## 3 Hydro-Mechanical Coupled Model for Fractured Rock Mass  
### Governing Equations  
A rock matrix is regarded as impermeable because of its extremely low permeability, and hydro-mechanical coupling only occurs in fractures. In this paper, the hydro-mechanical coupled model in a fractured rock mass has been discussed as follows:  
Compression deformation of a fracture will occur due to normal stress, which leads to variation in the permeability with a change in the fracture's aperture. In this paper, it is assumed that the normal stiffness of fractures is constant [67,68]. Under the action of normal stress, the fracture aperture \(\mathrm{e}\) is calculated by the following formula:  
\[\mathrm{e}=\mathrm{e}_{0}+\sigma_{\mathrm{n}}/\mathrm{K}_{\mathrm{n}} \tag{7}\]  
where \(\mathrm{e}_{0}\) is the initial aperture, \(\sigma_{\mathrm{n}}\) is the normal stress of the fracture surface, the compressive stress is negative, and \(\mathrm{K}_{\mathrm{n}}\) is the normal stiffness of the fracture. The intrinsic permeability of fracture \(\mathrm{K}\) is calculated by the following formula [69,70]:  
\[\mathrm{K}=\frac{\rho\mathrm{ge}^{2}}{12\mu} \tag{8}\]  
where \(\rho\) is the fluid density, \(\mathrm{g}\) is the gravitational acceleration, and \(\mu\) is the dynamic viscosity coefficient of the fluid. It should be noted that Formula (8) is valid only when the fracture aperture \(\mathrm{e}\) is greater than 0; when the aperture is less than 0, that is, when the fracture surface is embedded, the inherent permeability tends to 0. Figure 7: Variation curve of Reynolds number with water pressure: (**a**) unfilled condition, (**b**) filled with 50 mesh quartz sand, (**c**) filled with 100 mesh quartz sand, and (**d**) filled with 200 mesh quartz sand. Equation (7) is introduced into Equation (8) to obtain:  
\[K=\frac{\rho g(e_{0}+\sigma_{n}/K_{n})^{2}}{12\mu} \tag{9}\]  
To remain consistent with the first-order term of [PERSON]'s law, Equation (4) is often expressed as:  
\[-\frac{dP}{dX}=\frac{\rho g}{K}v+\beta\rho v^{2} \tag{10}\]  
where P is the water pressure, \(v\) is the fluid velocity, and \(\beta\) is the non-Darcy coefficient. It is related to the permeability and porosity of fractured media and is inversely proportional to the power of permeability [71; 72]. \(\rho\) is the fluid density. \[v=-\frac{1}{\left(1/K+\frac{\rho v}{g}\right)}J \tag{11}\]  
where J is the hydraulic gradient. In the following equation, \(K_{F}\) is defined as the Forchheimer-type non-Darcy permeability:  
\[K_{F}=1/(1/K+\beta v/g) \tag{12}\]  
Equation (6) is introduced into Equation (9) to obtain the following:  
\[K_{F}=1/\left(12\mu/\rho g\left(e_{0}+\sigma_{n}/K_{n}\right)^{2}+\beta v/g\right) \tag{13}\]  
The influence of the normal stress of a fracture changes the permeability of a fracture, which will lead to changes in the flow field and water pressure distribution in the fracture, and these changes have an effect on the stress field. The stress balance equation of a rock mass can be expressed by the principle of virtual work:  
\[\int\limits_{v}\!\delta\epsilon^{T}d\sigma dV-\int\limits_{v}\!\delta u^{T}dfdV -\int\limits_{S}\!\delta u^{T}dtdS=0 \tag{14}\]  
where t is the surface force, \(f\) is the physical strength, \(\delta\epsilon\) and \(\delta u\) are the virtual displacement and virtual strain, respectively. The constitutive model for the fracture is stated as follows:  
\[d\sigma^{\prime}=D_{e}(d\epsilon-d\epsilon_{s}) \tag{15}\]  
where \(D_{e}\) is the elastic matrix and \(d\epsilon_{s}\) is the normal strain of fracture caused by water pressure. The specific expression is stated as follows:  
\[\mathrm{K}=\frac{\rho\mathrm{ge}^{2}}{12\mu} \tag{8}\]0  
where \(m=\left[1,1,1,0,0,0\right]^{T}\), \(\overline{P}\) is the fluid pressure, and \(K_{S}\) is the compression modulus of the rock mass.