## 2 Methodology
### Governing Equations of the Solid-Fluid-Heat Coupling Process  
Deep shale gas reservoirs are characterized by high temperature and high pressure. The injection and flowback of fracturing fluid during the fracturing process will cause huge heat loss to the formation, especially in the area near the hydraulic fractures. At this time, the effect of temperature on gas properties must be considered. Considering the thermal convection and self-diffusion flow mechanisms, the shale gas flow control equation in the shale matrix can be obtained:  
\[\frac{\partial\left(\phi\rho_{fg}\right)}{\partial t}-\
abla\cdot\left(D_{e} \
abla\rho_{fg}+\rho_{fg}\beta_{T}\
abla T\right)=R_{n}+Q_{f} \tag{1}\]  
Figure 1: Changes caused by fracture closure. where \(\phi\) is the effective porosity of rock. \(\rho_{fg}\) is the density of free gas, kg/m\({}^{3}\). \(D_{e}\) is the effective self-diffusivity of gas, m\({}^{2}\)/s. \(\beta_{T}\) is the thermal conductivity, m\({}^{2}\)/(s\(\cdot\)K). \(T\) is the temperature, K.  
\(Q_{f}\) is the mass transfer rate between the shale matrix and the fracture:  
\[Q_{f}=-D_{e}\
abla\rho_{fg}\cdot\mathbf{n}_{f} \tag{2}\]  
where \(\mathbf{n}_{f}\) is the norm vector of the fracture surface. \(R_{n}\) is the net desorption rate per unit volume of shale:  
\[R_{n}=R_{d}\rho_{ag}-R_{f}(1-\alpha)\rho_{fg} \tag{3}\]  
where \(R_{d}\) is the desorption rate of adsorbed gas, 1/s. \(\rho_{ag}\) is the density of adsorbed gas, kg/m\({}^{3}\). \(R_{f}\) is the adsorption rate of free gas, 1/s. \(\alpha=\rho_{ag}/\rho_{agm}\) is the saturation of adsorption gas. \(\rho_{agm}\) is the density when the concentration of adsorbed gas reaches the maximum, kg/m\({}^{3}\). Ignoring the temperature change of the fracture surface and the presence of adsorbed gas, the self-diffusion flow equation in the fracture is  
\[\rho_{fg}\phi_{f}Z_{p}\frac{\partial p}{\partial t}-\
abla\cdot\left(\rho_{fg }Z_{p}D_{ef}\
abla p\right)=Q_{f} \tag{4}\]  
where \(\phi_{f}\) is the porosity of the fracture. \(Z_{p}\) is the gas compression factor under pressure. \(p\) is the pressure of the gas, Pa. \(D_{ef}\) is the self-diffusiveness of gas in the fractures, m\({}^{2}\)/s. The limited heat transfer efficiency between a gas and solid produces local thermal nonequilibrium [30]. Considering the heat exchange brought by the gas flow, the governing equation of the gas temperature field can be expressed as [31]  
\[\phi\rho_{fg}C_{g}\frac{\partial T_{g}}{\partial t}-\eta_{g1}\
abla^{2}T_{g}- \eta_{g2}\
abla^{2}p=Q_{Tf} \tag{5}\]  
\[\eta_{g1}=\phi k_{g}+\beta_{T}T\left(Z_{T}/Z_{p}-\rho_{fg}C_{g}\right) \tag{6}\]  
\[\eta_{g2}=-T_{g}\left[\beta_{T}+\left(Z_{T}/Z_{p}-\rho_{fg}C_{g}\right)\frac{ D_{e}}{p}\right] \tag{7}\]  
\[Q_{Tf}=\frac{q_{sf}}{1-\phi}\left(T_{s}-T_{g}\right) \tag{8}\]  
where \(C_{g}\) is the specific heat of the gas, J/(kg\(\cdot\)K). \(T_{g}\) is the temperature of the gas, K. \(k_{g}\) is the thermal conductivity of the gas, W/(K\(\cdot\)m). \(q_{sf}\) is the interstitial convective heat transfer coefficient, J/(m\({}^{3}\cdot\)K\(\cdot\)s). \(T_{s}\) is the temperature of the rock, K.  
The governing equation of the rock temperature field is  
\[\rho_{s}C_{s}\frac{\partial T_{s}}{\partial t}+\
abla(-k_{s}\
abla T_{s})=Q_ {Ts} \tag{9}\]  
Figure 1:0  
where \(\rho_{s}\) is the density of the rock, kg/m\({}^{3}\). \(C_{s}\) is the specific heat of the rock, J/(kg\(\cdot\)K). \(k_{s}\) is the thermal conductivity of the rock, W/(K\(\cdot\)m). Considering the quasi-static process of solid deformation, the stress equilibrium equation is  
Figure 1:1where \(g\) is the gravitational acceleration, m/s\({}^{2}\). \(\sigma=C:\varepsilon-\alpha p\) is the total stress considering fluid-solid coupling. When the gas cannot enter the fracture from the proppant embedding area, the relationship between the effective leakage surface area of the fracture and the proppant embedding is  
Figure 1:2