# Solid-Gas Coupling Model for Coal-Rock Mass Deformation and Pressure Relief Gas Flow in Protection Layer Mining

Authors: Zhuohui Zhu; Tao Feng; Zhigang Yuan; Donghai Xie; Wei Chen
Journal: Advances in Civil Engineering
 (2018)

---

## Abstract

The solid-gas coupling model for mining coal-rock mass deformation and pressure relief gas flow in protection layer mining is the key to determine deformation of coal-rock mass and migration law of pressure relief gas of protection layer mining in outburst coal seams. Based on the physical coupling process between coal-rock mass deformation and pressure-relief gas migration, the coupling variable of mining coal-rock mass, a part of governing equations of gas seepage field and deformation field in mining coal-rock mass, is introduced. Then, a new solid-gas coupling mathematical model reflecting the coupling effects of gas adsorption/desorption, gas pressure, and coal-rock mass deformation on the mining coal-rock mass deformation and pressure relief gas flow is established combined with the corresponding definite conditions. It lays a theoretical foundation for the numerical calculation of the deformation of mining coal-rock mass and the migration law of gas under pressure relief in the outburst coal seam group.


---

## Body

## 1. Basic Hypothesis

After mining protection seam, the initial rock stress balance is broken and the coal-rock mass deforms, which destroys the dynamic balance of original gas adsorption and desorption, and part of gas desorbs and diffuses to the mining fissure and gas pressure reduces, which makes the coal-rock mass pressure state changed; namely, deformation of mining coal-rock mass and flow of pressure relief gas are a multiphysical field coupling process. Therefore, the establishment of a solid-gas coupling mathematical model which can describe the deformation of mining coal-rock mass and pressure relief gas flow is the key to study the deformation of coal-rock mass and the migration law of gas under pressure relief in protection layer mining.Due to the solid-gas coupling effect of coal-rock mass deformation and pressure relief gas flow are complicated, which involves seepage mechanics, rock mechanics, elastic-plastic mechanics and many other subjects [1–5]. The existing research results found that seepage pressure affects the stress field; moreover, the stress field in surrounding rock results in the change of permeability of rock masses [3, 5]. Based on the previous studies [1–5], the following hypotheses are introduced:(1)
The solid-gas coupling system consists of both solid phase (coal-rock mass) and gas phase.(2)
The solid skeleton deformation of coal-rock mass is small, which can be regarded as homogeneous and isotropic linear elastic medium; the coal seam gas can be regarded as an ideal gas, and its flow process can be regarded as approximately isothermal process, which obeys the Darcy seepage law.(3)
The coal seam is saturated by single-phase gas, and the original gas pressure is uniformly distributed; gas content in coal seam follows the modified Langmuir equation.(4)
The effective stress of coal-rock mass skeleton follows the modified Terzaghi effective stress principle.


## 2. Control Equations of Deformation Field of Mining Coal-Rock Mass

The control equations of displacement and stress fields of mining coal-rock mass deformation are mainly composed of constitutive equation, geometric equation, and equilibrium equation.


### 2.1. Constitutive Equation

For the single-phase saturated linear elastic porous media, the effective stressσii′ (tension is positive), the pore pressure p (compression is positive), and the total stress σij follow the modified Terzaghi effective stress principle [6, 7]:(1)E=∑i=1mSi,where δij is the Kronecker symbol (when i = j, δij=1, and when i ≠ j, δij=0) and αα≤1 is the Biot coefficient, whose value depends on the compressibility of coal-rock mass and is calculated as follows:(2)α=1−KKs,where K is the bulk modulus of coal-rock mass skeleton (Pa) and Ks is the matrix bulk modulus of coal-rock mass (Pa).Previous studies have shown that [8, 9] adsorption/desorption will cause the bulk strain of coal matrix, and it can be calculated combined with the generalized Hooke’s law:(3)σij′=2Gεij+2Gv1−2vεkkδij−Kεsδij,where v is the Poisson’s ratio of drainage, G is the shear modulus (Pa), and εs is the matrix bulk strain caused by coal adsorption/desorption; the experimental results show that the relationship between εs and p meets the Langmuir type curve [8, 9]:(4)εs=εLpp+pL,where εL is the bulk strain constant when the pore pressure is infinite and pL is the pore pressure when the bulk strain is 0.5εL (Pa).


### 2.2. Geometric Equation

Assuming that the deformation of coal-rock mass is small, the geometric equation is [10](5)εij=12ui,j+uj,i,where ui is the displacement in the i direction.


### 2.3. Equilibrium Equation

According to the virtual displacement principle of elastic mechanics, the momentum equation of coal-rock mass skeleton under quasi-static condition is [11](6)∫ΩδεijσijdV−∫ΩfiδuidV−∫Sf¯iδuidS=0,where f¯i and fi are the boundary surface force and the bulk force of the skeleton, δui and δεij are the coordination of virtual displacement variation and virtual strain variation, and Ω and S are the regions occupied by solid skeleton and the known boundary of stress.It can be obtained according to (5) and the symmetry of strain tensor [12]:(7)δεij=δ12ui,j+uj,i=δui,j.Substituting (1) into (6), we get(8)∫Ωδεijσij′−αpδijdV−∫ΩfiδuidV−∫Sf¯iδuidS=0.According to the Green-Gauss formula, the first item on the left of (8) is(9)∫Ωδui,jσij′⁡−⁡αpδijdV=∫Sδuiσij′−αpδijnjdS−∫Ωσij,j′−αpjδijδuidV.The equilibrium equation of porous medium of coal-rock mass can be obtained by (8) and (9):(10)σij′−αpj+fi=0.Connecting the constitutive equation (3), the geometric equation (5), and the equilibrium equation (10), the governing equations which take the solid-gas coupling deformation field under the effect of coal adsorption/desorption deformation, stress, and pore pressure into account can be obtained:(11)Gui,jj+G1−2vuj,ji−αpi−Kεs,i+fi=0.


## 2.1. Constitutive Equation

For the single-phase saturated linear elastic porous media, the effective stressσii′ (tension is positive), the pore pressure p (compression is positive), and the total stress σij follow the modified Terzaghi effective stress principle [6, 7]:(1)E=∑i=1mSi,where δij is the Kronecker symbol (when i = j, δij=1, and when i ≠ j, δij=0) and αα≤1 is the Biot coefficient, whose value depends on the compressibility of coal-rock mass and is calculated as follows:(2)α=1−KKs,where K is the bulk modulus of coal-rock mass skeleton (Pa) and Ks is the matrix bulk modulus of coal-rock mass (Pa).Previous studies have shown that [8, 9] adsorption/desorption will cause the bulk strain of coal matrix, and it can be calculated combined with the generalized Hooke’s law:(3)σij′=2Gεij+2Gv1−2vεkkδij−Kεsδij,where v is the Poisson’s ratio of drainage, G is the shear modulus (Pa), and εs is the matrix bulk strain caused by coal adsorption/desorption; the experimental results show that the relationship between εs and p meets the Langmuir type curve [8, 9]:(4)εs=εLpp+pL,where εL is the bulk strain constant when the pore pressure is infinite and pL is the pore pressure when the bulk strain is 0.5εL (Pa).


## 2.2. Geometric Equation

Assuming that the deformation of coal-rock mass is small, the geometric equation is [10](5)εij=12ui,j+uj,i,where ui is the displacement in the i direction.


## 2.3. Equilibrium Equation

According to the virtual displacement principle of elastic mechanics, the momentum equation of coal-rock mass skeleton under quasi-static condition is [11](6)∫ΩδεijσijdV−∫ΩfiδuidV−∫Sf¯iδuidS=0,where f¯i and fi are the boundary surface force and the bulk force of the skeleton, δui and δεij are the coordination of virtual displacement variation and virtual strain variation, and Ω and S are the regions occupied by solid skeleton and the known boundary of stress.It can be obtained according to (5) and the symmetry of strain tensor [12]:(7)δεij=δ12ui,j+uj,i=δui,j.Substituting (1) into (6), we get(8)∫Ωδεijσij′−αpδijdV−∫ΩfiδuidV−∫Sf¯iδuidS=0.According to the Green-Gauss formula, the first item on the left of (8) is(9)∫Ωδui,jσij′⁡−⁡αpδijdV=∫Sδuiσij′−αpδijnjdS−∫Ωσij,j′−αpjδijδuidV.The equilibrium equation of porous medium of coal-rock mass can be obtained by (8) and (9):(10)σij′−αpj+fi=0.Connecting the constitutive equation (3), the geometric equation (5), and the equilibrium equation (10), the governing equations which take the solid-gas coupling deformation field under the effect of coal adsorption/desorption deformation, stress, and pore pressure into account can be obtained:(11)Gui,jj+G1−2vuj,ji−αpi−Kεs,i+fi=0.


## 3. Governing Equation of Gas Flow in Mining Coal-Rock Mass

The governing equation of gas flow in coal-rock mass combines the continuity equation, state equation, gas content equation, and Darcy law of motion and eliminates some variables to deduct differential equations. Migration morphology of gas is closely related to the pore structure, but the pore structure of coal seam is complicated; when studying migration law of gas, people are often based on different types of gas diffusion or the primary and secondary roles of seepage to simplify the pore structure of coal seam into pure diffusion pore medium model, pure permeability fracture medium model, and pore-fissure dual-medium model with coexistence of diffusion and permeation. The pore-fissure dual-medium model has been accepted and used by many scholars, and it assumes gas migration in coal seam is a continuous process in two steps: diffusion and permeation are a cascade process that the gas diffuses from the matrix to the fissure in the form of diffusion and then flows through the fissures into the tunnel or borehole in the form of the Darcy seepage.The continuity equation of gas diffusion motion is as follows [13, 14]:(12)∂C∂t=−∇mc−Qs,where mc is the mass diffusion flux (kg/(m2·t)), Qs is the exchange mass source of the pore system flowing to the fracture system (kg/(m3·s)), and C is the quality of adsorbed gas in unit bulk coal:(13)C=ρgaρcVLpp+pL,where ρga is the gas density under the standard state (kg/m3), ρc is density of coal-rock mass (kg/m3), and VL is the Langmuir bulk constant (kg/m3).According to the law of mass conservation, the continuity equation of gas seepage in fracture can be obtained [13–15]:(14)∂ρgϕ∂t=−∇⋅ρgqg+Qs,where ρg is the gas density (kg/m3), ϕ is the porosity of coal-rock mass, and qg is the velocity of gas seepage (m/s).The gas flow is divided into two open systems: diffusion and seepage. In fact, there is a coupling effect between gas diffusion and seepage, and the adsorption and desorption of gas only change in the state of matter, and it does not cause the change of gas quality in representative bulk units. Therefore, in order to facilitate the analysis and solution, the mass conservation equation of gas flow can be obtained by adding (12) and (14):(15)∂ρgϕ∂t+∂C∂t+∇⋅ρgqg+mc=0.For low permeability coal-rock mass, it can be assumed that the gas is desorbed and diffused into the fracture from the matrix instantaneously, and (15) can be further simplified:(16)∂m∂t+∇⋅ρgqg=0.In (16), m=ρgϕ+ρgaρcVLp/p+pL, the first item on the right shows the free gas content in the coal-rock mass, and the second item represents the adsorbed gas content in the coal.The coal gas, whose flow process could be assumed to be isothermal, is considered as the actual gas, and then the state equation of the gas can be obtained:(17)ρg=MgRTp,where Mg is the molar mass of gas, R is the universal gas constant, and T is the absolute temperature.The outward coal-rock mass gas flow from the coal seam crack accord with the Darcy seepage law:(18)qg=−kμ⋅∇p,where k is the permeability of the coal seam, μ is the dynamic viscosity of gas (Pa·s), and ∇p is the gas pressure gradient (Pa/m).Substituting (17) and (18) into (16), we get(19)ϕ+ρcpaVLpLp+pL2∂p∂t+p∂ϕ∂t−∇⋅kμp∇p=0,where pa is the standard atmospheric pressure (Pa). In (19), the permeability k depends on the porosity of coal ϕ, and the porosity of coal ϕ is a function of the volumetric strain εv of coal-rock mass and the volumetric strain εs of matrix. Therefore, (11) and (19) are connected by the porosity of coal ϕ and physical parameters of permeability k.


## 4. The Structure of Solid-Gas Coupling Mathematical Model for Coal-Rock Mass Deformation and Pressure Relief Gas Flow in Protection Seam Mining

### 4.1. Solid-Gas Coupling Variables in Mining Coal-Rock Mass

Traditional classical percolation mechanics holds that in the pore pressure change process, the solid skeleton of porous media does not produce any elastic-plastic deformation, so the porosity of porous mediaϕ is a constant. However, under the effect of surrounding rock stress and gas pressure, the solid skeleton of coal will deform in different degrees, which will cause the change of effective stress and lead to the dynamic change of porosity and permeability of coal-rock mass. When it comes to the deformation of coal-rock mass in protection layer mining and the solid-gas coupling of pressure relief gas flow, the dynamic changes of porosity, permeability, and other parameters of coal-rock mass must be taken into account. So, it is necessary to establish the theoretical calculation model for the dynamic changes of the basic physical parameters such as porosity and permeability of coal-rock mass.


#### 4.1.1. Dynamic Variation Model of Coal-Rock Mass Porosity

The bulkV of porous medium in coal-rock mass is composed of the bulk of solid phase Vs and the bulk of pore volume Vp, namely, V = Vs + Vp. The definition of porosity is ϕ=Vp/V, and the bulk strain εv of coal-rock mass can be obtained by (11) [16, 17]:(20)εv=ΔVV=−1Kσ¯−αp+εs,(21)ΔVpVp=−1Kpσ¯−βp+εs,where σ¯=−σkk/3, β=1−Kp/Ks, and Kp is the module of pore bulk.It is assumed that the strain caused by coal adsorption is equal to the amount of pore variation. Without considering the adsorption effect of gas, the bulk change of porous media meets the equivalent law of Betti-Maxwell:(22)∂V∂pσ¯=∂Vp∂σ¯p,and therefore,(23)Kp=ϕαK.Then, deducted from the definition of porosity(24)ΔVV=ΔVsVs+Δϕ1−ϕ,(25)ΔVpVp=ΔVsVs+Δϕϕ1−ϕ.Connecting (20), (21), (24) and (25), we get(26)Δϕ=ϕ1K−1Kpσ¯−p.Substituting (20) and (23) into (26), we get(27)Δϕ=α−ϕεv+pKs−εs.When the pore pressure of coal-rock mass isp0, the initial porosity is ϕ0, and the initial bulk strain is 0; then, the porosity can be expressed as(28)ϕ=11+S1+S0ϕ0+αS−S0.In formula (28), S=εv+p/Ks−εs and S0=p0/Ks−εLp0/p0+pL.


#### 4.1.2. Dynamic Change Model of Coal-Rock Mass Permeability

Permeability, a physical parameter reflecting the difficult degree of gas flow in coal seam, is important for studying coupling process between deformation of mining coal-rock mass and gas flow. The relationship between the permeability of coal seam and the bulk strain of coal can be deduced by the Kozeny-Carman equation:(29)kk0=11+εvϕϕ03.Neglecting the bulk deformation of coal, (29) can be simplified as(30)kk0=ϕϕ03.Connecting (28) and (30), we get(31)kk0=11+S1+S0+αϕ0S−S03.


### 4.2. Solid-Gas Coupling Mathematical Model of Mining Coal Deformation and Pressure Relief Gas Flow

According to the antecedent, (11) is the control equation of mining coal deformation; substituting (4) into (11), we get(32)Gui,jj+G1−2vuj,ji−αpi−KεLpLp+pL2pi+fi=0.It can be obtained from (28):(33)∂ϕ∂t=α−ϕ1+S∂εv∂t+1Ks∂p∂t−εLpLp+pL2∂p∂t.Substituting (33) into (19), the control equation of the pressure relief gas of the mining coal can be obtained:(34)ϕ+ρcpaVLpLp+pL2+α−ϕp1+SKs−α−ϕεLpLp1+Sp+pL2∂p∂t−∇⋅kμp∇p=−α−ϕp1+S∂εv∂t.Formulas (32) and (34) constitute a gas-solid coupling mathematical model of coal seam deformation and pressure relief gas flow in protection seam mining.


### 4.3. Solid-Gas Coupling Mathematical Model of Rock Mass Deformation and Pressure Relief Gas Flow in Protective Seam Mining

When the rock mass is affected by mining, deformation and cracks occur, the change of rock porosity will lead to the change of permeability, and the pressure relief gas will flow under the effect of gas pressure gradient. The flow of gas in the mining rock mass is different from that of the mining coal, mainly in the absence of adsorption of the rock mass on the gas. Therefore, there is no need to consider the deformation of bulk expansion caused by the gas diffusion process and gas adsorption/desorption, so it is necessary to modify the solid-gas coupling mathematical model of mining coal deformation and pressure relief gas flow.To sum up, the governing equation of mining rock mass deformation can be expressed as(35)Gui,jj+G1−2vuj,ji−αpi+fi=0.Similarly, the control equation of pressure relief gas of mining coal can be expressed as(36)ϕ+α−ϕp1+S1Ks∂p∂t−∇⋅kμp∇p=−α−ϕp1+S1∂εv∂t,where S1=εv+p/Ks.


### 4.4. Definite Conditions

The control equations (32) and (34–36) constitute the complicated solid-gas coupling mathematical model of the mining coal-rock mass deformation and the pressure relief gas flow. For a particular solution, additional conditions must be added, that is, initial conditions and boundary conditions. Definite conditions of the model include the boundary conditions of the deformation field of the mining coal-rock mass and the initial conditions and boundary conditions of the gas seepage [17–22].


#### 4.4.1. Boundary Conditions of Mining Coal-Rock Mass Deformation Field

The first kind of boundary condition: the surface force of the solid skeleton of the coal-rock mass is known as(37)σij⋅nj=f¯ix,y,z,t,t∈0,+∞.The second kind of boundary condition: the surface displacement of the solid skeleton of the coal-rock mass is known as(38)uix,y,z,t=gi¯x,y,z,t,t∈0,+∞,where nj is the unit normal vector of the surface solid skeleton of coal-rock mass and gi¯ and f¯i are the known surface displacement functions and known surface stress distribution functions.


#### 4.4.2. Initial Conditions and Boundary Conditions of Gas Seepage

(1) Initial Conditions of Gas Seepage
(39)
p
x
,
y
,
z
t
=
0
=
p
¯
x
,
y
,
z
,
q
x
,
y
,
z
t
=
0
=
q
¯
x
,
y
,
z
.(2) Boundary Conditions of Gas Seepage. The first kind of boundary condition (the Dirichlet condition):(40)px,y,z,t=p¯x,y,z,t,t∈0,+∞.The second kind of boundary condition (the Neumann condition):(41)∂px,y,z,t∂n=Q¯x,y,z,t,t∈0,+∞.The third kind of boundary condition (equal flow at the interface):(42)K∂p∂n1=K′∂p′∂n′2,where p¯ and Q¯ are the distribution function and the fluid flow function of the pressure p and time t at the point (x, y, z) of the outer boundary, respectively, and q¯ is the initial flow value at the outer boundary.


## 4.1. Solid-Gas Coupling Variables in Mining Coal-Rock Mass

Traditional classical percolation mechanics holds that in the pore pressure change process, the solid skeleton of porous media does not produce any elastic-plastic deformation, so the porosity of porous mediaϕ is a constant. However, under the effect of surrounding rock stress and gas pressure, the solid skeleton of coal will deform in different degrees, which will cause the change of effective stress and lead to the dynamic change of porosity and permeability of coal-rock mass. When it comes to the deformation of coal-rock mass in protection layer mining and the solid-gas coupling of pressure relief gas flow, the dynamic changes of porosity, permeability, and other parameters of coal-rock mass must be taken into account. So, it is necessary to establish the theoretical calculation model for the dynamic changes of the basic physical parameters such as porosity and permeability of coal-rock mass.


### 4.1.1. Dynamic Variation Model of Coal-Rock Mass Porosity

The bulkV of porous medium in coal-rock mass is composed of the bulk of solid phase Vs and the bulk of pore volume Vp, namely, V = Vs + Vp. The definition of porosity is ϕ=Vp/V, and the bulk strain εv of coal-rock mass can be obtained by (11) [16, 17]:(20)εv=ΔVV=−1Kσ¯−αp+εs,(21)ΔVpVp=−1Kpσ¯−βp+εs,where σ¯=−σkk/3, β=1−Kp/Ks, and Kp is the module of pore bulk.It is assumed that the strain caused by coal adsorption is equal to the amount of pore variation. Without considering the adsorption effect of gas, the bulk change of porous media meets the equivalent law of Betti-Maxwell:(22)∂V∂pσ¯=∂Vp∂σ¯p,and therefore,(23)Kp=ϕαK.Then, deducted from the definition of porosity(24)ΔVV=ΔVsVs+Δϕ1−ϕ,(25)ΔVpVp=ΔVsVs+Δϕϕ1−ϕ.Connecting (20), (21), (24) and (25), we get(26)Δϕ=ϕ1K−1Kpσ¯−p.Substituting (20) and (23) into (26), we get(27)Δϕ=α−ϕεv+pKs−εs.When the pore pressure of coal-rock mass isp0, the initial porosity is ϕ0, and the initial bulk strain is 0; then, the porosity can be expressed as(28)ϕ=11+S1+S0ϕ0+αS−S0.In formula (28), S=εv+p/Ks−εs and S0=p0/Ks−εLp0/p0+pL.


### 4.1.2. Dynamic Change Model of Coal-Rock Mass Permeability

Permeability, a physical parameter reflecting the difficult degree of gas flow in coal seam, is important for studying coupling process between deformation of mining coal-rock mass and gas flow. The relationship between the permeability of coal seam and the bulk strain of coal can be deduced by the Kozeny-Carman equation:(29)kk0=11+εvϕϕ03.Neglecting the bulk deformation of coal, (29) can be simplified as(30)kk0=ϕϕ03.Connecting (28) and (30), we get(31)kk0=11+S1+S0+αϕ0S−S03.


## 4.1.1. Dynamic Variation Model of Coal-Rock Mass Porosity

The bulkV of porous medium in coal-rock mass is composed of the bulk of solid phase Vs and the bulk of pore volume Vp, namely, V = Vs + Vp. The definition of porosity is ϕ=Vp/V, and the bulk strain εv of coal-rock mass can be obtained by (11) [16, 17]:(20)εv=ΔVV=−1Kσ¯−αp+εs,(21)ΔVpVp=−1Kpσ¯−βp+εs,where σ¯=−σkk/3, β=1−Kp/Ks, and Kp is the module of pore bulk.It is assumed that the strain caused by coal adsorption is equal to the amount of pore variation. Without considering the adsorption effect of gas, the bulk change of porous media meets the equivalent law of Betti-Maxwell:(22)∂V∂pσ¯=∂Vp∂σ¯p,and therefore,(23)Kp=ϕαK.Then, deducted from the definition of porosity(24)ΔVV=ΔVsVs+Δϕ1−ϕ,(25)ΔVpVp=ΔVsVs+Δϕϕ1−ϕ.Connecting (20), (21), (24) and (25), we get(26)Δϕ=ϕ1K−1Kpσ¯−p.Substituting (20) and (23) into (26), we get(27)Δϕ=α−ϕεv+pKs−εs.When the pore pressure of coal-rock mass isp0, the initial porosity is ϕ0, and the initial bulk strain is 0; then, the porosity can be expressed as(28)ϕ=11+S1+S0ϕ0+αS−S0.In formula (28), S=εv+p/Ks−εs and S0=p0/Ks−εLp0/p0+pL.


## 4.1.2. Dynamic Change Model of Coal-Rock Mass Permeability

Permeability, a physical parameter reflecting the difficult degree of gas flow in coal seam, is important for studying coupling process between deformation of mining coal-rock mass and gas flow. The relationship between the permeability of coal seam and the bulk strain of coal can be deduced by the Kozeny-Carman equation:(29)kk0=11+εvϕϕ03.Neglecting the bulk deformation of coal, (29) can be simplified as(30)kk0=ϕϕ03.Connecting (28) and (30), we get(31)kk0=11+S1+S0+αϕ0S−S03.


## 4.2. Solid-Gas Coupling Mathematical Model of Mining Coal Deformation and Pressure Relief Gas Flow

According to the antecedent, (11) is the control equation of mining coal deformation; substituting (4) into (11), we get(32)Gui,jj+G1−2vuj,ji−αpi−KεLpLp+pL2pi+fi=0.It can be obtained from (28):(33)∂ϕ∂t=α−ϕ1+S∂εv∂t+1Ks∂p∂t−εLpLp+pL2∂p∂t.Substituting (33) into (19), the control equation of the pressure relief gas of the mining coal can be obtained:(34)ϕ+ρcpaVLpLp+pL2+α−ϕp1+SKs−α−ϕεLpLp1+Sp+pL2∂p∂t−∇⋅kμp∇p=−α−ϕp1+S∂εv∂t.Formulas (32) and (34) constitute a gas-solid coupling mathematical model of coal seam deformation and pressure relief gas flow in protection seam mining.


## 4.3. Solid-Gas Coupling Mathematical Model of Rock Mass Deformation and Pressure Relief Gas Flow in Protective Seam Mining

When the rock mass is affected by mining, deformation and cracks occur, the change of rock porosity will lead to the change of permeability, and the pressure relief gas will flow under the effect of gas pressure gradient. The flow of gas in the mining rock mass is different from that of the mining coal, mainly in the absence of adsorption of the rock mass on the gas. Therefore, there is no need to consider the deformation of bulk expansion caused by the gas diffusion process and gas adsorption/desorption, so it is necessary to modify the solid-gas coupling mathematical model of mining coal deformation and pressure relief gas flow.To sum up, the governing equation of mining rock mass deformation can be expressed as(35)Gui,jj+G1−2vuj,ji−αpi+fi=0.Similarly, the control equation of pressure relief gas of mining coal can be expressed as(36)ϕ+α−ϕp1+S1Ks∂p∂t−∇⋅kμp∇p=−α−ϕp1+S1∂εv∂t,where S1=εv+p/Ks.


## 4.4. Definite Conditions

The control equations (32) and (34–36) constitute the complicated solid-gas coupling mathematical model of the mining coal-rock mass deformation and the pressure relief gas flow. For a particular solution, additional conditions must be added, that is, initial conditions and boundary conditions. Definite conditions of the model include the boundary conditions of the deformation field of the mining coal-rock mass and the initial conditions and boundary conditions of the gas seepage [17–22].


### 4.4.1. Boundary Conditions of Mining Coal-Rock Mass Deformation Field

The first kind of boundary condition: the surface force of the solid skeleton of the coal-rock mass is known as(37)σij⋅nj=f¯ix,y,z,t,t∈0,+∞.The second kind of boundary condition: the surface displacement of the solid skeleton of the coal-rock mass is known as(38)uix,y,z,t=gi¯x,y,z,t,t∈0,+∞,where nj is the unit normal vector of the surface solid skeleton of coal-rock mass and gi¯ and f¯i are the known surface displacement functions and known surface stress distribution functions.


### 4.4.2. Initial Conditions and Boundary Conditions of Gas Seepage

(1) Initial Conditions of Gas Seepage
(39)
p
x
,
y
,
z
t
=
0
=
p
¯
x
,
y
,
z
,
q
x
,
y
,
z
t
=
0
=
q
¯
x
,
y
,
z
.(2) Boundary Conditions of Gas Seepage. The first kind of boundary condition (the Dirichlet condition):(40)px,y,z,t=p¯x,y,z,t,t∈0,+∞.The second kind of boundary condition (the Neumann condition):(41)∂px,y,z,t∂n=Q¯x,y,z,t,t∈0,+∞.The third kind of boundary condition (equal flow at the interface):(42)K∂p∂n1=K′∂p′∂n′2,where p¯ and Q¯ are the distribution function and the fluid flow function of the pressure p and time t at the point (x, y, z) of the outer boundary, respectively, and q¯ is the initial flow value at the outer boundary.


## 4.4.1. Boundary Conditions of Mining Coal-Rock Mass Deformation Field

The first kind of boundary condition: the surface force of the solid skeleton of the coal-rock mass is known as(37)σij⋅nj=f¯ix,y,z,t,t∈0,+∞.The second kind of boundary condition: the surface displacement of the solid skeleton of the coal-rock mass is known as(38)uix,y,z,t=gi¯x,y,z,t,t∈0,+∞,where nj is the unit normal vector of the surface solid skeleton of coal-rock mass and gi¯ and f¯i are the known surface displacement functions and known surface stress distribution functions.


## 4.4.2. Initial Conditions and Boundary Conditions of Gas Seepage

(1) Initial Conditions of Gas Seepage
(39)
p
x
,
y
,
z
t
=
0
=
p
¯
x
,
y
,
z
,
q
x
,
y
,
z
t
=
0
=
q
¯
x
,
y
,
z
.(2) Boundary Conditions of Gas Seepage. The first kind of boundary condition (the Dirichlet condition):(40)px,y,z,t=p¯x,y,z,t,t∈0,+∞.The second kind of boundary condition (the Neumann condition):(41)∂px,y,z,t∂n=Q¯x,y,z,t,t∈0,+∞.The third kind of boundary condition (equal flow at the interface):(42)K∂p∂n1=K′∂p′∂n′2,where p¯ and Q¯ are the distribution function and the fluid flow function of the pressure p and time t at the point (x, y, z) of the outer boundary, respectively, and q¯ is the initial flow value at the outer boundary.


## 5. Conclusions

The solid-gas coupling effect of coal-rock mass deformation and pressure relief gas flow in protective layer mining is very complicated, involving rock mechanics, seepage mechanics, elastic mechanics, and other subjects. Based on the existing research results, the seepage control equation of mining coal-rock mass and the control equation of deformation field are established including solid-gas coupling variable of mining coal-rock mass, namely, the dynamic change equation of coal-rock mass porosity and permeability. According to the coupling equations, the corresponding definite conditions are put forward, and the solid-gas coupling mathematical model of the deformation of coal-rock mass and pressure relief gas flow in the protection layer mining are built. The details are as follows:(1)
The control equations (32) and (35) of the coal-rock mass deformation in protection layer mining that contain the coupling term which reflects the pore gas pressure and gas adsorption/desorption effect is established, which can only be solved by connecting with the governing equations of gas seepage field.(2)
The governing equations (34) and (36) of the flow field of the pressure relief gas flow in the mining coal-rock mass that include the coupling term reflecting the effect of coal-rock mass deformation is established, and it is necessary to connect with the control equation of mining coal-rock deformation field to be solved.(3)
The combination of the solid-gas coupling control equation of mining coal-rock mass deformation and pressure relief gas flow with definite conditions is established, which constitute the solid-gas coupling mathematical model of coal-rock mass deformation and pressure relief gas flow, and it lays a foundation for numerical simulation research of coal-rock mass deformation and pressure relief gas flow law in following protection layer mining. Numerical simulations research will be showed in other papers.



---

*Source: 5162628-2018-04-02.xml*