# A Dual Fractal Poroelastic Model for Characterizing Fluid Flow in Fractured Coal Masses

Authors: Guannan Liu; Dayu Ye; Feng Gao; Jishan Liu
Journal: Geofluids
 (2020)

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## Abstract

In the process of coalbed methane exploitation, the fracture and pore structure is the key problem that affects the permeability of coalbed. At present, the coupling effect of fracture and pore structure and in situ stress is seldom considered in the study of coal seam permeability. In this paper, the fractal seepage model is coupled with coal deformation, and the adsorption expansion effect is considered. A multifield coupling model considering the influence of matrix and fracture structure is established. Then, the influence of pore structure parameters of main fracture on macropermeability is analyzed, including (1) fractal dimension of fracture length, (2) maximum fracture length, (3) fractal dimension of throat diameter, and (4) fractal dimension of throat bending. At the same time, the simulation results are compared with the results of Darcy’s uniform permeability model. The results show that the permeability calculated by the proposed model is significantly different from that calculated by the traditional cubic model. Under the action of in situ stress, when the porosity and other parameters remain unchanged, the macropermeability of coal is in direct proportion to the fractal dimension of coal fracture length, the fractal dimension of throat diameter, and the maximum fracture length and in inverse proportion to the fractal dimension of coal throat curvature.


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## Body

## 1. Introduction

Coalbed methane is one of the main high-quality clean energy sources in the world. In the process of coalbed methane extraction, the fracture and pore network of coal seam is the main space of its migration [1, 2]. Quantitative analysis of the influence of fracture pore structure of coal seam on the macropermeability is the key to improve the extraction rate of coalbed methane and ensure the safety of production [3–5].The fracture network of coal is highly complex [6–8]. In recent years, researchers have found that a large number of coal and coal fractures have fractal characteristics. In nature, the distribution, length, opening, and orientation of fractures are often random and disordered. It is a great challenge to find its analytical solution. Fractal geometry theory has successfully studied the percolation, heat, and electric conduction of fluids in porous media, as well as the physical quantities related to the roughness of porous media surface, nanofluids, pool boiling, and dendriform bifurcation network [9–11]. Fractal geometry is considered as an effective method to quantitatively describe the structure of fracture network.Combined with the first law of thermodynamics and fractal description, Deinert et al. [12, 13] simplified the force balance analysis process of capillary force. And the relationship between capillary force and saturation is established by introducing pore volume and fractal dimension of pore surface. Xu et al. [14–16] analyzed the percolation and heat conduction properties of bifurcated tree networks based on fractal geometry. The bifurcated tree network is an ideal symmetric network, which is very similar to the actual tree branching network in structure. Ishibashi et al. [17] used fractal theory to systematically study the flow characteristics and opening of fractures under different pressures. The results show that the contact area depends on the fracture opening; the fracture permeability meets the power law distribution. By introducing the fractal dimension of pore space, Costa [18] obtained the relationship between Kozeny-Carman equation constant and fractal permeability and porosity. Guarracino [19] established the fracture network permeability model by using the Sierpinski blanket fractal model and predicted the conductivity of hydraulic fracturing. Miao et al. [20–22] extended the theory of fractal porous media to fracture network and established the fractal model of fracture network and dual porous media, discussing the influence of fracture length fractal dimension, pore throat tortuosity fractal dimension, and other structural parameters on permeability. Based on the fractal theory, Wu [23] et al. obtained the expansion deformation of double porosity coal and the nonuniform expansion deformation caused by the pore distribution. Denis et al. [24] observed the deformation of coal after two months of injecting carbon dioxide with a pressure of 3.8 MPa and found that the average deformation of coal is 0.34%.However, the above research method did not take into account the influence of in situ stress and gas adsorption and desorption phenomenon, which may cause deformation of coal body and change the permeability of coal seam. Therefore, based on fractal permeability models, we establish a multifield coupling model of single porosity seepage [25]. In this work, we extend the coupling model of fractal seepage to dual porosity. The influence of structure parameters on seepage is analyzed: (1) fractal dimension of fracture length, (2) maximum fracture length, (3) fractal dimension of pore diameter, and (4) fractal dimension of tortuosity.


## 2. Governing Equations

The fractal permeability model is deduced by considering the structural parameters and coupling them with the gas seepage model of adsorption deformation. In order to simplify the calculation of partial differential equations, we make the following assumptions [26]:
(1)
Coal deformation satisfies linear elasticity(2)
The influence of heterogeneity and anisotropy is not considered(3)
The adsorption deformation is small(4)
The viscosity of gas is a constant(5)
The gas in the pores is saturated(6)
All streams of flow are in parallel and unmixed layers


### 2.1. Governing Equations of Gas Flow

It is well known that coal seam resource storage can be divided into adsorption and free phases according to the existing state, in which the total quality governing equation can be expressed as [23]
(1)mm=ρϕgmm+ρgaρcVLpmpm+PL,(2)mf=ρϕgff+ρgaρcVLpfpf+PL,where the mass of gases in matrix and fracture is expressed by mm and mf and gas density is ρgm and ρgf. ρga is the gas density under standard conditions; ρc is the coal density.According to the of conservation of mass, the equation of gas quality control can be expressed as(3)∂m∂t+∇⋅ρgqg=Qs,where t is the total time, ρg is the gas density, and qg is the velocity of Darcy flow. We neglect the effect of gravity on the whole; the flow rate of Darcy flow can be expressed as follows:
(4)q̇g=−kμ∇p.In Equation (4), q̇g is the derivative of qg. The simultaneous formula Equations (1), (2), (3), and (4) shows that the mass conservation equation of gases can be expressed as follows:
(5)ϕm+ρcpaVLPLpm+PL2∂pm∂t+pm∂ϕm∂t−∇⋅kmμpm∇pm=Qs,ϕf+ρcpaVLPLpf+PL2∂pf∂t+pf∂ϕf∂t−∇⋅kfμpf∇pf=−Qs,where pa is atmospheric pressure, pa=101.325kPa.


### 2.2. Governing Equations of Coal Deformation

Based on the assumption at the beginning of this chapter, we can simplify the relationship between strain and displacement into the following equation [27]:
(6)εij=12ui,j+uj,i,where εij is the component of the total strain in different directions; ui,j and uj,i are the displacement components. Therefore, the deformation equilibrium equation can be expressed as
(7)σij+fi=0,where σij is the component of the stress tensor and fi is the component of the force. The relationship between adsorption and induced deformation can be expressed as follows:
(8)εij=12Gσij−16G−19kσkkδij+α3Kpδij+εs3δij,where εs is the gas adsorption strain. According to the force element analysis, σkk=σ11+σ22+σ33. G is the shear modulus of elasticity. K is the bulk modulus of coal, and there are K=E/31−2ν. ν is Poisson’s ratio of coal. α is the Biot coefficient, α=1−K/Ks, where Ks is the bulk modulus of coal particles and δij is the Crohneck function.By simplifying Equation (8) and combining Equations (6) and (7), we can obtain that
(9)Gui,kk+G1−2μuk,ki−αpi−Kεs,i+fi=0.Equation (9) is the governing equation of coal deformation.


### 2.3. Porosity Model

As a porous medium, the volume of coal can be divided into two parts: pore volume and coal volume. Under the influence of adsorption deformation, the pore structure undergoes corresponding deformation, which satisfies the Langmuir equation [23]:
(10)εs=εLpPL+p.Therefore, the volume strain equation of coal can be expressed as(11)ΔVV=−1Kσ¯−αp+εs.Thus,(12)ΔVpVp=−1Kpσ¯−1−KpKsp+εs,where εL is Langmuir’s volumetric strain, Vp is the volume of pore in medium coal, Vc is the volume of coal, and σ¯ is the average stress.If the strain of coal attracted by the solution is assumed to have the same effect as the pore strain mentioned above, that is,(13)Kp=ϕαK.At the same time, according to the definition of porosity, we can get the following two equations:(14)ΔVV=ΔVcVc+Δϕ1−ϕ,(15)ΔVpVp=ΔVcVc+Δϕϕ1−ϕ.Combine Equations (11), (12), (14), and (15), the overall volume strain is 0 in the initial state. We can obtain that porosity can be expressed as
(16)ϕ=11+S1+S0ϕ0+αS−S0,where S=εv+p/Ks−εs and S0=p0/Ks+εLp0/p0+pL. When S≪1 and S0≪1, Equation (16) can be simplified as follows:
(17)ϕ=ϕ0+αεv+p−p0Ks+εLPLp0−pp0+PLp+PL.


### 2.4. Fractal Permeability Model

In this section, based on fractal geometry, a coal permeability model considering fracture and microstructure is established. Based on this model, the influence of pore structure on macropermeability in time and space is analyzed.The fractal dimension of pore throat is a fraction between 1 and 2. With the increase of fractal dimension, the coverage and complexity of pore throat also increase correspondingly. According to the fractal model proposed by Miao et al. [21], the fractal power relation can be expressed as follows:
(18)N≥λ∝λ−Df,where λ is the equivalent radius of the pore, λmin≤λ≤λmax. N is the pore number of the object of study. Df is the fractal dimension of the pore in the region. For two-dimensional object of study, 1<Df<2. Therefore, Equation (18) can also be rewritten as follows:
(19)N≥λ=kcλ−Df.In Equation (19), kc is a positive proportional coefficient. From Equation (19), it can be concluded that the differential equation for the number of voids in the range of λ,λ+dλ is
(20)dNλ=−kcDfλ−Df−1dλ.Therefore, combined with Equation (20), the regional pore probability density function can be expressed as follows:
(21)fλ=dNλNtdλ=−kcNtDfλ−Df−1.Among them,Nt is the total number of regional holes. The normalized probability density function can be obtained as follows:
(22)−∫λminλmaxkcNtDfλ−Df−1dλ=1.Ifλmin≪λmax, Equation (22) can be simplified as follows:
(23)kc=NtλminDf.Meanwhile, natural pore network usually satisfiesλmin/λmax≤10−2. According to the fractal distribution of porous media, the number of pores can be expressed as
(24)Ntλ=λmaxλminDf.Combine Equations (23) and (24), we can obtain that
(25)kc=λmaxDf.By substituting Equation (25) into Equation (22), we can obtain that
(26)dNλ=−DfλmaxDfλ−Df−1dλ.Equation (26) is a fractal power expression of pore size. Yang et al. concluded that the length of throat satisfies the following fractal distribution:
(27)Ltλ=λ1−DTL0DT,where λ is the length of the throat, and Ltλ is the length of the throat along the flow direction. DT is the fractal dimension of the throat tortuosity, 1<DT<2.According to Mitra et al. [28], the relationship between pore size and fractal dimension is as follows:
(28)Df=2−lnϕlnλmin/λmax.The velocity in the throat can be calculated by Hagen-Poiseuille formula:(29)qλ=π128ΔPLtλλ4μ,where ΔP is the pressure difference between the inlet and outlet of the throat. Combining Equation (27) and (29), we can obtain that the total flow control equation of microthroat is
(30)Q=−∫λminλmaxqλdNλ=π128ΔPμAL0L01−DTADf3+DT−Dfλmax3+DT1−λminλmaxDfλminλmax3+DT−2Df.Because the two-dimensional throat pore satisfies1<DT<2 and 1<Df<2, it can be obtained 3+DT−2Df>0. Therefore, we can obtain that 0<λmin/λmax3+DT−2Df<1. So Equation (30) can be simplified to
(31)Q=−∫λminλmaxqλdNλ=π128ΔPμAL0L01−DTADf3+DT−Dfλmax3+DT,where A is the cross-sectional area of the throat. According to Darcy’s, we can obtain that
(32)Q=kAμΔpL0.Therefore, the fractal governing equations of pore permeability can be obtained by combining Equations (31) and (32):
(33)k=μL0QΔPA=π128L01−DTADf3+DT−Dfλmax3+DT.The cross-sectional area of the matrix element is [23]
(34)Am=Apϕm=πDpλmax242−Dp1−ϕmϕm.And the probability density of fracture is expressed as(35)−dNlNt=kNtDfl−Df+1dl.In Equation (35), Nt represents the total number of fracture in the coal. The normalization of probability density function can be obtained as follows:
(36)kNtlmin−Df−lmax−Df=kNt1lminDf1−lminlmaxDf=1.whenlmin≪lmax, Equation (36) can be simplified as follows:
(37)k=NflminDf.From Equations (34) and (37), we can obtain that
(38)βDllmax21−ϕf2−Dfϕf=n⋅πDfλmax21−ϕm42−Dpϕm.Ignoring the interaction between fluid in the matrix pore and fluid in the fracture network [29], if the flow rate into the fracture wall equals the flow rate out of the fracture wall, the total flow rate through the fracture network is as follows:
(39)Q=Qt,m+Qf=nπ128L01−DtμΔpL0Dp3+Dp−DTλmax3+DT+β312μDf1−cos2αsin2α4−DfΔpL0lmax4.Thus,(40)Qt,m=nQm=nπ128L01−DtμΔpL0Dp3+Dp−DTλmax3+DT.By Darcy’s law, it can be obtained that(41)Q=KAfμΔpL0.Combining Equations (39) and (41), we can obtain that
(42)K=π128L01−DtAmDp3+Dp−DTλmax3+DT+β312AfDf1−cos2αsin2α4−Dflmax4.Based on the assumption that the coal matrix is impermeable, i.e., satisfiesϕm=0, therefore, Equation (42) can be simplified as follows:
(43)Kf=β312AfDf1−cos2αsin2α4−Dflmax4.Equation (43) is the fractal governing equation for fracture permeability. Among them, the permeability of the dynamic permeability fractal model is affected by both pore and fracture, but the total permeability is not simply the sum of them.


### 2.5. Mass Transfer between Pores and Fractures

The mass transfer function between the fracture system and the pore system can be expressed as [30]
(44)Q=ωPm−Pf,where ω=aVρmKm/μ, ρm is the volume of gas, and Km is the permeability of the pore system. μ is the viscosity of fluid, and a is the shape factor of matrix fracture transfer. Pm is the pressure of pore structure, and Pf is the pressure of fracture structure.In the coupling process of the above physical fields, firstly, the pressure in the volume strain equations is governed by the coal porosity and permeability equations, and the volume stress and volume strain determine the distribution of coal porosity; furthermore, the permeability of the structure-permeability model affects the coal permeability and porosity equations, and the coal porosity on the contrary determines the permeability equations of the structure-permeability model.Coupling and unification between multiple physical fields are achieved through four parameters: pressure, porosity, strain, and stress. The coupling diagram of the fractal seepage model is shown in Figure1.Figure 1
Coupling relation of the fractal model.


## 2.1. Governing Equations of Gas Flow

It is well known that coal seam resource storage can be divided into adsorption and free phases according to the existing state, in which the total quality governing equation can be expressed as [23]
(1)mm=ρϕgmm+ρgaρcVLpmpm+PL,(2)mf=ρϕgff+ρgaρcVLpfpf+PL,where the mass of gases in matrix and fracture is expressed by mm and mf and gas density is ρgm and ρgf. ρga is the gas density under standard conditions; ρc is the coal density.According to the of conservation of mass, the equation of gas quality control can be expressed as(3)∂m∂t+∇⋅ρgqg=Qs,where t is the total time, ρg is the gas density, and qg is the velocity of Darcy flow. We neglect the effect of gravity on the whole; the flow rate of Darcy flow can be expressed as follows:
(4)q̇g=−kμ∇p.In Equation (4), q̇g is the derivative of qg. The simultaneous formula Equations (1), (2), (3), and (4) shows that the mass conservation equation of gases can be expressed as follows:
(5)ϕm+ρcpaVLPLpm+PL2∂pm∂t+pm∂ϕm∂t−∇⋅kmμpm∇pm=Qs,ϕf+ρcpaVLPLpf+PL2∂pf∂t+pf∂ϕf∂t−∇⋅kfμpf∇pf=−Qs,where pa is atmospheric pressure, pa=101.325kPa.


## 2.2. Governing Equations of Coal Deformation

Based on the assumption at the beginning of this chapter, we can simplify the relationship between strain and displacement into the following equation [27]:
(6)εij=12ui,j+uj,i,where εij is the component of the total strain in different directions; ui,j and uj,i are the displacement components. Therefore, the deformation equilibrium equation can be expressed as
(7)σij+fi=0,where σij is the component of the stress tensor and fi is the component of the force. The relationship between adsorption and induced deformation can be expressed as follows:
(8)εij=12Gσij−16G−19kσkkδij+α3Kpδij+εs3δij,where εs is the gas adsorption strain. According to the force element analysis, σkk=σ11+σ22+σ33. G is the shear modulus of elasticity. K is the bulk modulus of coal, and there are K=E/31−2ν. ν is Poisson’s ratio of coal. α is the Biot coefficient, α=1−K/Ks, where Ks is the bulk modulus of coal particles and δij is the Crohneck function.By simplifying Equation (8) and combining Equations (6) and (7), we can obtain that
(9)Gui,kk+G1−2μuk,ki−αpi−Kεs,i+fi=0.Equation (9) is the governing equation of coal deformation.


## 2.3. Porosity Model

As a porous medium, the volume of coal can be divided into two parts: pore volume and coal volume. Under the influence of adsorption deformation, the pore structure undergoes corresponding deformation, which satisfies the Langmuir equation [23]:
(10)εs=εLpPL+p.Therefore, the volume strain equation of coal can be expressed as(11)ΔVV=−1Kσ¯−αp+εs.Thus,(12)ΔVpVp=−1Kpσ¯−1−KpKsp+εs,where εL is Langmuir’s volumetric strain, Vp is the volume of pore in medium coal, Vc is the volume of coal, and σ¯ is the average stress.If the strain of coal attracted by the solution is assumed to have the same effect as the pore strain mentioned above, that is,(13)Kp=ϕαK.At the same time, according to the definition of porosity, we can get the following two equations:(14)ΔVV=ΔVcVc+Δϕ1−ϕ,(15)ΔVpVp=ΔVcVc+Δϕϕ1−ϕ.Combine Equations (11), (12), (14), and (15), the overall volume strain is 0 in the initial state. We can obtain that porosity can be expressed as
(16)ϕ=11+S1+S0ϕ0+αS−S0,where S=εv+p/Ks−εs and S0=p0/Ks+εLp0/p0+pL. When S≪1 and S0≪1, Equation (16) can be simplified as follows:
(17)ϕ=ϕ0+αεv+p−p0Ks+εLPLp0−pp0+PLp+PL.


## 2.4. Fractal Permeability Model

In this section, based on fractal geometry, a coal permeability model considering fracture and microstructure is established. Based on this model, the influence of pore structure on macropermeability in time and space is analyzed.The fractal dimension of pore throat is a fraction between 1 and 2. With the increase of fractal dimension, the coverage and complexity of pore throat also increase correspondingly. According to the fractal model proposed by Miao et al. [21], the fractal power relation can be expressed as follows:
(18)N≥λ∝λ−Df,where λ is the equivalent radius of the pore, λmin≤λ≤λmax. N is the pore number of the object of study. Df is the fractal dimension of the pore in the region. For two-dimensional object of study, 1<Df<2. Therefore, Equation (18) can also be rewritten as follows:
(19)N≥λ=kcλ−Df.In Equation (19), kc is a positive proportional coefficient. From Equation (19), it can be concluded that the differential equation for the number of voids in the range of λ,λ+dλ is
(20)dNλ=−kcDfλ−Df−1dλ.Therefore, combined with Equation (20), the regional pore probability density function can be expressed as follows:
(21)fλ=dNλNtdλ=−kcNtDfλ−Df−1.Among them,Nt is the total number of regional holes. The normalized probability density function can be obtained as follows:
(22)−∫λminλmaxkcNtDfλ−Df−1dλ=1.Ifλmin≪λmax, Equation (22) can be simplified as follows:
(23)kc=NtλminDf.Meanwhile, natural pore network usually satisfiesλmin/λmax≤10−2. According to the fractal distribution of porous media, the number of pores can be expressed as
(24)Ntλ=λmaxλminDf.Combine Equations (23) and (24), we can obtain that
(25)kc=λmaxDf.By substituting Equation (25) into Equation (22), we can obtain that
(26)dNλ=−DfλmaxDfλ−Df−1dλ.Equation (26) is a fractal power expression of pore size. Yang et al. concluded that the length of throat satisfies the following fractal distribution:
(27)Ltλ=λ1−DTL0DT,where λ is the length of the throat, and Ltλ is the length of the throat along the flow direction. DT is the fractal dimension of the throat tortuosity, 1<DT<2.According to Mitra et al. [28], the relationship between pore size and fractal dimension is as follows:
(28)Df=2−lnϕlnλmin/λmax.The velocity in the throat can be calculated by Hagen-Poiseuille formula:(29)qλ=π128ΔPLtλλ4μ,where ΔP is the pressure difference between the inlet and outlet of the throat. Combining Equation (27) and (29), we can obtain that the total flow control equation of microthroat is
(30)Q=−∫λminλmaxqλdNλ=π128ΔPμAL0L01−DTADf3+DT−Dfλmax3+DT1−λminλmaxDfλminλmax3+DT−2Df.Because the two-dimensional throat pore satisfies1<DT<2 and 1<Df<2, it can be obtained 3+DT−2Df>0. Therefore, we can obtain that 0<λmin/λmax3+DT−2Df<1. So Equation (30) can be simplified to
(31)Q=−∫λminλmaxqλdNλ=π128ΔPμAL0L01−DTADf3+DT−Dfλmax3+DT,where A is the cross-sectional area of the throat. According to Darcy’s, we can obtain that
(32)Q=kAμΔpL0.Therefore, the fractal governing equations of pore permeability can be obtained by combining Equations (31) and (32):
(33)k=μL0QΔPA=π128L01−DTADf3+DT−Dfλmax3+DT.The cross-sectional area of the matrix element is [23]
(34)Am=Apϕm=πDpλmax242−Dp1−ϕmϕm.And the probability density of fracture is expressed as(35)−dNlNt=kNtDfl−Df+1dl.In Equation (35), Nt represents the total number of fracture in the coal. The normalization of probability density function can be obtained as follows:
(36)kNtlmin−Df−lmax−Df=kNt1lminDf1−lminlmaxDf=1.whenlmin≪lmax, Equation (36) can be simplified as follows:
(37)k=NflminDf.From Equations (34) and (37), we can obtain that
(38)βDllmax21−ϕf2−Dfϕf=n⋅πDfλmax21−ϕm42−Dpϕm.Ignoring the interaction between fluid in the matrix pore and fluid in the fracture network [29], if the flow rate into the fracture wall equals the flow rate out of the fracture wall, the total flow rate through the fracture network is as follows:
(39)Q=Qt,m+Qf=nπ128L01−DtμΔpL0Dp3+Dp−DTλmax3+DT+β312μDf1−cos2αsin2α4−DfΔpL0lmax4.Thus,(40)Qt,m=nQm=nπ128L01−DtμΔpL0Dp3+Dp−DTλmax3+DT.By Darcy’s law, it can be obtained that(41)Q=KAfμΔpL0.Combining Equations (39) and (41), we can obtain that
(42)K=π128L01−DtAmDp3+Dp−DTλmax3+DT+β312AfDf1−cos2αsin2α4−Dflmax4.Based on the assumption that the coal matrix is impermeable, i.e., satisfiesϕm=0, therefore, Equation (42) can be simplified as follows:
(43)Kf=β312AfDf1−cos2αsin2α4−Dflmax4.Equation (43) is the fractal governing equation for fracture permeability. Among them, the permeability of the dynamic permeability fractal model is affected by both pore and fracture, but the total permeability is not simply the sum of them.


## 2.5. Mass Transfer between Pores and Fractures

The mass transfer function between the fracture system and the pore system can be expressed as [30]
(44)Q=ωPm−Pf,where ω=aVρmKm/μ, ρm is the volume of gas, and Km is the permeability of the pore system. μ is the viscosity of fluid, and a is the shape factor of matrix fracture transfer. Pm is the pressure of pore structure, and Pf is the pressure of fracture structure.In the coupling process of the above physical fields, firstly, the pressure in the volume strain equations is governed by the coal porosity and permeability equations, and the volume stress and volume strain determine the distribution of coal porosity; furthermore, the permeability of the structure-permeability model affects the coal permeability and porosity equations, and the coal porosity on the contrary determines the permeability equations of the structure-permeability model.Coupling and unification between multiple physical fields are achieved through four parameters: pressure, porosity, strain, and stress. The coupling diagram of the fractal seepage model is shown in Figure1.Figure 1
Coupling relation of the fractal model.


## 3. Model Validation

In order to verify the correctness of the fractal permeability model established in the preceding section, we apply the fractal permeability models to represent field experimental data [28]. The calculation model is shown in Figure 2, where the pore pressure is 6.2 MPa. The upper boundary is affected by 15 MPa pressure, and the left, right, and lower boundaries are hinged. The ambient pressure is atmospheric pressure, and the left, right, and lower boundaries are no flow. In the simulation of COMSOL Multiphysics, the input parameters are shown in Table 1.Figure 2
Field data verification model.Table 1
Parameters of the simulation model.
ParameterValueYoung’s modulus of coal,EMPa2713Young’s modulus of coal grains,EsMPa8139Poisson’s ratio of coal,ν0.339Density of coal,ρckg/m31.25×103Density of methane,ρgkg/m30.717Methane dynamic viscosity,μPa⋅s1.84×10‐5Langmuir pressure constant,PLMPa6.109Langmuir volume constant,VLm3/kg0.015Langmuir volumetric strain constant,εL0.02295Initial porosity of pore system,φm00.00804Initial porosity of pore system,φf00.02Considering that the deformation of coal seam caused by the flow field in time and space evolution is nonlinear, COMSOL Multiphysics is used to calculate the coupled equations by the method of multiphysical field coupling. Select the data of field mining under different mines to explore the change of permeability under different pressures. The model fitting results are compared with the experimental data, as shown in Figure3. From Figure 3, it can be concluded that the proposed structure-gas fractal coupling model is in good agreement with the measured data, which verifies the correctness of the fractal model.Figure 3
Comparisons between the calculated results of the model and field data.


## 4. Numerical Experiments

### 4.1. Boundary and Initial Conditions

In order to study the performance of the new model under the influence of coupling multiprocess, we apply the model to a common case: constant confining pressure condition. The calculation model is rectangular, and the size is0.1m×0.05m. The bottom of the model is constrained by the fixed end, and the rest of the boundary is affected by the pressure of 100 psia. The initial pore pressure of the model is 6.2 Mpa, there is no gas flow at the bottom, and the other three boundary pressures are 0.1 atm. The boundary and initial conditions are shown in Figure 4.Figure 4
Schematic diagram of boundary and initial conditions.The property parameters of coal and gas are shown in Table2. Most of the values are derived from previous experimental results [29, 30]; unreported parameters are replaced by contemporary literature.Table 2
Property parameters of coal and gas.
ParameterValueYoung’s modulus of coal,EMPa2713Young’s modulus of coal grains,EsMPa8139Poisson’s ratio of coal,ν0.339Density of coal,ρckg/m31.25×103Density of methane,ρgkg/m30.717Methane dynamic viscosity,μPa⋅s1.84×10‐5Langmuir pressure constant,PLMPa6.109Langmuir volume constant,VLm3/kg0.015Langmuir volumetric strain constant,εL0.02295Initial porosity of coal,ϕ00.00804Aperture ratioλmin/λmax0.006Initial permeability of coal,k0m23.7996×10‐17


### 4.2. Discussion of the Simulation Results

#### 4.2.1. Stress and Deformation Evolution of Coal

Based on the above coupled governing equations and the above parameters, the stress distribution and deformation of coal in 20, 40, 100, and 500 days are simulated. The simulation results are shown in Figure5.Figure 5
Evolution of stress distribution in coal.As shown in Figure5, with the passage of time, the stress in the central gas-carrying area of coal body increases gradually, and the deformation of coal body increases gradually. The bottom is constrained by the fixed end, and the stress near it is the largest. The stress of the other three boundaries is much smaller than that at the bottom because of the expansion deformation. The minimum stress occurs at the two corners of the top. With the passage of time, the stress at the bottom decreases gradually, while the stress at the left and right edges decreases slowly with the passage of time.


#### 4.2.2. Pressure Evolution of Pore

According to the equation established in this paper and the parameters mentioned above, the distribution of pore pressure in 20, 40, 100, and 500 days is studied. The pore pressure distribution obtained by the model simulation is shown in Figure6.Figure 6
Evolution of pore pressure.From Figure6, it can be seen that with the passage of time, the pore pressure of coal body decreases as a whole. Because the bottom is constrained by the fixed end and there is no gas flow out, the high-pressure zone finally concentrates on the bottom center. Because the upper, left, and right boundaries have gas flow due to the pressure difference, and the force is not changed, with the passage of time, the low-pressure zone at the edge gradually diffuses to the center. The results of coal body simulation show that the pore pressure decreases gradually from the top, left, and right to the center.


#### 4.2.3. Dynamic Evolution of Permeability in Upper Boundary

Based on this model, the permeability evolution of the fracture and pore system at the upper boundary is studied. With the same model parameters, the evolution of the upper boundary fracture and pore system permeability with time and space is shown in Figures7 and 8.Figure 7
Evolution of permeability in the upper boundary fracture system.Figure 8
Evolution of permeability in the upper boundary pore system.From Figures7 and 8, we can conclude that fracture and pore permeability shows a trend of low edge and high middle permeability at different time. However, with the passage of time, fracture and pore permeability shows a downward trend as a whole. As can be seen from Figures 5 and 6, with the passage of time, the stress of top coal body decreases gradually, and the pore stress decreases gradually. At the same time, the central pressure of coal decreases with the flow of gas, and the permeability of fracture and pore also shows a downward trend with the passage of time.


#### 4.2.4. Effect of Adsorption Strain and Volume Strain on Porosity

The effect of adsorption strain and volume strain on porosity is shown in Figure9. It can be concluded from the graph that the volume strain and adsorption strain of the matrix have a great influence on the porosity. With the decrease of pore pressure and adsorption pressure, the influence of pore volume strain is not obvious. At the same time, we can conclude that the effect of adsorption-induced volumetric strain is more significant than that of volumetric mechanical volumetric strain.Figure 9
Effect of adsorption strain and volume strain on porosity.


#### 4.2.5. Evolution of Fractal Dimensions of Fractures and Pores with Time

Meanwhile, based on the above model, we obtained the evolution of fractal dimension of fracture and pore with time. We choose the center point of the model (0.025, 0.05) as the representative point of analysis. We study the evolution of fractal dimension of pore and fracture at this point with time. The evolution ofDf and Dp at this location over time is shown in Figures 10 and 11.Figure 10
Evolution of the fractal dimension of fracture length at the location (0.025, 0.05) by the fractal model.Figure 11
Evolution of the fractal dimension of throat diameter at the location (0.025, 0.05) by the fractal model.Combining with Figures10 and 11, we concluded that the evolution curve of fractal dimension in the center of the model decreases with time. Comparing Figures 1 and 11 with Figure 5, the fractal dimension of the model center is proportional to the stress. This is because the stress of coal increases gradually under the action of long-time pressure. As a high-stress area of coal, the longer the compression time, the larger the deformation of pore and fracture structure.


#### 4.2.6. Influence of Adsorption Parameters on Coal Seam Permeability

Based on the model above, we explore the effect of adsorption parameters on coal seam permeability under the condition that other coal seam parameters remain unchanged. We select three points on the axis of the model (0.025, 0.025; 0.025, 0.05; 0.025, 0.075) as the representative points of the analysis. And the simulation results are shown in Figures12, 13, and 14.Figure 12
Influence of Langmuir volume strain constantεL on the fractal dimension Df of fracture length.Figure 13
Influence of Langmuir pressure constantPL on the fractal dimension Df of fracture length.Figure 14
Influence of Langmuir volume constantVL on fracture length fractal dimension Df.The Langmuir volume constant (VL) is the maximum adsorption capacity under the condition of pore fracture equilibrium of coal seam. So, as it increases, the volume, number, and surface area of small fractures increases. And the fractal dimension also increases. Langmuir pressure (PL) is the corresponding pressure when the adsorption capacity reaches half of Langmuir volume (VL). Because of the increase of Langmuir pressure, micropores and small fractures close; the fractal dimension eventually decreases. Besides, with the increase of Langmuir’s volume strain constant (εL), the volume expansion degree of pores and fractures in coal seam increases, new pores and fractures are produced, and the fractal dimension of coal seam also increases.


## 4.1. Boundary and Initial Conditions

In order to study the performance of the new model under the influence of coupling multiprocess, we apply the model to a common case: constant confining pressure condition. The calculation model is rectangular, and the size is0.1m×0.05m. The bottom of the model is constrained by the fixed end, and the rest of the boundary is affected by the pressure of 100 psia. The initial pore pressure of the model is 6.2 Mpa, there is no gas flow at the bottom, and the other three boundary pressures are 0.1 atm. The boundary and initial conditions are shown in Figure 4.Figure 4
Schematic diagram of boundary and initial conditions.The property parameters of coal and gas are shown in Table2. Most of the values are derived from previous experimental results [29, 30]; unreported parameters are replaced by contemporary literature.Table 2
Property parameters of coal and gas.
ParameterValueYoung’s modulus of coal,EMPa2713Young’s modulus of coal grains,EsMPa8139Poisson’s ratio of coal,ν0.339Density of coal,ρckg/m31.25×103Density of methane,ρgkg/m30.717Methane dynamic viscosity,μPa⋅s1.84×10‐5Langmuir pressure constant,PLMPa6.109Langmuir volume constant,VLm3/kg0.015Langmuir volumetric strain constant,εL0.02295Initial porosity of coal,ϕ00.00804Aperture ratioλmin/λmax0.006Initial permeability of coal,k0m23.7996×10‐17


## 4.2. Discussion of the Simulation Results

### 4.2.1. Stress and Deformation Evolution of Coal

Based on the above coupled governing equations and the above parameters, the stress distribution and deformation of coal in 20, 40, 100, and 500 days are simulated. The simulation results are shown in Figure5.Figure 5
Evolution of stress distribution in coal.As shown in Figure5, with the passage of time, the stress in the central gas-carrying area of coal body increases gradually, and the deformation of coal body increases gradually. The bottom is constrained by the fixed end, and the stress near it is the largest. The stress of the other three boundaries is much smaller than that at the bottom because of the expansion deformation. The minimum stress occurs at the two corners of the top. With the passage of time, the stress at the bottom decreases gradually, while the stress at the left and right edges decreases slowly with the passage of time.


### 4.2.2. Pressure Evolution of Pore

According to the equation established in this paper and the parameters mentioned above, the distribution of pore pressure in 20, 40, 100, and 500 days is studied. The pore pressure distribution obtained by the model simulation is shown in Figure6.Figure 6
Evolution of pore pressure.From Figure6, it can be seen that with the passage of time, the pore pressure of coal body decreases as a whole. Because the bottom is constrained by the fixed end and there is no gas flow out, the high-pressure zone finally concentrates on the bottom center. Because the upper, left, and right boundaries have gas flow due to the pressure difference, and the force is not changed, with the passage of time, the low-pressure zone at the edge gradually diffuses to the center. The results of coal body simulation show that the pore pressure decreases gradually from the top, left, and right to the center.


### 4.2.3. Dynamic Evolution of Permeability in Upper Boundary

Based on this model, the permeability evolution of the fracture and pore system at the upper boundary is studied. With the same model parameters, the evolution of the upper boundary fracture and pore system permeability with time and space is shown in Figures7 and 8.Figure 7
Evolution of permeability in the upper boundary fracture system.Figure 8
Evolution of permeability in the upper boundary pore system.From Figures7 and 8, we can conclude that fracture and pore permeability shows a trend of low edge and high middle permeability at different time. However, with the passage of time, fracture and pore permeability shows a downward trend as a whole. As can be seen from Figures 5 and 6, with the passage of time, the stress of top coal body decreases gradually, and the pore stress decreases gradually. At the same time, the central pressure of coal decreases with the flow of gas, and the permeability of fracture and pore also shows a downward trend with the passage of time.


### 4.2.4. Effect of Adsorption Strain and Volume Strain on Porosity

The effect of adsorption strain and volume strain on porosity is shown in Figure9. It can be concluded from the graph that the volume strain and adsorption strain of the matrix have a great influence on the porosity. With the decrease of pore pressure and adsorption pressure, the influence of pore volume strain is not obvious. At the same time, we can conclude that the effect of adsorption-induced volumetric strain is more significant than that of volumetric mechanical volumetric strain.Figure 9
Effect of adsorption strain and volume strain on porosity.


### 4.2.5. Evolution of Fractal Dimensions of Fractures and Pores with Time

Meanwhile, based on the above model, we obtained the evolution of fractal dimension of fracture and pore with time. We choose the center point of the model (0.025, 0.05) as the representative point of analysis. We study the evolution of fractal dimension of pore and fracture at this point with time. The evolution ofDf and Dp at this location over time is shown in Figures 10 and 11.Figure 10
Evolution of the fractal dimension of fracture length at the location (0.025, 0.05) by the fractal model.Figure 11
Evolution of the fractal dimension of throat diameter at the location (0.025, 0.05) by the fractal model.Combining with Figures10 and 11, we concluded that the evolution curve of fractal dimension in the center of the model decreases with time. Comparing Figures 1 and 11 with Figure 5, the fractal dimension of the model center is proportional to the stress. This is because the stress of coal increases gradually under the action of long-time pressure. As a high-stress area of coal, the longer the compression time, the larger the deformation of pore and fracture structure.


### 4.2.6. Influence of Adsorption Parameters on Coal Seam Permeability

Based on the model above, we explore the effect of adsorption parameters on coal seam permeability under the condition that other coal seam parameters remain unchanged. We select three points on the axis of the model (0.025, 0.025; 0.025, 0.05; 0.025, 0.075) as the representative points of the analysis. And the simulation results are shown in Figures12, 13, and 14.Figure 12
Influence of Langmuir volume strain constantεL on the fractal dimension Df of fracture length.Figure 13
Influence of Langmuir pressure constantPL on the fractal dimension Df of fracture length.Figure 14
Influence of Langmuir volume constantVL on fracture length fractal dimension Df.The Langmuir volume constant (VL) is the maximum adsorption capacity under the condition of pore fracture equilibrium of coal seam. So, as it increases, the volume, number, and surface area of small fractures increases. And the fractal dimension also increases. Langmuir pressure (PL) is the corresponding pressure when the adsorption capacity reaches half of Langmuir volume (VL). Because of the increase of Langmuir pressure, micropores and small fractures close; the fractal dimension eventually decreases. Besides, with the increase of Langmuir’s volume strain constant (εL), the volume expansion degree of pores and fractures in coal seam increases, new pores and fractures are produced, and the fractal dimension of coal seam also increases.


## 4.2.1. Stress and Deformation Evolution of Coal

Based on the above coupled governing equations and the above parameters, the stress distribution and deformation of coal in 20, 40, 100, and 500 days are simulated. The simulation results are shown in Figure5.Figure 5
Evolution of stress distribution in coal.As shown in Figure5, with the passage of time, the stress in the central gas-carrying area of coal body increases gradually, and the deformation of coal body increases gradually. The bottom is constrained by the fixed end, and the stress near it is the largest. The stress of the other three boundaries is much smaller than that at the bottom because of the expansion deformation. The minimum stress occurs at the two corners of the top. With the passage of time, the stress at the bottom decreases gradually, while the stress at the left and right edges decreases slowly with the passage of time.


## 4.2.2. Pressure Evolution of Pore

According to the equation established in this paper and the parameters mentioned above, the distribution of pore pressure in 20, 40, 100, and 500 days is studied. The pore pressure distribution obtained by the model simulation is shown in Figure6.Figure 6
Evolution of pore pressure.From Figure6, it can be seen that with the passage of time, the pore pressure of coal body decreases as a whole. Because the bottom is constrained by the fixed end and there is no gas flow out, the high-pressure zone finally concentrates on the bottom center. Because the upper, left, and right boundaries have gas flow due to the pressure difference, and the force is not changed, with the passage of time, the low-pressure zone at the edge gradually diffuses to the center. The results of coal body simulation show that the pore pressure decreases gradually from the top, left, and right to the center.


## 4.2.3. Dynamic Evolution of Permeability in Upper Boundary

Based on this model, the permeability evolution of the fracture and pore system at the upper boundary is studied. With the same model parameters, the evolution of the upper boundary fracture and pore system permeability with time and space is shown in Figures7 and 8.Figure 7
Evolution of permeability in the upper boundary fracture system.Figure 8
Evolution of permeability in the upper boundary pore system.From Figures7 and 8, we can conclude that fracture and pore permeability shows a trend of low edge and high middle permeability at different time. However, with the passage of time, fracture and pore permeability shows a downward trend as a whole. As can be seen from Figures 5 and 6, with the passage of time, the stress of top coal body decreases gradually, and the pore stress decreases gradually. At the same time, the central pressure of coal decreases with the flow of gas, and the permeability of fracture and pore also shows a downward trend with the passage of time.


## 4.2.4. Effect of Adsorption Strain and Volume Strain on Porosity

The effect of adsorption strain and volume strain on porosity is shown in Figure9. It can be concluded from the graph that the volume strain and adsorption strain of the matrix have a great influence on the porosity. With the decrease of pore pressure and adsorption pressure, the influence of pore volume strain is not obvious. At the same time, we can conclude that the effect of adsorption-induced volumetric strain is more significant than that of volumetric mechanical volumetric strain.Figure 9
Effect of adsorption strain and volume strain on porosity.


## 4.2.5. Evolution of Fractal Dimensions of Fractures and Pores with Time

Meanwhile, based on the above model, we obtained the evolution of fractal dimension of fracture and pore with time. We choose the center point of the model (0.025, 0.05) as the representative point of analysis. We study the evolution of fractal dimension of pore and fracture at this point with time. The evolution ofDf and Dp at this location over time is shown in Figures 10 and 11.Figure 10
Evolution of the fractal dimension of fracture length at the location (0.025, 0.05) by the fractal model.Figure 11
Evolution of the fractal dimension of throat diameter at the location (0.025, 0.05) by the fractal model.Combining with Figures10 and 11, we concluded that the evolution curve of fractal dimension in the center of the model decreases with time. Comparing Figures 1 and 11 with Figure 5, the fractal dimension of the model center is proportional to the stress. This is because the stress of coal increases gradually under the action of long-time pressure. As a high-stress area of coal, the longer the compression time, the larger the deformation of pore and fracture structure.


## 4.2.6. Influence of Adsorption Parameters on Coal Seam Permeability

Based on the model above, we explore the effect of adsorption parameters on coal seam permeability under the condition that other coal seam parameters remain unchanged. We select three points on the axis of the model (0.025, 0.025; 0.025, 0.05; 0.025, 0.075) as the representative points of the analysis. And the simulation results are shown in Figures12, 13, and 14.Figure 12
Influence of Langmuir volume strain constantεL on the fractal dimension Df of fracture length.Figure 13
Influence of Langmuir pressure constantPL on the fractal dimension Df of fracture length.Figure 14
Influence of Langmuir volume constantVL on fracture length fractal dimension Df.The Langmuir volume constant (VL) is the maximum adsorption capacity under the condition of pore fracture equilibrium of coal seam. So, as it increases, the volume, number, and surface area of small fractures increases. And the fractal dimension also increases. Langmuir pressure (PL) is the corresponding pressure when the adsorption capacity reaches half of Langmuir volume (VL). Because of the increase of Langmuir pressure, micropores and small fractures close; the fractal dimension eventually decreases. Besides, with the increase of Langmuir’s volume strain constant (εL), the volume expansion degree of pores and fractures in coal seam increases, new pores and fractures are produced, and the fractal dimension of coal seam also increases.


## 5. Influences of Structural Parameters on Permeability

In this paper, based on the fractal geometry method, a dual pore seepage coupling model is established, in which the maximum fracture length, the fractal dimension of fracture length, the fractal dimension of pore size, and the fractal dimension of pore throat tortuosity are mainly considered, and the proposed model does not contain empirical parameters.


### 5.1. Influence of Fractal Dimension of Fracture Length

Under the influence of adsorption stress and in situ stress, we calculated the change of coal permeability with the fractal dimension of coal fracture length. Figure15 shows the difference results between the fractal model established in this paper and the classical cubic model. When the number of fractures is constant, the two-scale fractal model established in this paper increases the permeability of coal seam with the increase of fractal dimension of fracture length. This cannot be obtained in the classical cubic model.Figure 15
Permeability evolution under different fractal dimension of fracture length.Because of the two-dimensional model, the fractal dimension is a fraction between 1 and 2. The closer the fractal dimension is to 2, the longer the fracture length span is, that is, the larger the number of large-scale fractures is. According to Figure15, the closer the fractal dimension is to 2, the higher the permeability of coal seam is, which is consistent with the actual seepage situation of coal seam. When the fractal dimension of fracture is 1.9, the permeability is about 55% different from that before the change.


### 5.2. Influence of Maximum Fracture Length

Under the same conditions, the mechanism of coal permeability varying with the maximum fracture length is discussed, as shown in Figure16.Figure 16
Permeability evolution under different maximum fracture length.As shown in Figure16, with the increase of maximum fracture length, the permeability of coal seam increases gradually. It is well known that the size of fracture is much larger than that of throats and pores. Therefore, the more fracture gas flows through, the less small-scale structures such as pore and throat experienced, and the smaller the resistance to flow. Therefore, when the fractal dimension, pore size, and other microstructure parameters are unchanged, the average size of gas migration channel increases with the increase of the maximum fracture length, the flow resistance of gas decreases, and the permeability increases. However, in the classical cubic model, the influence of this factor on macropermeability is usually neglected.


### 5.3. Influence of Fractal Dimension of Throat Diameter

Similarly, we studied the change of permeability of the coal pore system with the fractal dimension of throat diameter. The two-scale fractal seepage model and the classical cubic model are presented in this paper. The trend diagram of the influence of pore fractal dimension on permeability is shown in Figure17.Figure 17
Permeability evolution under different fractal dimension of throat diameter.From Figure17, it can be seen that in the two-scale fractal seepage model established in this paper, with the increase of fractal dimension of throat diameter, the permeability of the coal pore system also shows an increasing trend. In the two-dimensional structure, with the increase of pore fractal dimension, the larger the number of large diameter throats, the smaller the flow resistance. Therefore, in this model, with the increase of pore fractal dimension, the higher the coal pore coverage, the greater the permeability of the coal pore system. This is beyond the description of the classical cubic model.


### 5.4. Influence of Fractal Dimension of Throat Tortuosity

The variation of permeability of the coal pore system under different throat fractal dimension is shown in Figure18. It shows that in the two-scale fractal model, with the increase of fractal dimension of throat tortuosity, the permeability of coal seam decreases gradually.Figure 18
Permeability evolution under different fractal dimension of throat tortuosity.In this study, a two-dimensional model is used. The fractal dimension of the larynx is between 1 and 2. The closer to 2, the greater the curvature of the larynx and the greater the resistance of the gas flow process. Therefore, with the increase of fractal dimension of throat, the permeability of the coal pore system shows a downward trend. With the change of the fractal dimension of the larynx, the difference of solution increased, so the new fractal model is closer to the real situation than the classical cubic model. It can be seen that although the fractal dimension of pore and throat directly affects the permeability of the whole model, the order of magnitude is much smaller than the fractal dimension and maximum fracture length at the fracture scale. Therefore, the pore size structure parameters have little influence on the overall permeability of coal samples.


## 5.1. Influence of Fractal Dimension of Fracture Length

Under the influence of adsorption stress and in situ stress, we calculated the change of coal permeability with the fractal dimension of coal fracture length. Figure15 shows the difference results between the fractal model established in this paper and the classical cubic model. When the number of fractures is constant, the two-scale fractal model established in this paper increases the permeability of coal seam with the increase of fractal dimension of fracture length. This cannot be obtained in the classical cubic model.Figure 15
Permeability evolution under different fractal dimension of fracture length.Because of the two-dimensional model, the fractal dimension is a fraction between 1 and 2. The closer the fractal dimension is to 2, the longer the fracture length span is, that is, the larger the number of large-scale fractures is. According to Figure15, the closer the fractal dimension is to 2, the higher the permeability of coal seam is, which is consistent with the actual seepage situation of coal seam. When the fractal dimension of fracture is 1.9, the permeability is about 55% different from that before the change.


## 5.2. Influence of Maximum Fracture Length

Under the same conditions, the mechanism of coal permeability varying with the maximum fracture length is discussed, as shown in Figure16.Figure 16
Permeability evolution under different maximum fracture length.As shown in Figure16, with the increase of maximum fracture length, the permeability of coal seam increases gradually. It is well known that the size of fracture is much larger than that of throats and pores. Therefore, the more fracture gas flows through, the less small-scale structures such as pore and throat experienced, and the smaller the resistance to flow. Therefore, when the fractal dimension, pore size, and other microstructure parameters are unchanged, the average size of gas migration channel increases with the increase of the maximum fracture length, the flow resistance of gas decreases, and the permeability increases. However, in the classical cubic model, the influence of this factor on macropermeability is usually neglected.


## 5.3. Influence of Fractal Dimension of Throat Diameter

Similarly, we studied the change of permeability of the coal pore system with the fractal dimension of throat diameter. The two-scale fractal seepage model and the classical cubic model are presented in this paper. The trend diagram of the influence of pore fractal dimension on permeability is shown in Figure17.Figure 17
Permeability evolution under different fractal dimension of throat diameter.From Figure17, it can be seen that in the two-scale fractal seepage model established in this paper, with the increase of fractal dimension of throat diameter, the permeability of the coal pore system also shows an increasing trend. In the two-dimensional structure, with the increase of pore fractal dimension, the larger the number of large diameter throats, the smaller the flow resistance. Therefore, in this model, with the increase of pore fractal dimension, the higher the coal pore coverage, the greater the permeability of the coal pore system. This is beyond the description of the classical cubic model.


## 5.4. Influence of Fractal Dimension of Throat Tortuosity

The variation of permeability of the coal pore system under different throat fractal dimension is shown in Figure18. It shows that in the two-scale fractal model, with the increase of fractal dimension of throat tortuosity, the permeability of coal seam decreases gradually.Figure 18
Permeability evolution under different fractal dimension of throat tortuosity.In this study, a two-dimensional model is used. The fractal dimension of the larynx is between 1 and 2. The closer to 2, the greater the curvature of the larynx and the greater the resistance of the gas flow process. Therefore, with the increase of fractal dimension of throat, the permeability of the coal pore system shows a downward trend. With the change of the fractal dimension of the larynx, the difference of solution increased, so the new fractal model is closer to the real situation than the classical cubic model. It can be seen that although the fractal dimension of pore and throat directly affects the permeability of the whole model, the order of magnitude is much smaller than the fractal dimension and maximum fracture length at the fracture scale. Therefore, the pore size structure parameters have little influence on the overall permeability of coal samples.


## 6. Conclusion

In this study, a new two-scale stress-flow fractal model is established, which combines gas flow, adsorption, and desorption with coal deformation process, and then quantitatively analyzes the influence of structural parameters such as coal fracture, pore, and throat on coal permeability. The classical model is based on the relationship between porosity and permeability, while the model proposed in this paper can consider the influence of fracture pore structure parameters on permeability under multifield coupling. Based on the numerical simulation results mentioned above, the following conclusions can be drawn:(i)
The effects of four parameters characterizing microstructure on coal permeability are discussed: (1) fractal dimension of throat diameter, (2) fractal dimension of throat tortuosity, (3) maximum fracture length, and (4) fractal dimension of fracture length. Permeability is directly proportional to fractal dimension of fracture length, maximum fracture length, and throat diameter and inversely proportional to fractal dimension of throat tortuosity(ii)
Among the four parameters used in characterization, fracture structure parameters have a greater impact on permeability, while pore scale structure parameters have a smaller impact on overall permeability. From the sketch figure of permeability and structural parameter evolution of the classical cubic model, we can conclude that the fracture-pore structure is an indispensable factor in the physical evolution of coal seam from the initial state to the final equilibrium state(iii)
Under the simulation parameters in this paper, with the passage of time, the stress in the central gas-carrying area of coal increases gradually, and the deformation of coal increases gradually; the stress at the bottom increases gradually, and the stress at the top decreases gradually; the pore pressure decreases gradually from the top, left, and right to the center, while the pore permeability of the top fracture decreases as a whole



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*Source: 2787903-2020-06-22.xml*