A permeability evolution model of coal particle from the perspective of adsorption deformation

As an important indicator, permeability can predict the gas drainage yield and prevent the mine gas disasters. We first reviewed our previous inversion method to investigate permeability coefficient of gas in coal particles; however, the relationship between permeability and adsorption pressure had not been summarized and explained theoretically. Here, a permeability evolution model including two crucial parameters of initial permeability and deformation coefficient was developed. The inversion gas permeability coefficients were converted into permeability, and then, the permeability of the same coal sample was fitted according to the evolution model. The results show that (i) the modeled results are matched reasonably well with the inversion permeability data, and thus, this model has been validated; and (ii) as volatile matter content increases, the initial permeability decreases exponentially, but the deformation coefficient basically grows in a linear trend. These two parameters are coupling to lead to a negative exponential decrease in permeability as gas pressure rises.

gas pressure and the effective stress, which submits to the law of effective stress. [11][12][13][14] The relationship between the gas pressure and the adsorption-induced strain can be described by the Langmuir-like equation. 15,16 The coal that adsorbs methane can be deformed, in which the adsorption-induced strain is approximately proportional to the gas adsorption capacity. 17,18 Therefore, gas pressure has a profound impact on permeability evolution. The permeability decreases first and then increases as the gas pressure rises under constant confining stress, 19 and the anisotropic permeability evolution model, Wang et al proposed in 2014, also confirmed this rule. 20 Pini et al had tested the turning point of permeability rebound at about 2 MPa. 21 Meng et al believed the coal matrix swelling had played a prevailing role in the permeability variation that is similar to roller coasters. 22 On the other hand, under constant effective stress conditions, the permeability should significantly decrease when increasing the gas pressure. 23, 24 Harpalani and Chen reported that, as pore pressure decreased from 6.2 MPa to 0.7 MPa, the permeability of coal sample increased by 17 times. 25 The macerals and pore-fracture systems of coal with different ranks are different, which is closely related to the change in coal strain and effective stress caused by matrix shrinkage. Li et al studied the dynamic process of adsorbed gas flow in coal of different ranks and pointed out that the permeability change rate was greatly affected by the maceral composition and initial permeability of coal. 26 Meng et al believe that under a certain effective stress, the permeability of low-rank coal is higher than that of high-rank coal, and the influence of methane adsorption-induced expansion strain on the permeability of high-rank coal is greater than that of low-rank coal. 27 The models mentioned above are all established for coal seams or cylindrical coal samples, 2 but the permeation behaviors of methane in coal particles are studied so little that the applicability of these models on coal particles needs further proof. In our previous work, 5 we proposed a methane adsorption model on the basis of Darcy's law and then solved by our self-developed software. Permeability coefficient was obtained through fitting simulation data with the adsorption experimental data. The calculative results show that the permeability coefficients have a close relationship with the adsorption pressures. However, this relationship failed to be explained or analyzed theoretically in the form of function in our last article. Thus, the main objective of this study was to address the evolution model of permeability of coal particle under the impact of adsorption pressure.
In this work, a permeability evolution model of coal particle was developed to investigate the permeability of coal particle in adsorption process. This model should be verified by matching the model's calculation values with our pervious inversion permeability data. Furthermore, the key factors of coal rank were quantitatively evaluated on how to influence the new evolution model.

| Effect of gas pressure on porosity
Coal swelling due to adsorbing methane gas is a well-known phenomenon, and it controls the permeability by changing the porosity. 28 Most permeability models utilize the empirical Langmuir-fit technique to deal with the effect of adsorption strain on porosity, 29 but these empirical models do not explain the principle of adsorption strain fundamentally. Therefore, the interaction between gas and coal is the key to more accurately describe the swelling strain induced by gas adsorption. 30 Wu considered that coal matrix expansion in adsorption process resulted from the decrease in surface pressure of pore in the coal. 31 The attractions between the surface molecules of coal matrix and the inside molecules decrease after the coal adsorbed methane gas, which leads coal matrix to swell. Wei et al held the same opinion. 32 According to this assumption, the reduction in surface pressure of coal induced by the adsorption can be expressed by the Gibbs adsorption equation as follows 29 : where ∆π is the variation of surface pressure from vacuum to the adsorbed state, N/m; Q is the gas adsorption capacity, m 3 /t; R m is the gas constant, 8.314 J/(mol·K); T is the temperature of coal, K; V m is the molar volume of gas, L/mol; S is the specific surface area of coal, m 2 /g; and p is the gas pressure, MPa.
The two sides of Equation (1) are integrated, and the following equation is obtained.
The gas adsorption of coal matrix satisfies the Langmuir equation.
where a and b are Langmuir adsorption constants, respectively, m 3 /t, MPa -1 .
Coal is assumed as an isotropic continuous elastic body with the same adsorbed deformation in all directions, which stays under a fully ideally constrained condition (ie, the work done by the surface pressure of coal is completely converted into the elastic energy). Accordingly, Wu had proposed an adsorption swelling model of coal matrix, 31 as follows: where ε v is the adsorption-induced strain; ρ v is the apparent density of coal, t/m 3 ; and K j is the elastic modulus of coal particles, MPa. At present, K j of coal cylinder is measured through macroscopic uniaxial compression experiment, and the sample size is usually around 50mm. There is still a big difference between the coal grains and the coal cylinder, and the size of the coal grains is very small. Although the K j of coal particles is real, it is still unrealistic to measure its specific value according to the current experimental technical means.
Effective stress refers to the force that causes the deformation of porous media. Its essence is that the porous media produces a certain degree of strain, resulting in the change of pore volume and porosity. Here, we focused on the adsorption-induced strain generated by effective stress to explore the expression of gas pressure and porosity. Put the Equation (4) into Equation (5) to get. Furthermore, Wu 31 considered that 1/3 of the total swelling strain was converted into the swelling stress at the contact interface in the adsorption process, and the other 2/3 was the inward swelling strain that reduced the volume of the fracture. The inward swelling strain (ε p ) can be calculated as.
The inward swelling strain has much greater magnitude than bulk strain, 23 and thus, it can be assumed that the coaladsorbing gas only causes inward swelling strain, but no change in the bulk strain of coal. Considering the impact of adsorption-induced strain on porosity of coal, the porosity model can be expressed as: where φ is the porosity of coal, %; V φ0 and V φ are the volume of pore fracture before and after deformation of coal, respectively, m 3 ; ΔV p is the adsorption-induced swelling volume, m 3 ; V v0 , and V v are the appearance volume before and after the deformation, respectively, m 3 ; and φ 0 is the initial porosity of coal, %.
By simultaneous Equations (7) and (8), the model of porosity affected by gas pressure can be established as

| Effect of gas pressure on permeability
Permeability is a basic parameter for evaluating the seepage characteristics of coalbed methane, which is directly related to the porosity. Current studies have focused on the effect of external factors on the permeability, such as in situ stress and pore pressure, 2 while ignoring the influences from the internal factors. However, the proposed inward swelling of coal matrix should overcome this defect and further rationalized the coupling relationship between permeability and porosity.
Generally, the relationship between permeability and porosity can be expressed by the cubic law as 6 where k is the permeability of coal, mD, mD = 10 -15 m 2 ; and k 0 is the initial permeability of coal at a reference pressure, mD.
By substituting the Equation (9) into Equation (10), the following is obtained.
Introduce a new parameter A as.
In above equation, A is defined as the deformation coefficient of the inward swelling strain of coal matrix, which is dimensionless and related to the coal properties, temperature, and the saturated adsorption capacity.
Then the Equation (11) can be simplified as The above equation is the permeability evolution model of coal particle that is affected by gas pressure and can be used to estimate the permeability's variation in adsorption process.

PERMEABILIT Y COEFFICIENTS
Coal body is a typical dual-porosity structure, which is composed of coal matrix and fracture. 33 Guo et al 34 believed that the limit of particle size is the same as the size of coal matrix block, and they pointed out that although the large coal block looks complete, it can be regarded as the aggregation of the small coal matrix/particles. Thus, many researchers proposed the theory that gas flow in coal flows from coal matrix (low permeability) to fractures (high permeability). 2,35,36 Previous studies have proposed a number of testing methods to examine the permeability of coal. Li et al had used geophysical logging data to predict permeability of coal seams of Qinshui coalfield. 37 Wang et al had measured the permeability of cylindrical coal samples through steady-state seepage test. 4 However, few involve the permeation behavior of methane in coal particles. Difficulties come from two aspects: (i) The particle sizes of coal particles are too small to measure their permeability through usual steady-state methods; (ii) pores play a leading role in controlling the permeability of small coal particles, but many models are designed for seepage in fractures. Therefore, our last article 38 had proposed a practical inversion method investigating gas permeability of coal particles, which is an unsteady-state test method.

| Coal sample properties
In our last article about the permeability inversion, 38 the experiments had been conducted on six coal samples including lignite, medium volatile bituminous, low volatile bituminous, and anthracite, and their basic properties are shown in Table 1.

| Quasi-constant pressure adsorption experiment
In previous work, we conducted a quasi-constant pressure isothermal adsorption experiment on the above six coal samples. 5 The experimental apparatus is shown in Figure 1. Open the valves V1, V3, V4, and V5, close the remaining valves, and fill the sample tube and reference tube with helium to test the air tightness of the experimental device. It is important to note that this work needs to be repeated before each set of experiments is carried out. Pour 6 g of coal particles (180μm~250μm) into the sample tube and dry them for 4 h. The volume of reference tube and the free volume of sample tube were measured subsequently. After the sample tube was vacuum-pumped, the adsorption test began. The coal samples continuously adsorbed methane gas throughout the experiment at 35°C, but the sample tube was intermittently filled with methane so that the gas pressure was maintained near the set value. Take the designated adsorption pressure of 0.5 MPa as an example, when gas pressure dropped by 1%, the sample tube was then connected to the reference tube by V5 valve to make its pressure recover to 0.5 MPa. Repeatedly inflate the sample tube to reach the constant pressure value of the sample tube. The pressure data were constantly collected to aggregate gas reduction. In this way, the curve of cumulative methane adsorption under one constant pressure can be obtained. This experimental approach has also been demonstrated in our previously published work. 5

| Methane adsorption model of coal particle and solving
Our previous work had proposed a methane adsorption model of coal particle based on Darcy's law as 38 where λ is the permeability coefficient of gas, m 2 /(MPa 2 ·s); P is the square of pressure of methane, P = p 2 , MPa 2 ; B is the free gas coefficient, m 3 /(t·MPa); ρ is the apparent density of coal, t/ m 3 ; and r is the radius of coal particle, m.

The initial and boundary conditions can be expressed as below:
where p w is the methane pressure outside the coal particle, MPa. The above adsorption model was transformed into a dimensionless equation in our previous article 38 and then was discretized through finite difference method (FDM). Subsequently, a solution software system was independently developed to perform iterative calculations on the dimensionless FDM mathematic equation.

| Inversion and results
In the inversion process, a methane adsorption experiment under quasi-constant pressure with specified coal particle size was conducted under a given pressure, and then, a permeability coefficient should be assumed to simulate the cumulative adsorption curve under the designated pressure. The gas permeability coefficient was finalized based on selecting assumed values for the permeability coefficients, which can allow the simulated curve with the quasi-pressure adsorption data (as shown in Figure 2). This inversion method has been successfully applied to our previous work. 38 As can be seen from Figure 2, for the AZ sample, the curve of λ = 0.0000049 is optimal to coincide with the whole measured data when the adsorption pressure was at 2.0 MPa. Therefore, the gas permeability coefficient of AZ sample was identified at 0.0000049 m 2 / (MPa 2 ·s) under 2.0 MPa. Similarly, the gas permeability coefficient of SY#9 sample under 0.5 MPa was obtained at 0.0000093 m 2 / (MPa 2 ·s), and that of YQW samples under 1.0 MPa was obtained at 0.000092 m 2 / (MPa 2 ·s). In the same way, the inversion gas permeability coefficients of each coal particle had been obtained separately and are shown in Table 2.

| Calculation of permeability
Permeability k is a parameter representing the capacity of porous medium itself to transmit fluid, which is related to porosity, particle size, pore geometry in the direction of fluid permeating, and arrangement of solid skeleton particles. The k is different from the gas permeability coefficient λ. The latter is the unit flow under the unit hydraulic gradient, representing the difficulty level of fluid passing through the pore skeleton. 39 The conversion relationship between them will be discussed as below: Here, gas flow in coal particle is considered to be obeyed to Darcy's law as: where v is gas seepage velocity, m/s; and μ is the dynamic viscosity, MPa·s. Zhou had proposed a conception of specific flow rate of methane (q, unit: m 3 ·m -2 ·s -1 ); its equation is as follows 40 : where q stands for the specific flow rate, m 3 /(m 2 ·s).
The transfer equation between gas permeability coefficient and permeability can be expressed as where p n represents the atmospheric pressure under standard conditions, MPa.
The dynamic viscosity of methane is greatly affected by ambient temperature but is scarcely influenced by pressure. The dynamic viscosity of methane can be calculated as where μ 0 is the dynamic viscosity of methane under T 0 K, MPa·s; C is a constant that is related to gas species. 41 Here, μ 0 = 1.03 × 10 -5 Pa·s, p n = 0.101325 MPa, T 0 = 273.15 K, C = 162 K (methane), and it can be calculated that the μ = 1.1423 × 10 -11 MPa·s when T = 308.15 K. According to Equation (18), combined with Table 2, the results of the permeability of coal particle under different conditions can be obtained and are shown in Table 3. Figure 3 shows the fitting results of the permeability of coal particle according to the permeability evolution model. The regression coefficients are shown in Table 4. We found that: (i) there is a negative correlation between permeability and adsorption pressure and (ii) the correlation coefficients of fittings are very high. In Table 4, we can find that correlation coefficients of five coal samples are more than 0.99, and the rest one reaches 0.98. Thus, the correctness of Equation (13) has been verified; that is, our permeability evolution model of coal particle is validated.

| Results and analysis
The adsorption pressure has a critical influence on the permeability variation. The larger the adsorption pressure, the smaller the permeability of coal particle. At present, there are three viewpoints in the mainstream to explain this phenomenon: (i) Coal matrix expands with the increase in adsorption pressure, resulting in narrowing of gas flow channel 42,43 ; (ii) the increased adsorption pressure causes more free gas to be adsorbed gas, which may block the micropores, 38 and (iii) a potential factor for the permeability reduction in the process of pressure rising may be the weakening of the Klinkenberg effect. 44 However, the fitting results in Figure 3 indicate that the solid skeleton swell of coal matrix may be the main factor of the permeability reduction. In the adsorption process, the coal matrix swells to make its volume increases, resulting in decreasing of the inner pore volume of coal particle; thus, increasing gas pressure leads to the flow path in coal particle narrower and narrower so that the permeability continuously decreases.
Wang used the method of gas desorption measurement and numerical simulation to determine the permeability of granular coal. 45 He did not consider the effect of adsorption expansion in the numerical simulation of coal particle permeable flow. Our work takes into account the adsorptive expansion strain. Specifically, our permeability evolution model of coal particle Gas permeability coefficient (m 2 ·MPa -2 ·s -1 ) is developed based on the energy balance between the work done by surface pressure and the variation of elastic energy of coal matrix. This theoretical model has the ability to describe the mechanism of the swelling strain induced by gas adsorption, which is more accurate than the empirical model. This provides a predictive basis for calculating the permeability of coal particle. In addition, the evolution model involve so fewer parameters that it can be more transparent and easy to understand.

| Effect of coal rank on permeability
Coal rank is adopted as a significant index to characterize the coalification degree, and a higher coal rank represents a more superior metamorphic grade. 46 Most countries use volatile matter content to classify the coal rank. 38 lower volatile matter content corresponds to a higher coal rank. Meanwhile, the aromatization and repolymerization of coal in the process of coal coalification lead to the rearrangement of coal structure, 49 which is closely related to the permeability of coal. Figure 4 reveals the relationship between the initial permeability of coal particle and the volatile matter content. Figure 4 shows that the volatile matter content has a negative exponential relationship with initial permeability. The initial permeability of WNT sample (Lignite) is the lowest at 0.0254 mD, while that of the YQW sample (Anthracite) is the largest, reaching 0.3176 mD, which is about 13 times as much as the lignite sample. The reasons may be as follows: (i) The surfaces of pore in lignite are mainly composed of some relatively loose space structures that contain high hydrogen and oxygen contents, long side chains but large spacing of aromatic sheets. These loose structures make the pore surfaces very bumpy, so the permeability of lignite is very small. However, the contents of hydrogen, oxygen, and the long side chains decrease as the coal rank grows so that the arrangement of aromatic sheets are more compact. 38,50 Hence, the higher the coal rank, the higher the initial permeability,(ii) with regard to the low-rank coal such as lignite, its metamorphism is very weak due to the low coalification, and thus, the pores and fractures are so poorly developed that it has a relatively low permeability. As the coal rank increases, the coal will undergo a long-term thermal metamorphism that promotes the formation of pores and microfractures network, so the permeability of high-rank coal is enormously improved. 46 Figure 5 illustrates the relationship between the deformation coefficient and the volatile matter content. As can be seen, the deformation coefficient basically presents a linear growth as the volatile matter content increases. Contrary to the change in initial permeability over coal rank, the WNT sample, a kind of lignite, has the highest volatile matter content, and its deformation coefficient is also the largest, but the anthracite samples have the smallest deformation coefficient. The variation of elastic modulus K j of coal sample may affect this phenomenon. According to the deformation coefficient's definition, shown in Equation (12), the deformation coefficient is inversely proportional to the elastic modulus. Shen et al obtained the elastic modulus of six coal ranks through a number of triaxial mechanical loading experiments (Shen et al, 2000). The results showed that the higher the metamorphism of the coal, the greater the elastic modulus, among which anthracite had the largest elastic modulus and was the least prone to mechanical deformation. Pan et al also reported that the lower the coal rank, the smaller the elastic modulus. 51 This also validates the results of Figure 5 to some extent; that is, the deformation coefficient is directly proportional to the volatile matter content. Furthermore, faced with the same adsorption pressure, a larger deformation coefficient corresponds to a lower permeability of coal particle. This further explains the permeability of low-rank coal sample is smaller.

| CONCLUSIONS
In this work, a permeability evolution model of coal particle was developed to address the effect of gas pressure on coal

F I G U R E 4
Relationship between the initial permeability of coal particle and the volatile matter content F I G U R E 5 Relationship between the deformation coefficient and the volatile matter content matrix deformation. This model had been verified by regression analysis of the inversion permeability of coal particle.
The main conclusions are summarized as follows: (i) A dynamic evolution model of coal permeability considering the impact of swelling strain was established. This model includes two main parameters, one is initial permeability of coal particle, and the other is deformation coefficient that reflects the variation characteristics of inward swelling strain of coal matrix. This proposed deformation coefficient was related to the properties of coal, the saturated adsorption capacity, and the temperature of coal.
(ii) The gas permeability coefficients of coal particles had been obtained in our previous work by inversion method. Here, these gas permeability coefficients were first converted into permeability, and then, the permeability of the same coal samples under different pressures was fitted according to the new permeability evolution equation. The results show that the theoretical calculations are matched well with the inversion permeability. Thus, the correctness of the evolution model has been verified. Furthermore, it also indicates that the solid skeleton swell of coal matrix is the main factor to reduce the permeability.
(iii) The effect of coal rank on the permeability evolution model has been investigated. The results indicate that, as the coal rank decreases, the initial permeability decreases exponentially, while the deformation coefficient basically grows in a linear trend. In addition, future work is needed to continue to explore the effect on permeability regarding the effective stresses that can only cause small deformations in coal particles.