# Double Unloading Gas Control Technology for Fracturing Soft Coal Seams in Overlying Key Strata
## 2 Method
### Establishing Numerical Models  
#### 2.4.1 Mathematical Modeling  
1. Seepage equations of motion  
The gas in the coal is mainly composed of two parts: free gas \(Q_{b}\) and adsorbed gas \(Q_{a}\). Therefore, the total gas storage capacity is  
\[Q=Q_{\alpha}+Q_{\rm b}=\frac{\varphi pM_{\rm c}}{RT}+\frac{abp}{1+bp}\frac{100- A-M}{100+31M}\rho_{s}\rho_{\rm a} \tag{6}\]  
where \(\varphi\) is the porosity of coal,%; \(p\) is the coal seam gas pressure, MPa; \(R\) is the gas molar constant, J/(mol\(\cdot\)K); \(T\) is the coal seam temperature, K; \(M_{\rm c}\) is the molar mass of methane, kg/mol; \(a\) is the maximum adsorbed gas amount of coal, m\({}^{3}\)/t; \(b\) is the adsorption constant, MPa\({}^{-1}\); \(M\) is the proportion of coal moisture, %; \(A\) is the proportion of coal ash content,%; \(\rho_{s}\) is the coal density, kg/m\({}^{3}\); and \(\rho_{a}\) is the gas density under standard conditions, kg/m\({}^{3}\).  
The gas flow in coal seam conforms to [PERSON]'s law of seepage, and the calculation equation is  
\[V=-\frac{k_{0}}{\mu}\
abla p \tag{7}\]  
where \(V\) is the gas seepage velocity, m/s; \(k_{0}\) is the initial permeability of the coal seam, and m\({}^{2}\); \(\mu\) is the gas dynamic viscosity coefficient, Pa s.  
Figure 3: Design and test of fracturing scheme for overlying strata.  
The flow of gas in coal seams conforms to the law of conservation of mass, and the calculation equation [37]:  
\[\frac{\partial Q}{\partial t}+\
abla\bullet(\rho V)=0. \tag{8}\]  
Combine Equations (6)-(8) to obtain the following formula:  
\[\frac{\partial(\frac{\varphi pM_{\rm c}}{RT}+\frac{abp}{1+pp}\frac{100-A-M}{100 +3 IM}\rho_{\rm s}\rho_{\rm a})}{\partial t}+\
abla(\frac{pM_{\rm c}}{RT})(- \frac{k_{0}}{\mu}\
abla p)=0. \tag{9}\]
2. Control equation for coal deformation Assuming that the coal seam is isotropic and considering the adsorption expansion effect, combined with [PERSON]'s effective stress formula and constitutive equation, the deformation control equation of the coal body can be obtained [38]: \[\frac{G}{(1-2v)}u_{j,ij}+Gu_{i,jj}+(a+\frac{2G}{3K_{\rm s}})p_{,i}+F_{i}-\{[ \frac{4G}{9V_{m}K_{\rm s}}-\frac{2(1-2v)}{3V_{m}}]\frac{ab\rho_{\rm s}RT}{1+bp} \}p_{,i}=0\] (10) where \(G\) is the shear modulus, MPa; \(u_{i}\) is the displacement of the coal body in the \(i\) direction, m; \(u_{i,jj}\), \(u_{j,ij}\) is the displacement component; \(F_{i}\) is the body stress, MPa/m\({}^{3}\); \(p_{,i}\) is the gas pressure in all directions, MPa; \(v\) is [PERSON]'s ratio of coal body; \(V_{m}\) is the molar volume of methane under standard conditions, L/mol; and \(K_{\rm s}\) is the volume modulus of the coal skeleton.
3. Other equations The dynamic model of coal seam porosity can be expressed as [39] \[\varphi=\frac{\varphi_{0}+\varepsilon_{v}}{1+\varepsilon_{v}}\] (11) where \(\varphi_{0}\) is the initial porosity of the coal seam, %; \(\varepsilon_{v}\) is the volumetric strain of the coal seam. \[k=\frac{k_{0}}{1+\varepsilon_{v}}(1+\frac{\varepsilon_{v}}{\varphi_{0}})^{3}\] (12) where \(k_{0}\) is the initial permeability of the coal seam.