# Impact of Dynamic Elastic Modulus on Coal Permeability: Modeling and Analysis

Authors: Jia Kong; Baiquan Lin; Chuanjie Zhu; Ting Liu; Xiangguo Kong
Journal: Geofluids
 (2022)

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## Abstract

Coalbed methane is a kind of high-quality clean energy. The underground gas drainage can achieve the efficient utilization of coalbed methane. Permeability is the key factor affecting gas drainage. At present, permeability research mainly considers the effect of gas adsorption-induced matrix swelling deformation as well as the effective stress. However, experiments indicate that the variation of pore pressure has an impact on the mechanical properties of coal, which further changes the coal permeability. Due to the differences between the theory and experimental results, firstly, the dynamic elastic modulus model was established in this paper. Then, an experiment was conducted to prove that the elastic modulus of coal-containing gas decreases with the increasing gas pressure. On this basis, a permeability model considering internal swelling deformation was constructed. Finally, the rationality of the model was validated through the laboratory experiment. Finally, through analyzing the influence of key parameters on the evolution law of permeability, we find that the parameters have three different impacts on permeability. The new permeability model is more accurate in predicting coalbed methane production capacity. The result can provide a theoretical basis for the research of the multifield coupling process during gas drainage.


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## Body

## 1. Introduction

Coalbed methane (CBM) is a kind of hydrocarbon gas existing in coal measure strata [1, 2]. It is not only the high-quality clean energy with high thermal efficiency but also a greenhouse gas, causing gas outbursts, gas exploration, and other accidents [3–6]. The CBM reserves in the world are estimated at 84-262 trillion cubic meters, about 50% of the conventional natural gas resources [7]. Gas control is the most commonly used method for commercial production of CBM, which is of great significance for economic development, environmental protection, mining safety, and efficient utilization of coalbed methane resources [8–10].Nowadays, gas control mainly adopts gas extraction, mine ventilation, etc. [11–17]. The permeability of coal reservoirs is the key factor that affects the adsorption/desorption, diffusion, and seepage of CBM. Therefore, mastering the evolution law of coal permeability is significant to methane extraction and the prevention of coal and gas outbursts. The previous studies focused on the influences of the competitive effect of gas adsorption-induced/desorption-induced matrix deformation and effective stress on permeability. Based on the theoretical study of uniaxial strain, Gray firstly established a coal permeability model which considered the matrix shrinkage [18]. Palmer and Mansoori developed the P&M model that incorporated the effects of matrix shrinkage and effective stress [19]. In addition, the P&M model matched well with the field test result of San Juan Basin and was widely used. Assuming that the volumetric strain caused by gas desorption changed the coal seam horizontal stress, which eventually altered the coal seam permeability, Shi and Durucan built the S&D model [20]. Based on the matchstick model, Liu and Rutqvist proposed the concept of a “matrix bridge” [21]. Under the condition of uniaxial strain and constant confining pressure, they introduced the internal swelling coefficient to quantify the interactive relationship between the matrix and fractures, and a new theoretical model was established.It is believed that coal permeability is affected by coal matrix swelling/shrinkage and effective stress [22–27]. The mechanical properties of coal have a significant impact on effective stress; therefore, coal permeability is also affected by the mechanical properties of coal. In the previous studies, the mechanical parameters of coal and rock samples were assumed to be constant. However, it has been suggested that the mechanical parameters of coal are variable, which is related to gas adsorption based on previous studies. Mishra and Dlamini studied the coal elastic modulus and deformation law when injecting gas into the coal [28]. They believed that the change of confining pressure caused by adsorption deformation is the main reason which gives rise to the variation of elastic modulus. Masoudian et al. found that CO2 reduces the coal elastic modulus and strength. Moreover, this influence is reversible [29]. After adsorbing gas, the change of dual pore structure enhances the plastic deformation capacity of the coal. Wang et al. found that methane changes the coal elastic modulus and strength mainly by affecting the effective confining pressure and quick release of gas expansion energy [30].Based on the previous studies, it is known that the mechanical properties of coal during methane extraction are dynamic. Although numerous models have been developed to characterize the permeability evolution, the impact of dynamic mechanical properties on coal permeability has not been completely understood and fully considered in the permeability models. The elastic modulus is a critical mechanical parameter of coal. In comparison with the previous studies, this paper proposes the third path to study the evolution law of coal permeability by introducing the concept of dynamic elasticity modulus, as shown in Figure1.Figure 1
Conceptual graph of gas-solid coupling paths of coal and gas.In this study, the triaxial seepage test system was established to conduct the triaxial compression tests and the coal permeability experiments under different methane pressure. Moreover, we analyzed the influence of methane pressure on elastic modulus and built a mathematical model of dynamic elastic modulus based on the experimental results. A dynamic coal permeability model was then established, and the model was validated through laboratory tests. The results suggest that the new permeability model is more accurate in predicting CBM production capacity and helps the reasonable gas extraction borehole arrangement in the coal mine. The results can also provide a theoretical basis for the research of the multifield coupling process during methane extraction.


## 2. Experimental Methodology

### 2.1. Sample Preparation

The two types of coal, including bitumite (GSXB) collected from the Xinbei coal mine located in the northwest of China and anthracite (GZLH) collected from the Linhua coal mine located in the southwest of China, are taken as the experimental coal samples for experimental studies of dynamic elastic modulus and dynamic permeability model. The maceral and vitrinite reflectance analyses of specimens are listed in Table1. For triaxial seepage tests, the samples with the size of Φ50×100mm were cored from large block coal, perpendicular to the bedding.Table 1
Raw coal properties.
TypeCoal rankVitrinite reflectanceRo,max (%)Proximate analysis (wt, %)Coal maceral composition (%)MadAdVdafFCadVILMGSXBBituminous1.452.519.7038.5455.4965.8931.51—2.6GZLHAnthracite2.862.1411.045.8183.7960.3335.37—4.3Notes:Mad: moisture, air-drying basis; Ad: ash yield, air-drying basis; Vdaf: volatile matter dry ash-free basis; FCad: fixed carbon content, air-drying basis; V: vitrinite; I: inertinite; L: liptinite; M: mineral.


### 2.2. Sample Characterization

As mentioned earlier in this paper, the mechanics of methane-saturated coal is closely related to its gas adsorption capacity, and the gas adsorption capacity of coal is affected by its pore structure characteristics [31–33]. To assess the pore structure characteristics of coal samples, a nitrogen adsorption/desorption experiment and a nuclear magnetic resonance (NMR) experiment were conducted.


#### 2.2.1. Nitrogen Adsorption/Desorption

To assess the pore structure characteristics of two samples, liquid nitrogen adsorption/desorption experiments were conducted. The liquid nitrogen adsorption and desorption curves are presented in Figure2. The pore structure characteristics of samples were measured using powdered coal particles that passed through an 18~25 millimeter. Figure 2 illustrates that the nitrogen adsorption of coal increases slowly under low relative pressure. As the pressure increases, the isotherms between the adsorption and desorption had a hysteresis effect. The appearance of the adsorption hysteresis loop mainly depends on the capillary condensation that occurs on the solid surface [34, 35]. Finally, the nitrogen adsorption of coal increases fast under high relative pressure. The pore structure characteristics of the two samples are shown in Table 2. The pore volume of GSXB is 0.009 mL/g and the specific surface area is 6.0835 m2/g which is relatively larger than 0.0055 mL/g and 2.3661 m2/g of GZLH, respectively, which can provide broad spaces for gas adsorption.Figure 2
Nitrogen adsorption/desorption curves.Table 2
Results of low-temperature nitrogen adsorption experiments.
Coal samplesGSXBGZLHPore volume (mL/g)0.0090.0055Specific surface area (m2/g)6.08352.3661Mean pore size (nm)5.91769.2980


#### 2.2.2. NMR Measurements

To analyze the coal pore distribution, nuclear magnetic resonance (NMR) experiments were conducted. In the magnetic field, the hydrogen atoms present within the fluid can be detected by the transversal relaxation time (T2); thus, the pore properties can be analyzed [36]. T2 can be expressed as [37]
(1)1T2=ρ×SV=ar,where ρ is a constant representing the transverse relaxation strength, S is the pore surface, V is the pore volume, r is the pore radius, and a is a pore shape factor.It can be seen from Equation (1) that larger pores have longer T2, and the higher the T2 spectrum, the more the pores. In addition, continuity between T2 spectrum peaks represents the pore connectivity [38]. For coals, T2<10ms corresponds to micropores and 100 ms corresponds to macropores [39].Figure3 illustrates the coal pore distributions of GSXB and GZLH. The T2 spectrum of GSXB consists of three separated peaks (i.e., P1, P2, and P3), indicating a large number of micropores and a small number of mesopores and macropores. Compared with the T2 spectrum of GSXB, GZLH has the same trend. In contrast, of GZLH is lower, P2′ of GZLH is roughly equivalent to GSXB, and P2′ is higher, which indicates that the GZLH has more macropore and mesopore. The free fluids mainly migrate in macropores, which may result in the higher permeability of GZLH. The continuity between T2 spectrum peaks of GZLH is better, which is a strong indication that micropores, mesopores, and all macropores are interconnected. In conclusion, GZLH has more macropores and fewer micropores, and this kind of unique feature makes GZLH have higher permeability and a lower adsorbability.Figure 3
Results of nuclear magnetic resonance (NMR) experiments.


### 2.3. Experimental Procedure

The experiment system was established, which consists of a loading system, strain monitoring system, and flow acquisition system, as is shown in Figure4.Figure 4
Schematic illustration of the saturation and triaxial setup.Prior to each test, the samples were dried at 100°C for 24 h in a drying oven, and then, they were put into the core holder. The sample was degassed in the core holder under vacuum conditions at 30°C for 24 h. In the process of the test, a constant confining pressure of 5.0 MPa was used for the triaxial tests. The water pressure inside the triaxial cell was provided from a pressure controller. The methane was supplied from a high-pressure CH4 cylinder with a maximum pressure of 13.0 MPa. According to the previous studies, the mechanical strength and elastic modulus of coal gradually decrease with the pore pressure increase in a certain range [29]. For example, the elastic modulus of coal does not change with increasing the CO2 pressure from 8.0 MPa to 16.0 MPa [40]. For the same reason, the elastic modulus of coal also changes in a certain methane pressure range. Besides, the adsorption capacity of coal with respect to carbon dioxide is greater than that of methane. Therefore, the experimental methane pressures were set as 0, 1, 2, 3, and 4 MPa. Finally, the elastic modulus tests and the permeability of methane-saturated coal tests were conducted.


## 2.1. Sample Preparation

The two types of coal, including bitumite (GSXB) collected from the Xinbei coal mine located in the northwest of China and anthracite (GZLH) collected from the Linhua coal mine located in the southwest of China, are taken as the experimental coal samples for experimental studies of dynamic elastic modulus and dynamic permeability model. The maceral and vitrinite reflectance analyses of specimens are listed in Table1. For triaxial seepage tests, the samples with the size of Φ50×100mm were cored from large block coal, perpendicular to the bedding.Table 1
Raw coal properties.
TypeCoal rankVitrinite reflectanceRo,max (%)Proximate analysis (wt, %)Coal maceral composition (%)MadAdVdafFCadVILMGSXBBituminous1.452.519.7038.5455.4965.8931.51—2.6GZLHAnthracite2.862.1411.045.8183.7960.3335.37—4.3Notes:Mad: moisture, air-drying basis; Ad: ash yield, air-drying basis; Vdaf: volatile matter dry ash-free basis; FCad: fixed carbon content, air-drying basis; V: vitrinite; I: inertinite; L: liptinite; M: mineral.


## 2.2. Sample Characterization

As mentioned earlier in this paper, the mechanics of methane-saturated coal is closely related to its gas adsorption capacity, and the gas adsorption capacity of coal is affected by its pore structure characteristics [31–33]. To assess the pore structure characteristics of coal samples, a nitrogen adsorption/desorption experiment and a nuclear magnetic resonance (NMR) experiment were conducted.


### 2.2.1. Nitrogen Adsorption/Desorption

To assess the pore structure characteristics of two samples, liquid nitrogen adsorption/desorption experiments were conducted. The liquid nitrogen adsorption and desorption curves are presented in Figure2. The pore structure characteristics of samples were measured using powdered coal particles that passed through an 18~25 millimeter. Figure 2 illustrates that the nitrogen adsorption of coal increases slowly under low relative pressure. As the pressure increases, the isotherms between the adsorption and desorption had a hysteresis effect. The appearance of the adsorption hysteresis loop mainly depends on the capillary condensation that occurs on the solid surface [34, 35]. Finally, the nitrogen adsorption of coal increases fast under high relative pressure. The pore structure characteristics of the two samples are shown in Table 2. The pore volume of GSXB is 0.009 mL/g and the specific surface area is 6.0835 m2/g which is relatively larger than 0.0055 mL/g and 2.3661 m2/g of GZLH, respectively, which can provide broad spaces for gas adsorption.Figure 2
Nitrogen adsorption/desorption curves.Table 2
Results of low-temperature nitrogen adsorption experiments.
Coal samplesGSXBGZLHPore volume (mL/g)0.0090.0055Specific surface area (m2/g)6.08352.3661Mean pore size (nm)5.91769.2980


### 2.2.2. NMR Measurements

To analyze the coal pore distribution, nuclear magnetic resonance (NMR) experiments were conducted. In the magnetic field, the hydrogen atoms present within the fluid can be detected by the transversal relaxation time (T2); thus, the pore properties can be analyzed [36]. T2 can be expressed as [37]
(1)1T2=ρ×SV=ar,where ρ is a constant representing the transverse relaxation strength, S is the pore surface, V is the pore volume, r is the pore radius, and a is a pore shape factor.It can be seen from Equation (1) that larger pores have longer T2, and the higher the T2 spectrum, the more the pores. In addition, continuity between T2 spectrum peaks represents the pore connectivity [38]. For coals, T2<10ms corresponds to micropores and 100 ms corresponds to macropores [39].Figure3 illustrates the coal pore distributions of GSXB and GZLH. The T2 spectrum of GSXB consists of three separated peaks (i.e., P1, P2, and P3), indicating a large number of micropores and a small number of mesopores and macropores. Compared with the T2 spectrum of GSXB, GZLH has the same trend. In contrast, of GZLH is lower, P2′ of GZLH is roughly equivalent to GSXB, and P2′ is higher, which indicates that the GZLH has more macropore and mesopore. The free fluids mainly migrate in macropores, which may result in the higher permeability of GZLH. The continuity between T2 spectrum peaks of GZLH is better, which is a strong indication that micropores, mesopores, and all macropores are interconnected. In conclusion, GZLH has more macropores and fewer micropores, and this kind of unique feature makes GZLH have higher permeability and a lower adsorbability.Figure 3
Results of nuclear magnetic resonance (NMR) experiments.


## 2.2.1. Nitrogen Adsorption/Desorption

To assess the pore structure characteristics of two samples, liquid nitrogen adsorption/desorption experiments were conducted. The liquid nitrogen adsorption and desorption curves are presented in Figure2. The pore structure characteristics of samples were measured using powdered coal particles that passed through an 18~25 millimeter. Figure 2 illustrates that the nitrogen adsorption of coal increases slowly under low relative pressure. As the pressure increases, the isotherms between the adsorption and desorption had a hysteresis effect. The appearance of the adsorption hysteresis loop mainly depends on the capillary condensation that occurs on the solid surface [34, 35]. Finally, the nitrogen adsorption of coal increases fast under high relative pressure. The pore structure characteristics of the two samples are shown in Table 2. The pore volume of GSXB is 0.009 mL/g and the specific surface area is 6.0835 m2/g which is relatively larger than 0.0055 mL/g and 2.3661 m2/g of GZLH, respectively, which can provide broad spaces for gas adsorption.Figure 2
Nitrogen adsorption/desorption curves.Table 2
Results of low-temperature nitrogen adsorption experiments.
Coal samplesGSXBGZLHPore volume (mL/g)0.0090.0055Specific surface area (m2/g)6.08352.3661Mean pore size (nm)5.91769.2980


## 2.2.2. NMR Measurements

To analyze the coal pore distribution, nuclear magnetic resonance (NMR) experiments were conducted. In the magnetic field, the hydrogen atoms present within the fluid can be detected by the transversal relaxation time (T2); thus, the pore properties can be analyzed [36]. T2 can be expressed as [37]
(1)1T2=ρ×SV=ar,where ρ is a constant representing the transverse relaxation strength, S is the pore surface, V is the pore volume, r is the pore radius, and a is a pore shape factor.It can be seen from Equation (1) that larger pores have longer T2, and the higher the T2 spectrum, the more the pores. In addition, continuity between T2 spectrum peaks represents the pore connectivity [38]. For coals, T2<10ms corresponds to micropores and 100 ms corresponds to macropores [39].Figure3 illustrates the coal pore distributions of GSXB and GZLH. The T2 spectrum of GSXB consists of three separated peaks (i.e., P1, P2, and P3), indicating a large number of micropores and a small number of mesopores and macropores. Compared with the T2 spectrum of GSXB, GZLH has the same trend. In contrast, of GZLH is lower, P2′ of GZLH is roughly equivalent to GSXB, and P2′ is higher, which indicates that the GZLH has more macropore and mesopore. The free fluids mainly migrate in macropores, which may result in the higher permeability of GZLH. The continuity between T2 spectrum peaks of GZLH is better, which is a strong indication that micropores, mesopores, and all macropores are interconnected. In conclusion, GZLH has more macropores and fewer micropores, and this kind of unique feature makes GZLH have higher permeability and a lower adsorbability.Figure 3
Results of nuclear magnetic resonance (NMR) experiments.


## 2.3. Experimental Procedure

The experiment system was established, which consists of a loading system, strain monitoring system, and flow acquisition system, as is shown in Figure4.Figure 4
Schematic illustration of the saturation and triaxial setup.Prior to each test, the samples were dried at 100°C for 24 h in a drying oven, and then, they were put into the core holder. The sample was degassed in the core holder under vacuum conditions at 30°C for 24 h. In the process of the test, a constant confining pressure of 5.0 MPa was used for the triaxial tests. The water pressure inside the triaxial cell was provided from a pressure controller. The methane was supplied from a high-pressure CH4 cylinder with a maximum pressure of 13.0 MPa. According to the previous studies, the mechanical strength and elastic modulus of coal gradually decrease with the pore pressure increase in a certain range [29]. For example, the elastic modulus of coal does not change with increasing the CO2 pressure from 8.0 MPa to 16.0 MPa [40]. For the same reason, the elastic modulus of coal also changes in a certain methane pressure range. Besides, the adsorption capacity of coal with respect to carbon dioxide is greater than that of methane. Therefore, the experimental methane pressures were set as 0, 1, 2, 3, and 4 MPa. Finally, the elastic modulus tests and the permeability of methane-saturated coal tests were conducted.


## 3. Modeling

### 3.1. Elastic Modulus Model

It is believed that gas adsorption on coal will result in the decrease of elastic modulus and strength of coal [41–43]. Therefore, it can be assumed that the variation of elastic modulus with pore pressure can be represented by the Langmuir model [29]. The elastic modulus defined in this work is different from that in the previous studies.
(2)ECH4=E−ΔE,where E is the elastic modulus of gas-free coal (E) and ECH4 is the elastic modulus of methane-saturated coal (ECH4). The decrease of elastic modulus caused by gas adsorption can be given as [29]
(3)ΔE=ΔEmax×PPE+P,where P is the gas pressure, ΔEmax is the maximum decrease in the elastic modulus (associated with the maximum adsorption), and PE is a curve-fitting parameter that is similar to the Langmuir pressure defined in Langmuir isotherm.


### 3.2. Permeability Model

To quantitatively analyze the influence of dynamic elastic modulus on permeability, the pore structure of coal was simplified. The typical simplified models mainly included the cube model, the sphere model, the capillary model, and the matchstick model [18, 44–46]. In this paper, the cube model containing a “coal matrix bridge” was proposed, which is composed of matrix blocks of size Lm and fractures with an aperture Lf. The matrix blocks are connected by the “coal matrix bridge” [47].


#### 3.2.1. Gas Adsorption-Induced Swelling Deformation

Due to the “coal matrix bridge”, the matrix deformation is impeded, which also reduces the fracture deformation. Therefore, only a part of gas adsorption-induced coal matrix deformation changes the fracture aperture, while the rest alters the volume of coal mass. In Figure5(c), the light grey area represents the volume before adsorbing. The dark grey area surrounded by the dotted line is the deforming volume after adsorbing. It can be seen that matrix deformation (∆LmS) consists of fracture deformation (∆LfS) and coal mass deformation ∆LbS. To quantify the adsorption-induced fracture deformation, the internal swelling coefficient f0<f<1 is introduced to represent the ratio of adsorption-induced fracture deformation to matrix deformation. Based on the simplified model,
(4)ΔLfS=fΔLmS,ΔLbS=1−fΔLmS.Figure 5
Schematic diagram of gas adsorption-induced internal swelling deformation. (a) Representative elementary volume (REV). (b) Coal structure model. (c) Internal swelling deformation caused by gas adsorption.In this paper, we assume that the changing size of the matrix contains fracture deformation and coal mass deformation. ForLf<<Lm, the strain increments of fracture volume and bulk volume caused by gas adsorption can be written as
(5)ΔεfS=−3ΔLfSLf=−3fΔLmSLf=−LmLffΔεmS,(6)ΔεbS=−3ΔLbSLb=−31−fΔLmSLm+Lf=1−fΔεmS.Under the unconstrained state, the matrix swelling strain caused by gas adsorption∆εmS satisfies the Langmuir-type equation [19, 20].
(7)ΔεmS=εLpmpL+pm−pm0pL+pm0.


#### 3.2.2. Deformation Caused by Effective Stress

The effective stress law of dual-porosity medium can be expressed as [48]
(8)σe=σ−αpf+βpmδij,where σe is effective stress (MPa); σ is the total stress (MPa); δij is Kronecker delta tensor; pf is fracture gas pressure (MPa); pm is matrix gas pressure (MPa); α and β are Biot effective stress coefficients of fracture and pore.
(9)α=1−KKf,β=1−KKm,where K is the bulk modulus of coal mass [K=E/31−2μ] (MPa); Km is the bulk modulus of matrix [Km=Em/31−2μ] (MPa); Kf is the bulk modulus of fracture [Kf=Ef/31−2μ] (MPa).The fracture deformation is the combined effect of gas adsorption and effective stress [18]. Therefore, the fracture deformation can be expressed as
(10)Δεf=ΔεfS+ΔεfE.The deformation influenced by effective stress is expressed as(11)ΔεfE=−ΔσeKf.Substitute Equations (5) and (11) into Equation (10); the fracture deformation can be written as
(12)Δεf=−LmLffεLpmpL+pm−pm0pL+pm0−ΔσeKf.


#### 3.2.3. Permeability Equation

Fracture porosity of simplified cube model can be defined as [49]
(13)ϕf=Lm+Lf3−Lm3Lm+Lf3≅3LfLm.Assume that the matrix size remains constant during coal deformation, and the volume strain of the fracture only changes the fracture aperture, we can obtain(14)ϕfϕf0=Lf+ΔLfLf=1+Δεf.Substitute Equation (12) into Equation (14), we obtain
(15)ϕfϕf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−αpf−pf0−βpm−pm0Kf.The permeability model is obtained based on Cubic law, it reads [50]
(16)kfkf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−αpf−pf0−βpm−pm0Kf3.Substitute Equations (2) and (3) into Equation (16), we obtain
(17)kfkf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−1−E−ΔE/3Kf1−2μpf−pf0−1−E−ΔE/3Km1−2μpm−pm0Kf3.


## 3.1. Elastic Modulus Model

It is believed that gas adsorption on coal will result in the decrease of elastic modulus and strength of coal [41–43]. Therefore, it can be assumed that the variation of elastic modulus with pore pressure can be represented by the Langmuir model [29]. The elastic modulus defined in this work is different from that in the previous studies.
(2)ECH4=E−ΔE,where E is the elastic modulus of gas-free coal (E) and ECH4 is the elastic modulus of methane-saturated coal (ECH4). The decrease of elastic modulus caused by gas adsorption can be given as [29]
(3)ΔE=ΔEmax×PPE+P,where P is the gas pressure, ΔEmax is the maximum decrease in the elastic modulus (associated with the maximum adsorption), and PE is a curve-fitting parameter that is similar to the Langmuir pressure defined in Langmuir isotherm.


## 3.2. Permeability Model

To quantitatively analyze the influence of dynamic elastic modulus on permeability, the pore structure of coal was simplified. The typical simplified models mainly included the cube model, the sphere model, the capillary model, and the matchstick model [18, 44–46]. In this paper, the cube model containing a “coal matrix bridge” was proposed, which is composed of matrix blocks of size Lm and fractures with an aperture Lf. The matrix blocks are connected by the “coal matrix bridge” [47].


### 3.2.1. Gas Adsorption-Induced Swelling Deformation

Due to the “coal matrix bridge”, the matrix deformation is impeded, which also reduces the fracture deformation. Therefore, only a part of gas adsorption-induced coal matrix deformation changes the fracture aperture, while the rest alters the volume of coal mass. In Figure5(c), the light grey area represents the volume before adsorbing. The dark grey area surrounded by the dotted line is the deforming volume after adsorbing. It can be seen that matrix deformation (∆LmS) consists of fracture deformation (∆LfS) and coal mass deformation ∆LbS. To quantify the adsorption-induced fracture deformation, the internal swelling coefficient f0<f<1 is introduced to represent the ratio of adsorption-induced fracture deformation to matrix deformation. Based on the simplified model,
(4)ΔLfS=fΔLmS,ΔLbS=1−fΔLmS.Figure 5
Schematic diagram of gas adsorption-induced internal swelling deformation. (a) Representative elementary volume (REV). (b) Coal structure model. (c) Internal swelling deformation caused by gas adsorption.In this paper, we assume that the changing size of the matrix contains fracture deformation and coal mass deformation. ForLf<<Lm, the strain increments of fracture volume and bulk volume caused by gas adsorption can be written as
(5)ΔεfS=−3ΔLfSLf=−3fΔLmSLf=−LmLffΔεmS,(6)ΔεbS=−3ΔLbSLb=−31−fΔLmSLm+Lf=1−fΔεmS.Under the unconstrained state, the matrix swelling strain caused by gas adsorption∆εmS satisfies the Langmuir-type equation [19, 20].
(7)ΔεmS=εLpmpL+pm−pm0pL+pm0.


### 3.2.2. Deformation Caused by Effective Stress

The effective stress law of dual-porosity medium can be expressed as [48]
(8)σe=σ−αpf+βpmδij,where σe is effective stress (MPa); σ is the total stress (MPa); δij is Kronecker delta tensor; pf is fracture gas pressure (MPa); pm is matrix gas pressure (MPa); α and β are Biot effective stress coefficients of fracture and pore.
(9)α=1−KKf,β=1−KKm,where K is the bulk modulus of coal mass [K=E/31−2μ] (MPa); Km is the bulk modulus of matrix [Km=Em/31−2μ] (MPa); Kf is the bulk modulus of fracture [Kf=Ef/31−2μ] (MPa).The fracture deformation is the combined effect of gas adsorption and effective stress [18]. Therefore, the fracture deformation can be expressed as
(10)Δεf=ΔεfS+ΔεfE.The deformation influenced by effective stress is expressed as(11)ΔεfE=−ΔσeKf.Substitute Equations (5) and (11) into Equation (10); the fracture deformation can be written as
(12)Δεf=−LmLffεLpmpL+pm−pm0pL+pm0−ΔσeKf.


### 3.2.3. Permeability Equation

Fracture porosity of simplified cube model can be defined as [49]
(13)ϕf=Lm+Lf3−Lm3Lm+Lf3≅3LfLm.Assume that the matrix size remains constant during coal deformation, and the volume strain of the fracture only changes the fracture aperture, we can obtain(14)ϕfϕf0=Lf+ΔLfLf=1+Δεf.Substitute Equation (12) into Equation (14), we obtain
(15)ϕfϕf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−αpf−pf0−βpm−pm0Kf.The permeability model is obtained based on Cubic law, it reads [50]
(16)kfkf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−αpf−pf0−βpm−pm0Kf3.Substitute Equations (2) and (3) into Equation (16), we obtain
(17)kfkf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−1−E−ΔE/3Kf1−2μpf−pf0−1−E−ΔE/3Km1−2μpm−pm0Kf3.


## 3.2.1. Gas Adsorption-Induced Swelling Deformation

Due to the “coal matrix bridge”, the matrix deformation is impeded, which also reduces the fracture deformation. Therefore, only a part of gas adsorption-induced coal matrix deformation changes the fracture aperture, while the rest alters the volume of coal mass. In Figure5(c), the light grey area represents the volume before adsorbing. The dark grey area surrounded by the dotted line is the deforming volume after adsorbing. It can be seen that matrix deformation (∆LmS) consists of fracture deformation (∆LfS) and coal mass deformation ∆LbS. To quantify the adsorption-induced fracture deformation, the internal swelling coefficient f0<f<1 is introduced to represent the ratio of adsorption-induced fracture deformation to matrix deformation. Based on the simplified model,
(4)ΔLfS=fΔLmS,ΔLbS=1−fΔLmS.Figure 5
Schematic diagram of gas adsorption-induced internal swelling deformation. (a) Representative elementary volume (REV). (b) Coal structure model. (c) Internal swelling deformation caused by gas adsorption.In this paper, we assume that the changing size of the matrix contains fracture deformation and coal mass deformation. ForLf<<Lm, the strain increments of fracture volume and bulk volume caused by gas adsorption can be written as
(5)ΔεfS=−3ΔLfSLf=−3fΔLmSLf=−LmLffΔεmS,(6)ΔεbS=−3ΔLbSLb=−31−fΔLmSLm+Lf=1−fΔεmS.Under the unconstrained state, the matrix swelling strain caused by gas adsorption∆εmS satisfies the Langmuir-type equation [19, 20].
(7)ΔεmS=εLpmpL+pm−pm0pL+pm0.


## 3.2.2. Deformation Caused by Effective Stress

The effective stress law of dual-porosity medium can be expressed as [48]
(8)σe=σ−αpf+βpmδij,where σe is effective stress (MPa); σ is the total stress (MPa); δij is Kronecker delta tensor; pf is fracture gas pressure (MPa); pm is matrix gas pressure (MPa); α and β are Biot effective stress coefficients of fracture and pore.
(9)α=1−KKf,β=1−KKm,where K is the bulk modulus of coal mass [K=E/31−2μ] (MPa); Km is the bulk modulus of matrix [Km=Em/31−2μ] (MPa); Kf is the bulk modulus of fracture [Kf=Ef/31−2μ] (MPa).The fracture deformation is the combined effect of gas adsorption and effective stress [18]. Therefore, the fracture deformation can be expressed as
(10)Δεf=ΔεfS+ΔεfE.The deformation influenced by effective stress is expressed as(11)ΔεfE=−ΔσeKf.Substitute Equations (5) and (11) into Equation (10); the fracture deformation can be written as
(12)Δεf=−LmLffεLpmpL+pm−pm0pL+pm0−ΔσeKf.


## 3.2.3. Permeability Equation

Fracture porosity of simplified cube model can be defined as [49]
(13)ϕf=Lm+Lf3−Lm3Lm+Lf3≅3LfLm.Assume that the matrix size remains constant during coal deformation, and the volume strain of the fracture only changes the fracture aperture, we can obtain(14)ϕfϕf0=Lf+ΔLfLf=1+Δεf.Substitute Equation (12) into Equation (14), we obtain
(15)ϕfϕf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−αpf−pf0−βpm−pm0Kf.The permeability model is obtained based on Cubic law, it reads [50]
(16)kfkf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−αpf−pf0−βpm−pm0Kf3.Substitute Equations (2) and (3) into Equation (16), we obtain
(17)kfkf0=1−3fεLϕf0pmpL+pm−pm0pL+pm0−σ−σ0−1−E−ΔE/3Kf1−2μpf−pf0−1−E−ΔE/3Km1−2μpm−pm0Kf3.


## 4. Model Validations

In this section, the validation of the elastic modulus model and permeability model is evaluated against the data sets measured in the laboratory.


### 4.1. Validation of Elastic Modulus Model

At a constant confining stress of 5.0 MPa, Equation (2) is adopted to fit the data sets measured in the laboratory of two types of coal samples. It can be seen from Figure 6 that the fitting result of GSXB and GZLH is successful, using a least-square regression method with an acceptable coefficient of determination (R2≈0.99). The result indicates that the new elastic modulus model can be used to predict the coal elastic modulus under the conditions of constant confining pressure. From Figure 6, we can see that the elastic modulus of CH4 saturated coal decreases as the pressure increases. However, the reduction becomes smaller as methane pressure. Based on previous research, it is believed that the reduction in elastic modulus results from the interaction of CH4 with the coal microstructure. The values of ΔEmax and PE of GZLH are estimated to be 2.60 GPa and 5.94 MPa and GSXB are estimated to be 3.28 GPa and 2.36 MPa, respectively. There is a difference between the two types of coal samples, which implies that ΔEmax will depend on the type, rank, molecular composition, etc. of the coal.Figure 6
Matching results of the dynamic elastic modulus model with laboratory test data under constant external stress conditions.


### 4.2. Validation of Permeability Model

Under the constant external stress condition, the increment of the mean stress equals zero.(18)σ−σ0=0.Moreover, the permeability evolution is governed by Equation (17). The fitting results are depicted in Figure 7. The parameters used in this paper are listed in Table 3.Figure 7
Matching results of the dynamic permeability model with laboratory test data under constant external stress conditions.
(a)(b)Table 3
Parameters used for data matching under constant external stress conditions.
TypeGSXBGZLHLangmuir strain constantεL0.012660.04Fracture porosity∅f00.0050.003Langmuir pressurepL3.85 MPa5.94 MPaBulk modulus of fractureKf9.5 MPa1.8 MPaBulk modulus of matrixKm14.3 MPa20 MPaPoisson’s ratioμ0.250.18From Figure7, we can see that the dynamic permeability model fits well with the two sets of data. The internal swelling coefficient f of GSXB and GZLH is 0.007898 and 0.2094, respectively. It means 8.98% and 20.94% of the matrix deformation caused by gas adsorption are used to alter the fracture permeability. In addition, it indicates that the internal swelling coefficient corresponding to different coal samples is different.


## 4.1. Validation of Elastic Modulus Model

At a constant confining stress of 5.0 MPa, Equation (2) is adopted to fit the data sets measured in the laboratory of two types of coal samples. It can be seen from Figure 6 that the fitting result of GSXB and GZLH is successful, using a least-square regression method with an acceptable coefficient of determination (R2≈0.99). The result indicates that the new elastic modulus model can be used to predict the coal elastic modulus under the conditions of constant confining pressure. From Figure 6, we can see that the elastic modulus of CH4 saturated coal decreases as the pressure increases. However, the reduction becomes smaller as methane pressure. Based on previous research, it is believed that the reduction in elastic modulus results from the interaction of CH4 with the coal microstructure. The values of ΔEmax and PE of GZLH are estimated to be 2.60 GPa and 5.94 MPa and GSXB are estimated to be 3.28 GPa and 2.36 MPa, respectively. There is a difference between the two types of coal samples, which implies that ΔEmax will depend on the type, rank, molecular composition, etc. of the coal.Figure 6
Matching results of the dynamic elastic modulus model with laboratory test data under constant external stress conditions.


## 4.2. Validation of Permeability Model

Under the constant external stress condition, the increment of the mean stress equals zero.(18)σ−σ0=0.Moreover, the permeability evolution is governed by Equation (17). The fitting results are depicted in Figure 7. The parameters used in this paper are listed in Table 3.Figure 7
Matching results of the dynamic permeability model with laboratory test data under constant external stress conditions.
(a)(b)Table 3
Parameters used for data matching under constant external stress conditions.
TypeGSXBGZLHLangmuir strain constantεL0.012660.04Fracture porosity∅f00.0050.003Langmuir pressurepL3.85 MPa5.94 MPaBulk modulus of fractureKf9.5 MPa1.8 MPaBulk modulus of matrixKm14.3 MPa20 MPaPoisson’s ratioμ0.250.18From Figure7, we can see that the dynamic permeability model fits well with the two sets of data. The internal swelling coefficient f of GSXB and GZLH is 0.007898 and 0.2094, respectively. It means 8.98% and 20.94% of the matrix deformation caused by gas adsorption are used to alter the fracture permeability. In addition, it indicates that the internal swelling coefficient corresponding to different coal samples is different.


## 5. Parameter Sensitivity Analysis

In this section, the parameter sensitivity for the dynamic permeability model under the condition of constant confining pressure was analyzed. GSXB coal sample was selected as the object, and we carried out the parameter sensitivity for its 6 most representative parameters (internal swelling coefficientf, Langmuir strain constant εL, initial fracture porosity ∅f0, Langmuir pressure constant PL, bulk modulus of fracture Kf, and bulk modulus of matrix Km).When analyzing the parameter sensitivity, due to the condition limitation, some parameters cannot be obtained from the field. To make the parameters more representative, we summarized some parameter ranges of the coal samples given by other papers. Then, we used them as the reference during sensitivity analysis. Table4 depicts the value and the possible ranges of each parameter.Table 4
Ranges of key parameters.
ReferenceLangmuir strain constant (εL)Langmuir pressure constant (PL) (MPa)Bulk modulus of fracture (Kf) (MPa)Seidle and Huitt [50]0.01363.451.95-16.13Harpalani and Chen [22]0.0273.45—Levine [51]0.0075-0.01364.3-6.3—Zahner [52]0.0051-0.04392.1-5.3—Mavor and Vaughn [53]0.01274.31—Palmer and Mansoori [19]0.01284.31—Shi and Durucan [54]0.012662.55, 4.313.45-3.76Shi and Durucan [20]0.012664.317.18-10.95Palmer and Gunter [55]0.0051-0.04392.1-5.31—Palmer [56]0.0333.45—Shi and Durucan [57]0.012664.311.72-8.62Liu and Rutqvist [21]0.01292.552.33Liu et al. [58]0.010754.1610.87Liu and Harpalani [59]——6.67-13.33Shi et al. [60]0.010754.162.06-5.15Range0.0051-0.04392.1-6.31.72-16.13In order to analyze the influence of different parameters on the evolution law of coal permeability, the other parameters were set as constant. The results are shown in Figure8.Figure 8
Effect of the different parameters on coal permeability.
(a)
Internal swelling coefficient(b)
Langmuir strain constant(c)
Initial fracture porosity(d)
Langmuir pressure constant(e)
Bulk modulus of fracture(f)
Bulk modulus of matrixTake the internal swelling coefficient (f) as an example; overall, the internal swelling coefficient (f) has a negative effect on the permeability by influencing the adsorption-induced matrix swelling deformation. As is shown in Figure 8(a), when f≤0.005, coal permeability continues to increase with the pressure. However, when f≥0.005, coal permeability increases at first and then decreases with the pressure (grey area in the figure). When f is low, the effect of adsorption-induced matrix swelling deformation on the permeability is small. Hence, effective stress plays a leading role in coal permeability. The effective stress increases with pressure, and permeability increases. When f is large enough, the effect of adsorption-induced matrix swelling deformation dominates permeability in the low-pressure range; therefore, the permeability decreases slightly. With the increase of gas pressure, the effect of effective stress is strengthened gradually and the permeability increases. Moreover, the larger pressure, the larger increase of permeability. When the pressure reaches a certain value, effective stress plays the dominant role, and the permeability increases with the pressure.The Langmuir strain constantεL mainly has a negative effect on the permeability by influencing the adsorption-induced matrix swelling deformation. The Langmuir pressure constant PL has a positive effect on the permeability by influencing the adsorption-induced matrix swelling deformation. The bulk modulus of fracture Kf and the bulk modulus of matrix Km have a positive effect on the permeability by influencing the effective stress. The initial fracture porosity ∅f0 has unknown effects on the permeability by influencing the adsorption-induced matrix swelling deformation, which needed further study.


## 6. Conclusions

(1)
The dynamic elastic modulus modelECH4=E−ΔE was obtained through model derivation. Its form indicates that coal elastic modulus shows linear relation with pore pressure. Moreover, with the gas pressure, the elastic modulus gets lower(2)
The dynamic permeability model under the constant confining pressure condition is obtained through model derivation. From the equation structure, it can be known that permeability is affected by both effective stress and adsorption-induced matrix swelling deformation. The third coupling path, namely, the dynamic change of the coal elastic modulus, affects the permeability by changing the effect of effective stress. The elastic modulus under the pore pressure condition is always lower than the initial elastic modulus, which leads to the enhancement of effective stress in the dynamic permeability model. Thus, the inflection point in the pore pressure-permeability figure is not obvious(3)
When validating the parameter sensitivity, different coefficients mainly have three effects, namely, positive effect (Langmuir pressure constantPL, bulk modulus of fracture Kf, and bulk modulus of matrix Km), negative effect (internal swelling coefficient f, Langmuir strain constant εL), and unknown effect (initial fracture porosity ∅f0) on the permeability by influencing the adsorption-induced matrix swelling deformation and effective stress



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