The Effect of Water on Gas-Containing Coal with Cyclic Loading: An Experimental Study

Abstract

In deep mining engineering, coal often is subjected to the influence of cyclic loading and water. In order to study the effect of water on gas-containing coal with cyclic loading, the pore evolution, permeability and energy dissipation of gas-containing coal under the effects of water and cyclic axial stress were analyzed in depth. The results showed that with the increase of water content, dissipation energy and permeability of coal samples decreased gradually. The presence of water weakened the increase in micropores and mesopores and promoted the increase in macropores during the cyclic loading process. Compared to dry coal samples, the ratio of micropores and mesopores in saturated coal samples decreased by 0.8872%, while the proportion of macropores increased by 0.0341%. On the basis of the above, a new formula for calculating the dissipation energy of gas-containing coal under the effects of water was proposed. A new damage variable based on energy dissipation was proposed. Finally, a theoretical model was derived to describe the permeability of gas-containing coal under the effects of water and cyclic axial stress.

1. Introduction

As the global economy is developing rapidly, various forms of energy consumption have increased dramatically, so it is imperative to develop deep resources [1,2]. In the deep mining process, the coal often undergoes repeated loading under the action of the mining stress field, which result from roadway excavation and support, periodic roof pressure and other engineering activities [3,4,5,6,7]. The gas flow characteristics and mechanical property for coal seams with the effect of cyclic stress are important factors affecting the safety of coal mining, and are also important references for studying underground mine disasters in coal mines [8,9]. At the same time gas and water are also widely found in coal seams. The presence of water and gas not only affects the mechanical properties of coal, but also leads to many underground mine disasters, for instance, rock burst, and mine water bursting [10,11,12,13,14]. During coal mining, the combined action of cyclic stress and water for gas-containing coal may stimulate dynamic instability events. So it is important to further analyze the influence of water for gas-bearing coal with cyclic stress, which includes seepage, deformation, and damage properties of gas-bearing coal. It will be helpful to recognize and understand the mechanical property and gas flow law of gas-bearing coal in the deep mining process, which will provide important guidance for coal and gas simultaneous extraction during the deep mining coal.

During the past few decades, various studies had developed about the mechanical property and gas-flow law of coal with cyclic loading. Yang et al. [15] experimentally studied fatigue damage of coal specimens under uniaxial cyclic stress, and discussed the internal relations between the number of cycles and dissipated energy. Li et al. [16] experimentally studied acoustic emission and dissipated energy of gas-containing coal with various cyclic loading stresses, and the differences of coal sample failures under the three stress paths were discussed. Fan and Liu [17] experimentally studied the permeability of coal specimens with cyclic loading by applying the pulse-decay method, and discussed the application of Biot’s coefficient to the coal reservoir. Yang et al. [18] experimentally studied the anisotropic permeability for coal samples with cyclic stress, investigated the internal connection between gas flow and external stress by applying a new power model. Zhang et al. [19] experimentally studied the seepage characteristics of broken coal with cyclic loading stress, and discussed particle deformation of broken coal specimens by using the Hertz contact deformation principle. All these research studies are carried out by using dried samples. However water exists widely in coal seams in the deep mining process. Therefore, it is very meaningful to investigate the influence of water on mechanical properties and the seepage characteristics of coal.

In recent years, many studies had developed about the mechanical property and gas flow law of coal with different water-containing states. Zhong et al. [9] experimentally investigated mechanical characteristics and pore structure evolution of water-bearing coal with cyclic-loading axial stress, and discussed the effect of cycle number and loading rate on saturated coal. Yin et al. [20] experimentally studied the gas flow law of coal samples under the condition of various water-containing conditions, and it is proposed that the gas flow in coal receives the joint action of effective stress and water content. Li et al. [21] established a gas flow model by considering effects of moisture on adsorption, and the gas flow law of water-bearing coal was well indicated. Yang et al. [22] experimentally studied expansion properties and the gas flow law for anthracite, including expanded clay with different water-containing conditions, and the influence of water on the anthracite, including expanded clay, was discussed. Zhang et al. [23] experimentally studied the stress sensitivity on coal samples with various water-containing conditions, and discussed the influence of water on the stress sensitivity coefficient and effective stress. The above studies either focus only on the effect of water on the mechanical characteristics of coal, or only on the influence of water on the gas flow law of coal with simple loading conditions. However, water and gas exist widely in coal seams, and coal seams will experience complex stress under the action of a mining stress field, such as cyclic loading and unloading. The joint action of water, gas, and cyclic loading may make coal seams cause complex damage, and lead to a lot of dynamic disasters. But studies about the effects of water on seepage characteristics and the mechanical property of gas-bearing coal with cyclic stress are very scarce. So it is of great significance to investigate the effects of water for gas-bearing coal with cyclic stress.

In this paper, pore evolution, permeability, and the dissipated energy of gas-containing coal with the effects of water and cyclic stress are analyzed in depth, and the inner mechanism of each are discussed. At the same time, a new permeability model is proposed for calculating the permeability evolution of coal with the effects of water and cyclic loading. The study will provide some guidance for the safety of mining coal in the deep mining process.

2. Materials and Methods

2.1. Basic Parameters for Coal Specimens

The coal specimens used in this paper were collected from a coal mine of the Sichuan Coal Group. The large coal blocks taken from the coal mine were drilled, and then cut and ground to standard cylindrical specimens. The basic physical parameters of coal specimens were measured before the experiment. Three coal specimens from the same coal block were selected to conduct experiments, labeled Coal 1, Coal 2 and Coal 3, respectively. At the same time, the basic parameters for three coal specimens were obtained, as shown in Table 1. We chose Coal 1 as the dry coal sample, Coal 2 as the semi-saturated coal sample, and Coal 3 as the saturated coal sample. We summarized the components and basic mechanical parameters for the coal specimens. The specific values are presented in Table 2 and Table 3, respectively.

2.2. Experimental Apparatus

In the course of the experiments, a thermal–fluid–solid coupling triaxial servo seepage testing apparatus for gas-bearing coal was applied to conduct the experiments of cyclic loading axial stress. The specific parameters for the experimental apparatus can be consulted in the research of Jiang et al. [25] and Yin et al. [20], and the experimental apparatus is shown in Figure 1. To investigate the effects of water on the internal structure of gas-bearing coal with cyclic loading, the pore distribution of coal specimens before and after cyclic loading experiments was measured by using a nuclear magnetic resonance analysis system. The nuclear magnetic resonance analysis apparatus is presented in Figure 2.

2.3. Experimental Scheme

By using the equipment in Section 2.2, we conducted the test for the influence of water on gas-bearing coal with cyclic stress. In the process of the experiments, the gas we introduced is CH₄, which is used to simulate the flow of coalbed methane in coal specimens. The detailed test procedures are as follows:

(1) Three coal specimens are scanned by the nuclear magnetic resonance analysis system. Then the specimens are placed in a drying chamber and dried in 65 °C. Coal 2 and Coal 3 are selected to soak in water until their weight does not change any more. Coal 2 is removed and placed in the air, so that it loses half of the water. Water saturation (S_w) was calculated by Equation (1). The water saturation of three coal samples is as shown in Table 4.

(1)$S_w={\displaystyle \frac{m_c-m_0}{m_s-m_0}}\times 100\%$

where S_w is the water saturation, m_s is the saturated weight, m₀ is the dry weight, and m_c is the weight of the coal sample.

(2) The specimen is placed on the loading device. The axial stress and radial stress are loaded at the same speed, so that the axial stress and radial stress reach hydrostatic pressure state (σ₁ = σ₂ = σ₃ = 5 MPa). At this time, CH₄ is introduced into the coal specimen for adsorption, and the gas pressure is 1 MPa.

(3) The outlet valve is opened after adsorption equilibrium. When we advance the working face, the abutment stress will change with the working face. The change in abutment stress caused by mining will make the abutment stress of coal seams in front of the working panel increase gradually. With the roadway support and stress redistribution, the abutment stress of coal seams in front of the working panel gradually decreases and returns to almost the same state as before advancing the working face. Therefore, as the working panel continuously advances, the coal seams far away from the working panel will be subject to repeated loading–unloading stresses. Based on the above engineering background, the axial stress is circularly loaded and unloaded when the gas flow of CH₄ is unchanged. The loading amplitude is 25 MPa, and the number of cycles is 20. The specific loading path is presented in Figure 3, and the flow chart for experimentation and analysis is shown in Figure 4.

(4) Three coal specimens after experiments are scanned by the nuclear magnetic resonance analysis system, and the experimental apparatus is cleaned up.

The experiments of cyclic loading axial stress are conducted for the coal specimens under the same conditions. It is assumed that the gas flow inside coal specimens obeys Darcy’s law. The permeability k can be obtained by the following formula [26,27,28,29,30,31].

(2)$k={\displaystyle \frac{2Q\mu p_{a}L}{A(p_1^2-p_2^2)}}$

where k is the permeability of coal specimens (m²), Q is the flow rate of methane (m³/s), µ is the kinematic viscosity of methane (Pa·s), L is the length of the coal specimens (m), A is the effective seepage area of coal specimens (m²), P_a is the standard atmospheric pressure (Pa), P₁ is the methane pressure for inlets of coal specimens (Pa), and P₂ is the methane pressure for outlets of coal specimens (Pa). In the experiments, the gas pressure of exhaust ports is the standard atmospheric pressure.

3. Experimental Results and Analysis

3.1. Evolution Law of Permeability and Deformation

3.1.1. Effect of Water Content on the Stress–Strain Curve with Cyclic Axial Stress

The stress–strain curve for coal specimens with cyclic loading is shown in Figure 5. From Figure 5, we can see that with the cyclic loading of the axial stress, the axial strain, radial strain and volumetric strain increase gradually. This indicates that the irreversible plastic deformation of coal specimens occurs during cyclic loading. Meanwhile, with the repeated loading of the axial stress, some plastic hysteresis loops are formed in the stress–strain curve. The area of the plastic hysteresis loop for the first cycle is the largest, while the area of the plastic hysteresis loop for the subsequent cycles is smaller than that of the first cycle, and the area of the plastic hysteresis loop for the subsequent cycles is almost the same.

In Figure 5, it is also shown that the irreversible plastic deformation in the first cycle is the largest, while with the repeated loading of axial stress, the irreversible plastic deformation in the subsequent cycles is almost the same. When the water content of the coal specimens increases, the plastic deformation gradually reduces, the area of plastic hysteresis loops continues to decrease, and the compression effect of axial stress on the coal samples tends to be less obvious. This is mainly due to two reasons. One is that most of the fractures in the coal specimens are compressed during the first cycle and the compression effect gradually weakens with the increase in cycle number. The other is that the pore water pressure in the coal specimens increases gradually, due to the compression effect on the fractures, and the pore water pressure resists the compression effect of axial stress. It can be concluded that the presence of water inhibits plastic deformation.

3.1.2. The Influence of Water on Seepage Behavior under Cyclic Loading

Axial stress–strain curves and permeability–strain curves for coal specimens are shown in Figure 6. When the axial stress is loaded and unloaded cyclically, the axial strain first increases and then decreases, and the permeability first decreases and then increases. In the early stage of cyclic loading, the strain and permeability fluctuate greatly, especially in the first cycle. As the number of cycles increases, the strain continuously increases, permeability continuously decreases, the sensitivity of permeability on axial stress weakens gradually. When the water content increases, the initial permeability gradually decreases. It is shown that the presence of water has obvious inhibition for gas flow. The main reason is that free water blocks the flow channel of gas.

3.2. The Evolution Law of Dissipated Energy

3.2.1. Calculation Principle of Dissipated Energy

During coal loading, mechanical energy inputs are simply converted into elastic potential energy, which is stored in rocks, while dissipation energy will be released when rocks are damaged [32]. It can be derived from the law of conservation of energy:

(3)$W=E_e+U_d$

where W is all the energy imported by the loading stress; E_e is the elastic strain energy; and U_d is the dissipated energy.

In the process of seepage experiments for coal samples, the formula that calculates dissipated energy for coal specimens should consider the influence of gas on loading stress [25]. In fact, it is the effective stress inside coal specimens that causes the deformation of coal specimens. The stress state of coal specimens under the joint action of gas pressure and loading stress is presented in Figure 7.

In Figure 7, σ₁ is the axial stress, σ₃ is the confining stress, σ′ is the abutment stress inside coal specimens, that is, the effective stress for coal specimens, F is the swelling stress resulting from temperature and adsorption, and p is gas pressure. In Figure 7, we first select the section A-A′ to be the research target; it is assumed that the area of the section A-A′ is S. From the principle of stress balance, we can obtain the following formula [30]:

(4)${\sigma }_{1}S=({\sigma }_1^{\prime }+F)(1-\varphi )S+p\varphi S$

By transforming Formula (4), we can obtain the calculation formula of axial effective stress:

(5)${\sigma }_1^{\prime }={\displaystyle \frac{{\sigma }_1-p\varphi }{1-\varphi }}-F$

The stress given rise to by gas adsorption is obtained by the following formula [21,33]:

(6)$F=\left[{\displaystyle \frac{3{\rho }_{c}RT{V}_L}{V_m}}\ln {\displaystyle \frac{p+p_L}{p_0+p_L}}-{\displaystyle \frac{48{\rho }_c{V}_L}{V_m{\rho }_{ads}}}(p-p_0-p_L\ln {\displaystyle \frac{p+p_L}{p_0+p_L}})\right]\exp (-\lambda m)$

where P is the gas pressure (MPa), ρ_c is the density of coal (g/cm³), V_m is the molar volume of gas, T is the thermodynamic temperature (K), R is the universal gas constant, P₀ is the initial methane pressure (MPa), P_L is the Langmuir adsorption pressure constant (MPa), V_L is the Langmuir adsorption volume constant (cm³/g), ρ_ads is the adsorption phase density (g/cm³), λ is the decay coefficient (dimensionless parameter) of coal adsorption to humidity, and m is the water content (%).

Similarly, to select the B-B′ section be the research target, we can obtain the calculation formula of radial effective stress:

(7)${\sigma }_2^{\prime }={\displaystyle \frac{{\sigma }_2-p\varphi }{1-\varphi }}-F$

(8)${\sigma }_3^{\prime }={\displaystyle \frac{{\sigma }_3-p\varphi }{1-\varphi }}-F$

Convert the formula of effective stress into tensor form:

(9)$\begin{array}{l}{\sigma }_{ij}^{\prime }={\displaystyle \frac{{\sigma }_{ij}}{1-\varphi }}-({\displaystyle \frac{p\varphi }{1-\varphi }}+({\displaystyle \frac{3{\rho }_{c}RT{V}_L}{V_m}}\ln {\displaystyle \frac{p+p_L}{p_0+p_L}}\\ -{\displaystyle \frac{48{\rho }_c{V}_L}{V_m{\rho }_{ads}}}(p-p_0-p_L\ln {\displaystyle \frac{p+p_L}{p_0+p_L}}))\exp (-\lambda m)){\delta }_{ij}\end{array}$

where ${\delta }_{ij}$ is the Kronecker symbol.

During cyclic loading axial stress, the energy for coal specimens is mostly the elastic strain energy and dissipated energy. In order to calculate conveniently, it is assumed that dissipation energy consists of two sections, namely, axial dissipation energy and radial dissipation energy. By overlaying these two dissipation energies, we can obtain the actual dissipation energy for coal specimens. During cyclic-loading axial stress, the axial dissipation energy can be obtained by the following equation:

(10)$U_{d1}=W_1-E_{e1}={\sigma }_1^{\prime }(\int d{\epsilon }_1^{-}-\int d{\epsilon }_1^{+})$

where ${\epsilon }_1^{-}$ and ${\epsilon }_1^{+}$ are axial strain values for the axial stress loading and unloading stages, respectively. As the confining pressure and methane pressure are maintained unchanged during the experiments, radial effective stress remains unchanged during the experiments.

In the same way, the radial dissipation energy will be obtained by the following equation:

(11)$U_{d3}=W_3-E_{e3}={\sigma }_3^{\prime }(\int d{\epsilon }_3^{-}-\int d{\epsilon }_3^{+})$

3.2.2. Analysis of Dissipation Energy

The dissipation energy of each cycle and the cumulative dissipation energy are calculated by using the formula in the previous section, as shown in Figure 8 and Figure 9, respectively.

In Figure 8, the dissipation energy of first cycle for three coal samples is the largest. When the cycle number increases, the dissipation energy generally shows a decreasing trend. And the dissipation energy trend stationary is when the cycle is nearing the end. With the increase in water content, the dissipation energy decreases gradually. Water has an obvious inhibition effect for dissipated energy of gas-bearing coal in the process of cycle loading. It is mainly caused by two reasons. First, the pore-water pressure in coal specimens increases gradually, due to the compression effect on fractures, and the pore-water pressure resists the compression effect of axial stress. Second, not only does the coal matrix absorb elastic strain energy, but the pore-water also absorbs a lot of elastic strain energy when the axial stress is loading. The joint action of these two causes leads to the inhibition effect of water on the dissipated energy.

As can be seen from Figure 9, when the cycle number increases gradually, the cumulative dissipation energy generally shows a linear increase, indicating that the damage degree for coal samples increases gradually. When the water content increases, the initial dissipation energy is reduced, and the increased rate of cumulative dissipation energy is reduced. From the above phenomenon, we can see that the dissipation energy can effectively characterize the damage degree of coal samples.

3.2.3. Damage Characterization Based on Dissipated Energy

The deformation characteristics and dissipated energy for the coal specimens with cyclic loading is actually caused by the damage to the coal specimens [25]. In order to express the degree of damage to the coal specimens, we propose a formula to characterize the damage variable D₁ for coal specimens:

(12)$D_1={\displaystyle \frac{2}{\pi }}\arctan {\displaystyle \sum _{n=1}^i{U}_{nd}}$

where U_nd is the dissipation energy of the nth cycle, n is the cycle number and, i is the maximum cycle number. When the dissipation energy is 0, even if the deviatoric stress increases continuously, the damage degree is still 0, indicating that the coal sample is the perfect elastic body. And when the dissipation energy increases, even if the deviatoric stress remains the same, the damage is still increasing continuously (creep damage, etc.). The above formula can reasonably characterize the damage variable.

The damage variables of three coal samples under cyclic loading will be obtained by Equation (12), as shown in Figure 10.

As we can see from Figure 10, when the cycle number increases gradually, the damage variables for coal samples gradually increase. The damage value of Coal 1 increases gradually in the form of logarithmic function when the cycle number increases. However, with the increase in water content, the damage value tends to gradually increase, linearly. We can conclude that the damage to the coal samples gradually reduces under cyclic loading when the water content increases. From the above results, it can be seen that the damage variable proposed in this paper can better characterize the damage evolution of the coal samples.

3.3. Pore Structure Evolution and Fractal Dimension Based on NMR

Through previous studies, we can obtain the relationship between transverse relaxation time and aperture, as follows:

(13)${\displaystyle \frac{1}{T_2}}=\rho ({\displaystyle \frac{S}{V}})$

where ρ is a constant characterizing the transverse relaxation strength. S/V is the ratio of pore surface area to pore volume. Thus, T₂ is linearly related to the pore size. That is to say, the longer the relaxation time, the larger the pore.

The relationship between S_v and D can be obtained by the following formula:

(14)$S_v={({\displaystyle \frac{T_{2\max }}{T_2}})}^{D-3}$

where S_v is the percentage of the sum of all pore volumes when the relaxation time is less than T₂.

The fractal dimension is calculated by the following formula:

(15)$\log (S_v)=(3-D)\log (T_2)+(D-3)log(T_{2\max })$

So the calculation formula of fractal dimension D is obtained through transforming Equation (15):

(16)$D={\displaystyle \frac{\log (S_v)}{\log (T_{2\max })-\log (T_2)}}+3$

3.3.1. Pore Structure Evolution

Figure 11 reveals the T₂ distribution for coal specimens before and after the experiments. The T₂ distributions for three coal specimens exist as three peaks, characterizing the micropore (first peak), mesopore (second peak) and macropore (third peak) inside coal specimens, respectively. Although often the second- and third-peak curves are connected and not easily distinguishable, it does not affect the analysis of the evolution laws of the second and third peaks. It can be easily concluded that coal specimens are mainly composed of micropores, while the proportion of mesopores and macropores is small.

As shown in Figure 11, the changes in the T₂ distribution of three coal specimens after cyclic loading are completely different. It is explained that water has an important influence on the pore structure evolution of coal specimens under cyclic stress. The first and second peaks of Coal 1 increased, while the third peak of Coal 1 decreased. The first and second peaks of Coal 2 increased, but the amplitude of increase is less than that of Coal 1, and the third peak of Coal 2 slightly increased. The first peak of Coal 3 moved slightly to the right, the second peak of Coal 3 moved slightly to the left, and the amplitudes of the first and second peaks were basically unchanged. The third peak of Coal 3 increased, and the amplitude of increase was greater than Coal 2. It can be concluded that the presence of water weakens the increase in micropores and mesopores and promotes the increase in macropores during the cyclic loading process.

The main reason is that gas is mainly adsorbed in micropores and mesopores. During cyclic loading of axial stress, it will cause gas desorption in micropores and mesopores, while the pore connectivity of micropores and mesopores is poor, thus forming a large gas pressure, causing the coal to produce tensile failure, and thus promoting the increase in the first and second peaks. The macropores are compressed during the cyclic loading process, which causes the third peak to decrease. As the water content increases, water weakens the adsorption of gas, thus weakening the increase in the first and second peaks. At the same time, the water in the macropores weakens the compression effect of axial stress, and the increase in the pore-water pressure causes tensile failure of the macropores.

3.3.2. Analysis of the Fractal Dimension

The relationship between log(S_v) and log(T₂/T_2max) of the coal specimens before and after the experiments is shown in Figure 12. We can obtain the fractal dimension of adsorption pores and seepage pores by using Equation (16). The values of fractal dimension for coal specimens before and after cyclic loading are calculated by using the slope of the curves in Figure 12, namely, the values of D_A, D_A′, D_S and D_S′.

To investigate the effect of water on fractal dimension under cyclic loading, the fractal dimensions of three coal specimens before and after cyclic loading are obtained and analyzed, as shown in Figure 12. By analyzing Figure 12, we can see that adsorption space and seepage space have completely different fractal characteristics. The fractal dimensions of adsorption space were all less than 1.5. However, the fractal dimensions for seepage space were all close to 3. After cyclic loading of axial stress, the fractal dimension of adsorption space for Coal 1 decreases from 1.16251 to 1.15051, and the fitting degrees are 0.81552 and 0.81637, respectively. The fractal dimension of adsorption space for Coal 2 reduces from 1.14584 to 1.14278, and the fitting degrees are 0.81961 and 0.81721, respectively. The fractal dimension of adsorption space for Coal 3 rises from 1.11177 to 1.21915, and the fitting degrees are 0.88934 and 0.80824, respectively. When the fractal dimension of the adsorption space decreases, the flatness of the adsorption space increases, which is helpful for gas adsorption. When the fractal dimension of the adsorption space increases, the flatness of the adsorption space decreases, which is helpful for gas desorption. The above results suggests that the presence of water promotes the increase in the fractal dimensions of the adsorption space under the condition of cyclic loading. That is, the presence of water reduces the area of adsorption space under cyclic loading of axial stress. At the same time, the fractal dimension of the seepage space decreases from 2.95764 to 2.93724 in Coal 1, with fitting degrees of 0.97045 and 0.92902; in Coal 2 it decreases from 2.95612 to 2.9513, with fitting degrees of 0.91904 and 0.90798; and in Coal 3 it rises from 2.96367 to 2.96521, with fitting degrees of 0.86785 and 0.89317. The reduction in fractal dimensions for seepage space indicates that the connectivity is improved after cyclic loading, which is helpful for coalbed methane extraction. Therefore, the presence of water promotes the increase in fractal dimensions of the seepage space under the condition of cyclic loading. That is, the presence of water reduces the connectivity of the seepage space under cyclic loading of axial stress.

3.4. Permeability Model

3.4.1. Permeability Model of Water-Containing Coal with Cyclic Loading

The fracture of the coal is the main channel for the gas flow, and also an important factor to determine coal permeability. The S-D model [34] is a widely used permeability analysis model, which is shown in Equation (17).

(17)$k_f=k_{f0}{e}^{-3C_f({\sigma }_e-{\sigma }_{e0})}$

where C_f is the cleat volume compressibility, σ_e is the effective stress, and σ_e0 is the initial effective stress.

We define the effective stress with cyclic loading of axial stress as:

(18)${\sigma }_e={\displaystyle \frac{{\sigma }_1^{\prime }+{\sigma }_2^{\prime }+{\sigma }_3^{\prime }}{3}}$

where ${\sigma }_1^{\prime }$ is the axial effective stress, ${\sigma }_2^{\prime }$ and ${\sigma }_3^{\prime }$ is the radial effective stress. In Section 3.2.1 we have discussed the calculation method of axial and radial effective stress. The axial and radial effective stress are calculated by Equation (9).

The fracture compressibility C_f is not unchanged, and has a strong dependence on the stress. When the coal sample is loaded in cyclic axial stress, the coal sample will be constantly compressed with the increase in cycle number, and the sensitivity of permeability to effective stress decreases gradually. In other words, the fracture compressibility C_f will constantly change and gradually stabilize. The curves of permeability and effective stress do not match during the cyclic loading stress, which shows that the permeability and the effective stress do not have a simple function relationship. Therefore, the dynamic evolution model of permeability with effective stress should be considered from the two stages of loading and unloading.

We define the relationship between fracture compressibility and the number of cycles in the loading process as follows:

(19)$C_{fL}=\alpha \ln (n)+C_{fL1}$

where C_fL is the fracture compressibility during the loading process, C_fL1 is the initial fracture compressibility during the loading process, n is the number of cycles, and α is a constant.

We define the relationship between fracture compressibility and the number of cycles in the unloading process as follows:

(20)$C_{fU}=\beta \ln (n)+C_{fU1}$

where C_fU is the fracture compressibility during the unloading process, C_fU1 is the initial fracture compressibility during the unloading process, n is the number of cycles, and β is a constant.

Finally, by using C_fL and C_fU instead of C_f in Equation (17), we can obtain the permeability model of water-containing coal with cyclic loading:

(21)$\left\{\begin{array}{l}{k}_{fL}=k_{fL0}{e}^{-3(\alpha \ln (n)+C_{fL1})({\sigma }_e-{\sigma }_{e0})}\cdots \cdots \cdots \mathrm{loading}\\ k_{fU}=k_{fU0}{e}^{-3(\beta \ln (n)+C_{fU1})({\sigma }_e-{\sigma }_{e0})}\cdots \cdots \cdots \mathrm{unloading}\end{array}\right.$

where k_fL is the permeability during loading stress, k_fU is the permeability during unloading stress, k_fL0 is the initial permeability of the nth cycle during the loading process, and k_fU0 is the initial permeability of the nth cycle during the unloading process.

3.4.2. Model Validation

By using the experimental data, we verify the permeability model with cyclic loading, and the parameters we used are determined in the lab. The model parameters obtained by matching data are shown in Table 5.

Figure 13 presents the curves of actual permeability and the model’s permeability with time. In Figure 13, we mainly analyzed the 1st cycle, 10th cycle and 20th cycle. The model parameters obtained by matching data are shown in Table 5. It can be seen from Figure 13 that the permeability model can better characterize the permeability evolution of coal specimens with cyclic loading. The parameters α and β obtained by matching data show that the fracture compressibility during cyclic loading gradually rises with the increase in water content. The permeability model proposed in this paper can express well the permeability of water-containing coal under cyclic loading conditions.

4. Conclusions

To study the effect of water on gas-containing coal with cyclic loading, the pore evolution, permeability and dissipated energy of gas-containing coal with the effects of water and cyclic stress are analyzed in depth, and the inner mechanism of each is discussed. By analyzing and discussing the experimental data, some conclusions are obtained, as follows.

Firstly, with the increase in water content, the plastic deformation is gradually reduced, the area of hysteresis loops gradually decreases and the compression effect of axial stress on coal samples tends to be less obvious. The presence of water is an obvious inhibition for gas flow.

Secondly, according to the principle of effective stress and the law of energy conservation, a new formula for calculating the dissipation energy of gas-bearing coal with the effects of water is proposed. The new damage variable based on dissipated energy is proposed. With the increase in water content, the dissipated energy decreases gradually. Water has an obvious inhibition effect on the dissipated energy of gas-bearing coal in the process of cycle loading.

Thirdly, the pore structure evolution and fractal dimension are analyzed based on NMR. Water has an important influence on the pore structure of coal specimens with cyclic loading. The presence of water weakens the increase in micropores and mesopores and promotes the increase in macropores during cyclic loading. Compared to dry coal samples, the ratio of micropores and mesopores in saturated coal samples decreased by 0.8872%, while the proportion of macropores increased by 0.0341%. The presence of water promotes the increase in the fractal dimension of the adsorption space and seepage space under cyclic loading of axial stress.

Finally, a new permeability model is developed to describe the permeability of gas-containing coal under the effects of water and cyclic axial stress. The establishment of this model proposes a new approach to explore the migration law of gas under cyclic stress and water. The parameters α and β obtained by matching data show that the fracture compressibility during cyclic loading gradually rises with the increase in water content.

Author Contributions

Conceptualization, Y.Y. and C.J.; methodology, Y.Y.; validation, Y.Y., D.H. and S.C.; formal analysis, Y.Y. and F.Y.; investigation, D.H.; resources, S.C.; writing—original draft preparation, Y.Y.; writing—review and editing, S.C.; All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Acknowledgments

This study was financially supported by the National Natural Science Foundation of China (52074044, 52274111, 52274221, 52404211), and Scientific Research Foundation of Key Laboratory of Mining Disaster Prevention and Control (MDPC202012).

Conflicts of Interest

Author Diandong Hou was employed by the company Yankuang Energy Group Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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Figures and Tables

Figure 1. The experimental apparatus of the cyclic loading test.

Figure 2. Nuclear magnetic resonance analysis apparatus.

Figure 3. Loading paths for cyclic loading tests.

Figure 4. The flow chart for experimentation and analysis.

Figure 5. The stress–strain curve for coal specimens with cyclic loading: axial strain (εa); radial strain (εr); volumetric strain (εv).

Figure 6. The evolution of axial stress and permeability with axial strain.

Figure 7. The stress state under the joint action of gas pressure and loading stress.

Figure 8. The dissipation energy of each cycle for three coal samples.

Figure 9. The cumulative dissipation energy of three coal samples.

Figure 10. The Evolution of damage variables under cyclic loading.

Figure 11. The T2 distribution for coal specimens before and after the experiments.

Figure 12. The fractal dimension of coal specimens before and after the experiments.

Figure 13. Permeability data-matching results under cyclic loading conditions.

Table 1

Basic parameters for coal specimens.

Specimens Number	Length (mm)	Diameter (mm)	Density (g/cm3)	Wave Velocity (m/s)
Coal 1	100.1	49.9	1.35	1573.6
Coal 2	99.9	49.9	1.32	1552.2
Coal 3	100.2	50.1	1.34	1568.5

Table 2

The components of coal seams [24].

Components	Kaolinite	Quartz	Calcite	Others
Content (%)	5.58	11.74	5.04	77.64

Table 3

Basic mechanical parameters for coal specimens [24].

Orientation	UCS (MPa)	E (GPa)	$\mathit{v}$	Tensile Strength (MPa)
Parallel to bedding	20.03	3.42	0.14	2.75
Perpendicular to bedding	25.28	2.95	0.22	1.95

UCS is uniaxial compressive strength, E is elastic modulus, and $v$ is Poisson’s ratio.

Table 4

Water saturation of three coal samples.

Coal Number	Coal 1	Coal 2	Coal 3
water saturation (%)	0	50	100

Table 5

Model parameters obtained by matching data.

	Coal 1	Coal 2	Coal 3
α	0.0051	0.0076	0.009
β	0.0013	0.0021	0.0029

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