## 3 Establishment of a Grouting Pipeline Blockage Model
### Establishment of Computational Models  
The slurry used in the grouting system has a density of 1700 kg/m\({}^{3}\), yield stress of 368.3 Pa, and viscosity of 0.5 Pa-s. The flow regime in the pipeline (laminar, turbulent, or transitional) is determined using the Reynolds number. The Reynolds number is calculated using the following equation [19]:  
\[R_{e}=\frac{v\rho d}{\mu}, \tag{1}\]  
In the equation, \(\rho\) represents the fluid density, \(v\) is the average velocity at the cross-section, \(\mu\) denotes the dynamic viscosity of the fluid, and \(d\) refers to the inner diameter of the filling pipeline.  
Given that the density of the filling slurry is 1700 kg/m\({}^{3}\), the pipeline transport velocity is selected as 2 m/s, and the diameter of the transport pipeline is 0.16 m.  
\[R_{e}=\frac{v\rho d}{\eta}=\frac{2\times 1700\times 0.16}{0.5}=1088\]  
The critical Reynolds number for a Newtonian fluid is 2100. For non-Newtonian Bingham slurry in engineering applications, the critical Reynolds number is variable and generally exceeds 2100. Therefore, the flow within the slurry pipeline remains in a laminar state. The motion of any substance must satisfy the mass conservation equation, which, in a Cartesian coordinate system, is expressed as follows:  
Equation for conservation of fluid mass:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho u_{i}}{\partial x_{i}}=0 \tag{2}\]  
Particle mass conservation equation:  
\[\frac{\partial\rho_{p}}{\partial t}+\frac{\partial\rho_{p}U_{pj}}{\partial x_ {j}}=\frac{\partial}{\partial x_{j}}\left(\frac{v_{p}}{\sigma_{p}}\frac{ \partial\rho_{p}}{\partial x_{j}}\right), \tag{3}\]  
Figure 6: Simulated pipeline distribution diagram.  
Figure 7: Simulated pipeline cross-section.  
In the equation, \(\rho\) represents the fluid density, xi is the coordinate component in the i-direction, and \(u_{i}\) denotes the velocity in the i-direction. Moreover, \(x_{i}\) refers to the Eulerian coordinates, with i = 1, 2, and 3; \(\rho\) represents the liquid phase density; \(\rho_{p}\) represents the solid phase density; \(u_{i}\) is the velocity vector of the liquid phase; \(U_{pj}\) denotes the velocity vector of the solid phase; and \(\
u_{p}\) is the turbulent diffusion coefficient.  
During the motion of the fluid, the momentum conservation equation also applies. Essentially, the momentum equation follows [PERSON] n's second law, which states that the rate of change in momentum at a fluid node is equal to the sum of all external forces acting on that node. The equation is expressed as follows:  
\[\frac{\partial}{\partial t}(\alpha_{i}\rho_{i}v_{i})+\
abla\cdot(\alpha_{i} \rho_{i}v_{i}v_{i})=\alpha_{i}\rho_{i}g+M_{i}-\alpha_{i}\
abla p+\
abla\cdot \tau_{i}, \tag{4}\]  
\[\tau_{i}=\alpha_{i}\mu_{i}\Big{(}\
abla v_{l}+\
abla v_{l}^{T}\Big{)}, \tag{5}\]  
\[\tau_{s}=\alpha_{s}\mu_{s}\Big{(}\
abla v_{s}+\
abla v_{s}^{T}\Big{)}+\alpha_ {s}\left(\lambda_{s}-\frac{2}{3}\mu_{s}\right)\
abla\cdot v_{s}, \tag{6}\]  
where  
\(\tau_{i}\) represents the shear stress tensor;  
\(\mu\) denotes the shear viscosity;  
\(M_{i}\) indicates the momentum exchange between the solid and liquid phases;  
\(\lambda_{s}\) represents the overall viscosity of the solid phase.