# Simulation of Gas Fracturing in Reservoirs Based on a Coupled Thermo-Hydro-Mechanical-Damage Model
## 2 The Mechanism of Hydraulic Fracturing under Temperature Field
### The Equation for Gas Equilibrium under the Action of a Temperature Field  
It is assumed that the numerical simulation model is constructed consisting of a solid matrix containing freely diffusible pore gases and pore spaces. Under these conditions, the mass balance equation for the gas can be expressed as follows:  
\[\frac{\partial m}{\partial t}+\
abla(\rho_{g}\mathbf{v}_{g})=Q_{s} \tag{7}\]  
where, \(m\) is the amount of gas, \(\rho_{g}\) is the density of the gas, \(q_{g}\) is the Darcy velocity of the gas, \(Q_{s}\) is the gas source, and \(t\) is the time. The amount of methane gas in the solid matrix can be described as:  
\[m=\rho_{g}\phi+\rho_{ga}\rho_{c}V_{sg} \tag{8}\]  
where, \(\phi\) is the porosity of the solid matrix, \(\rho_{ga}\) is the density of the gas under the standard condition, \(\rho_{c}\) is the density of the solid matrix, and \(V_{sg}\) is the content of the adsorbed gas. The content of adsorbed gas \(V_{sg}\) can be calculated by:  
\[V_{sg}=\frac{V_{L}p}{p+P_{L}}\exp[-\frac{c_{2}}{1+c_{1}p}(T_{ar}+T-T_{t})] \tag{9}\]  
In Equation (9), Tar denotes the absolute reference temperature of the rock formation in the absence of stress, while Tt signifies the temperature reference for gas adsorption/desorption experiments. Therefore, \((T_{ar}+T)\) represents the overall temperature of the rock formation. \(c_{1}\) denotes the pressure coefficient, while \(c_{2}\) represents the temperature coefficient. For ideal gases, it is usually expressed by the ideal gas equation:  
\[\rho_{g}=\frac{M_{g}}{R(T_{ar}+T)}p \tag{10}\]  
where, \(\rho_{g}\) is the density of the gas, \(p\) is the gas pressure, and \(M_{g}\) is the molar mass of the gas. The density of a gas in the standard state can be deduced from the ideal gas equation, which can be denoted by (11):  
\[\rho_{ga}=\frac{M_{g}p_{a}}{RT_{a}} \tag{11}\]  
The gas flow equation incorporating the non-Darcy Forchheimer model is employed. \[-\
abla p=\frac{k_{e}}{\mu}\mathbf{v}_{g}+\frac{c_{f}\rho_{g}}{\sqrt{k}}\mathbf{v}_{g }\abs{\mathbf{v}_{g}} \tag{12}\]  
where, \(k_{e}\) is the permeability of the medium, \(\mu\) is the dynamic viscous velocity of the gas, \(p\) is the gas pressure, \(\abs{\mathbf{v}_{g}}\) is the absolute value of the velocity vector, and \(\rho_{g}\) is the density of the gas mixture. \(c_{f}\) is the drag coefficient, which is dimensionless and represented as:  
\[c_{f}=\frac{1.75}{\sqrt{150e^{3}}} \tag{13}\]Equation (12) can be alternatively expressed in a form that aligns with [PERSON]'s law:  
\[\mathbf{v}_{g}=-\delta\frac{k_{e}}{\mu}\
abla p \tag{14}\]  
where, \(\delta\) is the correction factor to which [PERSON]'s law applies, which is defined as:  
\[\delta=\frac{1}{1+\frac{\sqrt{E_{g}}}{\mu}c_{f}\rho_{g}\big{|}\mathbf{v}_{g}\big{|}} \tag{15}\]  
For models that generally include porosity, there are:  
\[\mathrm{d}\phi=\frac{1}{K}(\alpha-\phi)(\mathrm{d}\overline{\sigma}+\mathrm{d }p) \tag{16}\]  
Regarding Equation (16), we can perform a transformation to obtain Equations (17) and (18):  
\[m=\rho_{g}\phi+\rho_{ga}\rho_{c}V_{sg} \tag{8}\]0  
\[m=\rho_{g}\phi+\rho_{ga}\rho_{c}V_{sg} \tag{8}\]1  
where the subscript 0 in the equation represents the initial state of the corresponding variable. Substituting Equation (4) into Equation (18) yields:  
\[m=\rho_{g}\phi+\rho_{ga}\rho_{c}V_{sg} \tag{8}\]2  
The effective permeability of a gas depends on the gas pressure to which the gas is subjected, as shown in Equation (20)  
\[m=\rho_{g}\phi+\rho_{ga}\rho_{c}V_{sg} \tag{8}\]3  
where, represents the gas permeability under extremely high gas pressure, where the Klinkenberg effect can be neglected. The parameter \"\(b\)\" in the equation is the Klinkenberg factor, which depends on the average free path of gas molecules. The relative mean free path of gas molecules is influenced by factors such as gas pressure, gas temperature, and molar mass. Thus, the factor \"\(b\)\" can be mathematically represented by Equation (21).