MPa), with corresponding desorption volumes of 2.49 mL/g (0.5 MPa), 3.77 mL/g (1.0 MPa), and 5.53 mL/g (1.5 MPa). At the same adsorption equilibrium pressure, the emissions of CH₄ replacement by CO₂ injection were 1.15, 1.20, and 1.07 times higher than those of N₂ injection, while the desorption volumes were 1.13, 1.31, and 1.11 times higher than those of N₂ injection. Hence, when subjected to identical injection pressure and injection time, the introduction of CO₂ into the coal seam results in a greater displacement of CH₄. Figure 7 also illustrates a comparison of CH₄ diffusion results between unidirectional and bidirectional diffusion. It can be found that whether N₂ or CO₂ is injected into the coal body, the cumulative CH₄ emission and desorption are significantly higher than the results of unidirectional diffusion of CH₄, which indicates that the gas injection replacement has a good effect of “pro-emission” of CH₄.

3.3. CH₄ Diffusion Experiments for Variable-Flow Gas Injection

3.3.1. Effect of Gas Injection Types and Injection Modes on CH₄ Diffusion

Figure 8 illustrates the effect of various injection gases and injection modes on the bidirectional diffusion of CH₄ under identical conditions at each adsorption equilibrium pressure. Figure 9 and Figure 10 present comparisons of CH₄ emissions and desorption across three gas injection types: sinusoidal-changed, step-changed, and constant flow. It is evident that CH₄ emissions and desorption exhibit an initial rapid increase over time, followed by a gradual attainment of equilibrium. This behavior aligns with the pattern observed during constant-flow gas injection to facilitate CH₄ diffusion. However, cumulative CH₄ emissions and desorption levels vary under different injection conditions.

In the context of the three injection methods, when CO₂ is used as the injected gas, both CH₄ emission and desorption are greater compared to when N₂ is employed. Furthermore, this difference becomes more pronounced as the pressure level increases. This phenomenon primarily arises from coal’s superior adsorption affinity for CO₂ over N₂. Free CH₄ was flushed out during N₂ injection, and the desorbed CH₄ was mainly contributed by the reduction of CH₄ partial pressure after N₂ injection, and only a small portion of the desorbed CH₄ was due to the competitive adsorption between N₂ and CH₄. However, after injecting CO₂, the dual effects of lowering CH₄ partial pressure and competitive adsorption between CO₂ and CH₄ enabled more CH₄ to be desorbed from the coal body compared with N₂ injection. Furthermore, at the same gas adsorption equilibrium pressure level for both N₂ and CO₂ injection, the extent of CH₄ emission and desorption followed this sequence: step-changed > sinusoidal-changed > constant flow, with the variable-flow gas injection method further enhancing CH₄ emission from coal. When CO₂ was employed as the injected gas, CH₄ emissions and desorption under the conditions of step-changed flow injection were significantly higher than those of other injection methods at each pressure level. In contrast, when N₂ served as the injected gas, the disparities in CH₄ emissions and desorption between sinusoidal-changed and step-changed injection were minimal at pressure levels of 0.5 MPa and 1.0 MPa. However, at a pressure level of 1.5 MPa, CH₄ emissions and desorption were notably higher in the step-changed injection method as opposed to the sinusoidal-changed injection method.

3.3.2. Analysis of the Effect of CH₄ Replacement by the Principle of Equal-Time and Equal-Quantity Gas Injection

After five cycles, the CH₄ emissions and desorption, following all three gas injection methods, exceeded 90%, while the CH₄ concentration dropped below 1%. Furthermore, as each group exhibited consistent experimental phenomena, we selected the experimental data from the initial five injection cycles, where the pre-adsorbed gas equilibrium pressure was maintained at 1.5 MPa, for analyzing the displacement effect.

The emissions of CH₄ can be divided into two components: the original free CH₄ flushed out and the adsorbed CH₄ desorbed. As illustrated in Figure 11, irrespective of whether N₂ or CO₂ is injected, and across the three injection modes, the CH₄ emissions during the five injection cycles follow this order: step-changed > sinusoidal-changed > constant flow. Similarly, the CH₄ desorption was ranked as follows: step-changed > sinusoidal-changed > constant flow. CH₄ emissions can be categorized into three stages. In Stage I, during the first cycle, CH₄ emissions significantly exceed those in subsequent cycles. Early in the cycle, a substantial portion of free-state CH₄ is flushed out, while in the later part, CH₄ adsorbed within the coal matrix pores desorbs, leading to an increasing desorption percentage. In Stage 2, in the second cycle, the decline in CH₄ emissions slows down, further reducing initial free-state CH₄, with a rapid increase in CH₄ desorption at this point. Stage 3 encompasses the third, fourth, and fifth cycles, where the decrease in CH₄ emissions levels off. By the fifth cycle, the ratio of CH₄ desorption to CH₄ emissions reaches approximately 90%. As indicated by Figure 11a, the ratio of N₂/CO₂ injection to CH₄ emission in each cycle exhibits significant divergence after the second cycle. The consumption order of injected gas for each volume of CH₄ emitted is as follows: constant flow > sinusoidal-changed > step-changed. This suggests that the variable-flow injection method offers a higher utilization rate of the injected gas and a more effective replacement effect. Notably, under each gas injection method, except for the first cycle, the quantity of injected CO₂ surpasses that of injected N₂ for the same volume of discharged CH₄. This phenomenon results from gradual coal particle adsorption of CO₂ over time, leading to an increase in the consumed CO₂ volume.

Figure 11b illustrates the order of CH₄ desorption as a percentage of CH₄ emission in each cycle under various gas injection methods, ranked from highest to lowest as follows: step-changed > sinusoidal-changed > constant flow. When N₂ is used as the injected gas, the percentage of CH₄ desorption exhibits a rapid increase from the first to the third cycle, reaching 87.69% (step-changed) and 80.86% (sinusoidal-changed) in the third cycle. Subsequently, it gradually rises from the fourth to the fifth cycle, peaking at 95.15% (step-changed) and 93.50% (sinusoidal-changed). Conversely, under constant flow conditions, the proportion of CH₄ desorption initially increases slowly from the first to the second cycle, experiences a rapid rise in the third to fourth cycle, yet consistently remains lower than that observed for step and sinusoidal variable flow, culminating at 91.47% in the fifth cycle. When CO₂ is utilized as the gas source, the trend of CH₄ desorption is consistent with that observed with N₂, with the notable difference being a halt in the rapid growth trend of CH₄ desorption in the fourth cycle, reaching percentages of 94.45% (step-changed) and 94.21% (sinusoidal-changed). Ultimately, the final CH₄ desorption percentages for all three gas injection methods stand at 97.74% (step-changed), 94.97% (sinusoidal-changed), and 93.49% (constant flow) when compared to N₂ injection. It is noteworthy that the differences in CH₄ desorption between gas injection methods, whether using N₂ or CO₂, decrease as the duration of gas injection progresses.

4. Analysis of the Applicability of the Gas Diffusion Model in Coal

4.1. Model of Gas Diffusion in Coal Particles

The model of coal particle gas diffusion serves as the foundation for investigating the laws of coal particle gas diffusion. Currently, the established model of coal particle gas diffusion primarily relies on Fick’s diffusion law. The conventional diffusion model categorizes the pores in coal as having unipore, bidisperse, or triple-pore structures. In this study, we selected the commonly employed unipore and bidisperse models for fitting experimental data, aiming to assess the suitability of the diffusion model for CH4 diffusion during gas injection in bidirectional diffusion processes.

4.1.1. Unipore Diffusion Model

The unipore model assumes that coal particles are spherical particles with an isotropic radius, r, and that the gas concentration surrounding these spherical coal particles remains constant. The derivation process is outlined in the existing literature [43], and the expression for the mass fraction of adsorbed/desorbed gas is as follows:

$${\displaystyle \frac{m_t}{m_{\infty }}}=1-{\displaystyle \frac{6}{{\pi }^2}}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}\exp \left(-n^2{\pi }^{2}t\frac{D}{r^2}\right)}$$

(2)

where m_t is the total amount of gas adsorption/desorption at time t (g); m_∞ is the total amount of gas adsorption/desorption after reaching equilibrium (g); r is the radius of spherical coal particles (m); D is the diffusion coefficient (m²/s); t is desorption/adsorption time (s); and the value of D/r² can be considered as the effective diffusion coefficient, D_e (1/s). In 1975, Crank [55] et al. provided calculations for a unipore model applicable to external concentration variations, considering the limitations of the assumptions of the unipore model. Coal possesses a non-uniform pore structure, and this inherent characteristic presents challenges in applying a single pore diffusion model; however, certain experiments [56,57,58] have demonstrated the applicability of this model in achieving a superior fit with diffusion data.

4.1.2. Bidisperse Diffusion Model

In contrast to the unipore model, which assumes the existence of a single pore type within the coal matrix, the bidisperse model postulates the presence of two distinct pore structures within the coal matrix: macropores and micropores, providing a more comprehensive depiction of gas diffusion within coal. This bidisperse model has found extensive application in predicting CBM production, resource assessment, etc. The model segregates the gas diffusion process within coal into two categories: macropore diffusion, characterized by high speed, and micropore diffusion, characterized by low speed.

For the macropore diffusion stage, the diffusion model can be expressed as follows:

$${\displaystyle \frac{m_a}{m_{a\infty }}}=1-{\displaystyle \frac{6}{{\pi }^2}}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}\exp \left(-n^2{\pi }^{2}t\frac{D_a}{{r_a}^2}\right)}$$

(3)

For the micropore diffusion stage, the diffusion model is then expressed as follows:

$${\displaystyle \frac{m_i}{m_{i\infty }}}=1-{\displaystyle \frac{6}{{\pi }^2}}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}\exp \left(-n^2{\pi }^{2}t\frac{D_i}{{r_i}^2}\right)}$$

(4)

In the above Equations (3) and (4), a and i denote the diffusion parameters corresponding to macropores and micropores, respectively; D_a/r_a² and D_i/r_i² can be written as the effective diffusion coefficients D_ae and D_ie (1/s), respectively.

Then the total overall adsorption/desorption of coal particles is the sum of the above two stages, which can be expressed by Equation (5):

$${\displaystyle \frac{m_t}{m_{\infty }}}={\displaystyle \frac{m_a+m_i}{m_{a\infty }+m_{i\infty }}}=\beta {\displaystyle \frac{m_a}{m_{a\infty }}}+\left(1-\beta \right){\displaystyle \frac{m_i}{m_{i\infty }}}$$

(5)

where $\beta ={\displaystyle \frac{m_{a\infty }}{m_{a\infty }+m_{i\infty }}}$ is the percentage of gas adsorbed/desorbed by the macropores to the total pore adsorption/desorption.

4.2. Diffusion Model Applicability Analysis

In this paper, the D_e in the unipore model and the parameters D_ae, D_ie, and β in the bidisperse model were individually adjusted to fit the diffusion behavior of CH₄ within the coal during experimentation. The value of n in Equations (2)–(4) was set to 5 in order to obtain the optimal fit for the diffusion parameters based on the experimental m_t/m_∞ curves and, thus, to carry out the analysis of the suitability of diffusion models. The optimal fitting parameters for each condition can be found in

Table 5. The optimal fitting parameters for each condition.

Type of Gas Injection	Adsorption Equilibrium Pressure/(MPa)	Unipore Model	Bidisperse Model
		De/(m2/s)	${\mathit{D}}_{\mathit{a}\mathit{e}}$/(m2/s)	${\mathit{D}}_{\mathit{i}\mathit{e}}$
CH4 unidirectional diffusion	0.5	8.39 × 10−10	2.41 × 10−10	3.27 × 10−11
	1.0	7.21 × 10−10	1.53 × 10−10	2.41 × 10−11
	1.5	8.07 × 10−10	9.32 × 10−11	1.94 × 10−11
Constant flow injection of N2 to replace CH4	0.5	3.78 × 10−10	6.17 × 10−11	1.47 × 10−11
	1.0	8.21 × 10−10	7.59 × 10−11	1.62 × 10−11
	1.5	7.06 × 10−10	4.03 × 10−11	1.22 × 10−11
Constant flow injection of CO2 to replace CH4	0.5	1.49 × 10−10	4.59 × 10−11	1.41 × 10−11
	1.0	1.66 × 10−10	2.69 × 10−11	1.21 × 10−11
	1.5	0.97 × 10−11	2.62 × 10−11	1.07 × 10−11
Sinusoidal-changed flow injection of N2 to replace CH4	0.5	4.95 × 10−11	1.08 × 10−11	2.31 × 10−12
	1.0	2.23 × 10−11	7.25 × 10−12	7.64 × 10−13
	1.5	1.92 × 10−11	5.25 × 10−12	6.14 × 10−13
Sinusoidal-changed flow injection of CO2 to replace CH4	0.5	7.81 × 10−11	1.23 × 10−11	2.11 × 10−12
	1.0	5.19 × 10−11	1.16 × 10−11	8.06 × 10−13
	1.5	3.55 × 10−11	9.96 × 10−12	6.00 × 10−13
Step-changed flow injection of N2 to replace CH4	0.5	5.98 × 10−11	1.28 × 10−11	3.11 × 10−12
	1.0	2.79 × 10−11	7.62 × 10−12	7.53 × 10−13
	1.5	2.16 × 10−11	6.86 × 10−12	6.44 × 10−13
Step-changed flow injection of CO2 to replace CH4	0.5	8.66 × 10−11	1.52 × 10−11	2.59 × 10−12
	1.0	5.54 × 10−11	1.26 × 10−11	7.56 × 10−13
	1.5	3.86 × 10−11	8.02 × 10−12	6.16 × 10−13

.

The applicability of unipore and bidisperse models for both unidirectional and bidirectional gas diffusion is separately analyzed. At each pre-adsorption pressure level, these two diffusion models exhibit a high degree of similarity in their respective simulation outcomes, whether for unidirectional or bidirectional diffusion. Therefore, this section presents a comparison between the simulated and experimental diffusion model curves at a pressure level of 0.5 MPa, as depicted in Figure 12. It is evident that, concerning the experimentally measured m_t/m_∞ curves, the simulated curves of the bidisperse model closely align with the measured curves, outperforming the unipore model. This superiority arises from the fact that, in contrast to the simplified pore structure assumption in the unipore model, the bidisperse model accounts for both macropores and micropores, offering a more accurate representation of the complex pore structure in the coal samples. Additionally, its corresponding diffusion coefficients for macropore and micropore pores provide a more comprehensive explanation of gas diffusion processes within the coal. Furthermore, the bidisperse model demonstrates a significantly enhanced simulation performance for CH₄ and CO₂ compared to N₂ in both unidirectional and bidirectional diffusion scenarios. This is due to the utilization of the bidisperse model, which considers gas adsorption within the coal body, enabling a more precise simulation of the transport behavior of easily adsorbed gases, such as CO₂ and CH₄, in the coal seam.

4.3. Effect of Different Gas Injection Methods on Diffusion Coefficients

It is well-established that the bidisperse model can better describe the CH₄ diffusion-desorption ratio in each experimental process. In this section, we carried out the analysis of the trend of D_ae, the effective macropore diffusion coefficient, and D_ie, the effective micropore diffusion coefficient, under varying pressure conditions for each method of gas injection, as shown in Figure 13. At three different pressure levels, when maintaining a constant flow displacement, the effective diffusion coefficients (D_ae and D_ie) of CH₄ exhibited a trend of initially increasing and subsequently decreasing as the injected gas source was N₂. Conversely, when CO₂ served as the injected gas source, the effective diffusion coefficients (D_ae and D_ie) of CH₄ demonstrated a consistent tendency to decrease with increasing pressure. When comparing the D_ae and D_ie values of CH₄ diffusion during N₂ and CO₂ injection, it is evident that the D_ae and D_ie values are greater in the case of N₂ injection compared to CO₂. In the context of the variable-flow injection method, it is observed that the diffusion coefficients (D_ae and D_ie) of CH₄ exhibit a negative correlation with gas pressure when both N₂ and CO₂ gases are injected. Unlike the constant flow gas injection method, the effective macropore diffusion coefficient (D_ae) of CH₄ during CO₂ injection notably exceeded that observed during N₂ injection. Additionally, the effective micropore diffusion coefficient (D_ie) exhibited a similar trend, surpassing that seen in N₂ injection scenarios when CO₂ was employed as the injected gas source at a pressure level of 0.5 MPa. Notably, D_ie remained nearly identical for CH₄ in both injection methods, i.e., at 1.0 and 1.5 MPa, underlining its consistent behavior.

Moreover, the unidirectional effective diffusion coefficients, D_ae and D_ie, for CH₄ were greater than those for bidirectional diffusion under the same pressure conditions. This suggests that the adsorption of injected N₂/CO₂ into coal particles reduces the desorption rate of pre-adsorbed CH₄ from coal particles, but Section 3 provides evidence supporting the conclusion that the “pro-emission” effect of gas injection and displacement can eliminate the influence of this inhibitory effect.

4.4. Limitations of Unipore and Bidisperse Diffusion Models

Section 4.3 reveals that although the bidisperse diffusion model provides a better fit for CH₄ bidirectional diffusion under gas injection than the unipore diffusion model, its predictions still deviate substantially from experimental CH₄ diffusion data during displacement. Specifically, at the initial phase, the unipore diffusion yields lower values than experimental measurements, whereas the bidisperse diffusion model produces higher values; at a later phase, the unipore diffusion model yields higher values, while the bidisperse diffusion model generates lower values relative to experimental data.

This paradox indicates that conventional unipore and bidisperse diffusion models inadequately describe CH₄ diffusion during gas injection displacement, primarily due to the following:

① Inherent limitation: Both models assume single-gas diffusion, thus failing to capture multi-component interactions during displacement.

② Homogeneity fallacy: Idealized assumptions of uniform coal/porosity [43] contradict natural heterogeneity.

③ Non-Fickian anomalies: Prior studies demonstrate that gas adsorption-induced coal swelling triggers matrix deformation, resulting in non-Fickian anomalous diffusion behavior [59,60,61], which existing single-/dual-pore models fundamentally fail to account for.

④ Dynamic diffusion coefficients: During gas injection displacement, the progressive intrusion of foreign gases into deep-seated pores within the coal matrix causes increasingly complex CH₄ diffusion pathways. This results in continuously variable diffusion coefficients of CH₄, which cannot be represented by one or several constant values—a phenomenon preliminarily interpreted in Section 4.3.

Consequently, novel diffusion models must be developed to characterize gas bidirectional behavior in displacement scenarios.

5. Discussion

Laboratory experiments have proved that compared with unidirectional diffusion and constant-flow gas injection for CH₄ replacement, variable-flow gas injection can enhance CH₄ emission and desorption and has a better effect of “pro-emission” of CH₄, which is a revelation for the application of variable-flow gas injection in the field of underground gas injection and replacement in coal mines:

(1) It can be inferred from Section 3.3.1 that, when employing variable-flow N₂ injection for replacement, and with adsorption equilibrium pressures at 0.5 MPa and 1.0 MPa, there is a small disparity in CH₄ emission and desorption between the step-changed flow and sinusoidal-changed flow injection methods. The distinction in CH₄ emissions between the two injection methods begins to manifest when the adsorption equilibrium pressure reaches 1.5 MPa. Conversely, under CO₂ injection conditions, CH₄ emission and desorption in the step-changed injection method exceed those in the sinusoidal-changed flow injection method across all pressure levels. These findings suggest that the step-changed flow N₂ injection technology is suited for coal seams characterized by high initial gas pressures or when high-pressure injection (≥1.5 MPa) is employed.

(2) Combined changes in CH₄ emissions, desorption, and desorption ratios in Section 3.3.2 reveal that, in scenarios with limited gas sources, step-changed flow and sinusoidal-changed flow gas injection methods can effectively discharge more CH₄ and enhance the desorption of adsorbed CH₄. Furthermore, when CO₂ is chosen as the gas source, these methods outperform their counterparts using N₂ as the gas source. However, in situations where the gas source is abundant or substantial, the advantages associated with variable-flow gas injection tend to diminish. Consequently, to fully harness the potential of variable-flow methods for pro-emission, reasonable consideration of the gas injection type and duration is imperative. Moreover, except for the first cycle, where the discharge of the same volume of CH₄ is observed, it is noteworthy that the quantity of CO₂ injected surpasses that of the injected N₂ gas. This phenomenon can be attributed to the gradual adsorption of CO₂ by coal particles over time, resulting in an increased consumption of CO₂. Furthermore, our findings indicate that employing a CO₂ injection for CH₄ replacement is more appropriate for enhancing coalbed methane production from surface wells within unrecoverable coal seams while simultaneously sequestering CO₂. Conversely, N₂ injection for CH₄ replacement is deemed suitable for the prevention and control of gas-related disasters in underground recoverable coal seams.

(3) As evidenced by Figure 4, Figure 5, Figure 6 and Figure 13, and Table 5, the diffusion coefficient of CH₄ under bidirectional diffusion conditions during gas injection displacement is lower than that observed in unidirectional CH₄ diffusion. The injection of foreign gases results in a reduction in the desorption rate of CH₄. This phenomenon arises primarily from two mechanisms: (i) Injected gases obstruct diffusion pathways within the coal matrix pores, increasing path tortuosity and thereby reducing the effective diffusion coefficient of CH₄. (ii) Gas injection pressure induces pore constriction, heightening diffusion resistance to CH₄. However, under sustained operation, while the CH₄ desorption rate decreases transiently during displacement (Figure 4, Figure 5 and Figure 6), maintaining system pressure over a sufficient duration enables competitive displacement of deeply adsorbed CH₄. This ultimately enhances cumulative CH₄ desorption significantly (Figure 7).

6. Conclusions

Laboratory experiments were conducted to investigate CH₄ diffusion patterns under two distinct gas injection modes: constant-flow and variable-flow displacement scenarios. Comparative analysis was performed between unidirectional and bidirectional diffusion processes to characterize CH₄ diffusion behavior. Under controlled experimental conditions, maintaining equivalent gas injection duration and volume, displacement efficiencies were quantitatively evaluated across both injection modes. Experimental data were subsequently applied to validate the CH₄ diffusion model’s accuracy under variable-flow conditions in coal matrices. The conclusions are as follows:

(1) Bidirectional diffusion enhanced by gas injection displacement demonstrates superior CH₄ liberation efficiency over unidirectional diffusion at equivalent pressure levels. The injected N₂/CO₂ gas can promote the adsorption and desorption of CH₄ in coal, and the larger the pre-adsorption equilibrium pressure, the larger the emission and desorption of CH₄. CO₂ exhibits superior displacement efficacy compared to N₂, achieving enhanced CH₄ recovery through the synergistic action of partial pressure reduction and competitive adsorption mechanisms.

(2) Under identical gas injection parameters (type and pressure), variable-flow displacement achieves superior CH₄ liberation compared to constant-flow operation, with ranking as step-variable > sinusoidal-variable > constant-flow regimes. When the gas injection source is CO₂, the CH₄ emission and desorption at the outlet are larger than that of N₂, and the higher the pressure level, the more obvious the difference.

(3) Analysis of equal time and quantity principles reveals the following CH₄ desorption-emission order under identical pressure and injection types: stepped-change > sinusoidal-variation > constant flow. However, cyclic progression shows three key trends: the CH₄ emission decline rate decreases, the desorption ratio accelerates, and the inter-method desorption differences diminish. From the second cycle onward, gas injection-to-CH₄ emission ratios demonstrate consumption efficiency as: constant flow > sinusoidal-variation > stepped-change, confirming variable-flow methods’ superior displacement effects. Additionally, CO₂ consistently exhibits higher consumption per CH₄ emission volume than N₂ across all cycles except the first.

(4) The bidisperse diffusion model demonstrates superior fitting accuracy to the unipore model for CH4 desorption ratios under both constant-flow and variable-flow displacement. For bidirectional diffusion of constant flow displacement, the effective diffusion coefficients D_ae and D_ie of CH₄ first increase and then decrease with the increase in pressure when N₂ is injected, while the effective diffusion coefficients D_ae and D_ie of CH₄ decrease with the increase in pressure when CO₂ is injected. For bidirectional diffusion with variable-flow, whether N₂ or CO₂ is injected, the D_ae and D_ie of CH₄ are negatively correlated with the gas pressure. Notably, CO₂ injection produces significantly higher macropore D_ae values than N₂ across pressure levels. For micropore D_ie at 0.5 MPa, CO₂ injection yields greater CH₄ diffusion coefficients compared to N₂, while values converge at 1.0 and 1.5 MPa pressures.