## 3 Engineering Application
### The Principle of Fluid-Solid Coupling Modeling  
FLAC3D software conducts fluid-structure coupling calculations based on the finite difference method. It defaults that the rock and soil mass are continuous media. The fluid seepage follows [PERSON]'s law and satisfies [PERSON]'s equation, mainly including the following equations:  
(1) Equilibrium equation  
For small deformation, the fluid particle equilibrium equation is:  
\[-q_{i,i}+q_{v}=\frac{\partial\varsigma}{\partial t} \tag{15}\]  
where \(q_{i,i}\) is seepage velocity (m/s); \(q_{v}\) is the volume fluid source intensity (s\({}^{-1}\)); \(\varsigma\) is the change in fluid volume per unit volume of porous media. \[\frac{\partial\varsigma}{\partial t}=\frac{1}{M}\frac{\partial p}{\partial t }+\alpha\frac{\partial\epsilon}{\partial t}-\beta\frac{\partial T}{\partial t} \tag{16}\]  
where \(M\) is the Biot modulus (N/m\({}^{2}\)); \(\alpha\) is the Biot coefficient; \(\beta\) is the coefficient of thermal expansion (\({}^{\circ}\)C\({}^{-1}\)), which considers liquid and solid particles. \(p\) is the pore water pressure (Pa); \(\epsilon\) is the volume strain; \(T\) is the temperature. The momentum balance equation is:  
\[\sigma_{ij,j}+\rho g_{i}=\rho\frac{\mathrm{d}v_{i}}{\mathrm{d}t} \tag{17}\]  
\[\rho=(1-n)\rho_{\mathrm{s}}+n\rho_{\mathrm{w}} \tag{18}\]  
Among them, \(\rho\) is the bulk density (kg/m\({}^{3}\)); \(\rho_{\mathrm{s}}\) and \(\rho_{\mathrm{w}}\) are the density of solid and liquid, respectively; \(n\) is porosity; \(g_{i}\) is the component of gravity acceleration (m/s\({}^{2}\)); \(v_{i}\) is the velocity component of the medium (m/s). \[q_{i}=-k\big{[}p-\rho_{\mathrm{f}}x_{j}g_{j}\big{]} \tag{19}\]  
Figure 2: The location of the Chenjiadian tunnel. where \(k\) is the permeability coefficient of the medium (m\({}^{2}\)/(Pa\(\cdot\)s)); \(\rho_{\rm f}\) is the fluid density (kg/m\({}^{3}\)); \(g_{\rm i}\) is the component of gravity acceleration (m/s\({}^{2}\)). (2) Constitutive equation  
The volume strain and the pore pressure of the fluid interact with each other. The change in the strain makes the pore pressure readjust, and the change in the pore pressure also affects the occurrence of the strain. The descriptive equation is:  
\[\Delta\sigma_{\rm ij}+\alpha\Delta p\delta_{ij}=H_{ij}(\sigma_{\rm ij},\Delta \tilde{\xi}_{ij}) \tag{20}\]  
Among them, \(\Delta\sigma_{\rm\bar{ij}}\) is the stress increment; \(\Delta p\) is the pore water pressure increment \(\delta_{\rm\bar{ij}}\) is the Kronecker factor. \(H_{\rm ij}\) is the given function. \(\Delta\tilde{\xi}_{\rm ij}\) is the total strain increment. (3) Compatibility equation  
The relationship between strain rate and velocity gradient is:  
\[\epsilon_{ij}=\frac{1}{2}(v_{i,j}+v_{j,i}) \tag{21}\]  
where \(v\) is the velocity of a point in the medium (m/s). (4) Boundary condition  
There are four types of boundary conditions in seepage calculation: (1) the given pore water pressure, (2) the given velocity vector outside the boundary normal direction, (3) the impervious boundary given by default in the program and (4) the pervious boundary. The form of pervious boundary is as follows:  
\[q_{\rm n}=h(p-p_{\rm e}) \tag{22}\]  
where \(q_{\rm n}\) is the velocity component in the normal direction outside the boundary, \(h\) is the leakage coefficient (m\({}^{3}\)/(N\(\cdot\)s)) and \(p_{\rm e}\) is the pore water pressure at the seepage outlet. (5) Time scale  
The fluid and mechanics processes are involved in the fluid-structure coupling calculation, and the time scales in these two states need to be considered. The characteristic time can generally characterize the size of the time scale. The characteristic time of the mechanical process is expressed as follows:  
\[t_{c}^{\rm m}=\sqrt{\frac{\rho}{K_{u}+4/3G}}L_{c} \tag{23}\]  
where \(K_{\rm u}\) is the undrained bulk modulus; \(G\) is the shear modulus; \(\rho\) is the density; \(L_{\rm c}\) is the feature length (the average size of the model).