## 2 Equations and Methods  
### Conservation Equations  
There is a fracture network in the reservoir after the hydraulic fracturing of rock or shale. Fluid (gas and water) migration in the porous media follows the laws of mass conservation and momentum conservation [28]. The mass equations of the gas phase and the water phase are:  
\[-\
abla(\frac{v_{g}}{B_{g}})+q_{sg}+q_{smfg}=\frac{\partial}{\partial t}( \frac{\varphi_{g}s_{g}}{B_{g}}) \tag{1}\]  
\[-\
abla(\frac{v_{w}}{B_{w}})+q_{sw}=\frac{\partial}{\partial t}(\frac{\varphi _{g}s_{w}}{B_{w}}) \tag{2}\]  
where \(v_{g}\) is the gas velocity, m/s; \(B_{g}\) is the gas volume coefficient, dimensionless; \(q_{sg}\) is the source term of formation gas, 1/s; \(q_{smfg}\) represents the gas source term that diffuses from the matrix into the fracture, 1/s; \(q_{f}\) is the average fracture porosity of the reservoir, dimensionless; \(s_{g}\) is the gas saturation, dimensionless; \(v_{w}\) is the water phase velocity, dimensionless; \(B_{w}\) is the volume coefficient of the water phase, dimensionless; \(q_{sw}\) is the formation water source, 1/s; and \(s_{w}\) is the gas saturation, dimensionless. It is generally believed that the migration process of fluid from the matrix to the fracture only includes the diffusion process of the gas phase and the diffusion process of the anhydrous phase.  
For the gas and water phases, the Darcy equations are written as:  
\[v_{g}=-\frac{k_{\infty}k_{rg}}{\mu_{g}}(\
abla P_{g}-\rho_{gg}\
abla h) \tag{3}\]  
\[v_{w}=-\frac{k_{\infty}k_{rw}}{\mu_{w}}(\
abla P_{w}-\rho_{wg}\
abla h) \tag{4}\]  
Figure 2: Idea of large-size proppant proposed by [PERSON] et al. [13]. (**a**) The upper part is the traditional fracture–proppant model, where the proppant pack consist of a great number of small-size proppants. (**b**) The lower part is the creative fracture–proppant model, where the proppant has a larger size and is arranged in the form of a discrete pillar in the fracture.  
where \(k_{\infty}\) is permeability, m\({}^{2}\); \(k_{sg}\) is the relative permeability of gas, dimensionless; \(P_{g}\) is the gas pressure, Pa; \(\mu_{g}\) is the gas viscosity, Pa-s; \(\rho_{g}\) is the gas density, kg/m\({}^{3}\); \(k_{\pi w}\) is the relative permeability of the water phase, dimensionless; \(P_{w}\) is the water phase pressure, Pa; \(\mu_{w}\) is the viscosity of the water phase, Pa-s; \(\rho_{w}\) is the density of the water phase, kg/m\({}^{3}\); and \(h\) is the reservoir depth, m. Combine Equations (1) and (3) to obtain the gas phase pressure control equation, as follows:  
\[\
abla\Big{[}\frac{k_{\infty}k_{rg}}{B_{g}\mu_{g}}(\
abla P_{g}-\rho_{g}g \
abla h)\Big{]}+q_{sg}+q_{smfg}=\frac{\partial}{\partial t}(\frac{\varphi_{ f}s_{g}}{B_{g}}) \tag{5}\]  
Combine Equations (2) and (4) to obtain the pressure control equation of the water phase, as follows:  
\[\
abla\Big{[}\frac{k_{\infty}k_{rw}}{B_{w}\mu_{w}}(\
abla P_{w}-\rho_{w}g \
abla h)\Big{]}+q_{sw}=\frac{\partial}{\partial t}(\frac{\varphi_{f}s_{w}}{ B_{w}}) \tag{6}\]  
Supplement the auxiliary equation, and the gas-liquid saturation meets the following relationship:  
\[s_{g}+s_{w}=1 \tag{7}\]  
The capillary pressure meets  
\[P_{cgw}=P_{g}-P_{w} \tag{8}\]  
So far, there are 4 unknown variables (\(P_{g}\), \(P_{w}\), \(s_{g}\), \(s_{w}\)) and 4 Equations (5)-(8). Hence, the equation can be solved.