## 2 Physical Modelling Experiment Design  
### Model Design  
The similarity simulation test system, developed by the China University of Mining and Technology (CUMT), in conjunction with various monitoring systems, was utilized for the experimental tests. The dimensions of the plane similarity model test bench were set at 250 cm \(\times\) 20 cm \(\times\) 180 cm. For this research, a mine in Central China was selected as the engineering background, with a coal seam depth of 148 m. Based on the pipeline's location and the test system dimensions, the geometric similarity ratio \(C_{1}\) was determined as follows:  
\[C_{1}=\frac{148\text{ cm}}{14,800\text{ cm}}=1{:}100 \tag{1}\]  
The materials used for the similarity model included river sand, calcium carbonate, and gypsum, with a bulk density of 16.2 kN/m\({}^{3}\), which was used to calculate the model weight similarity ratio as follows:  
\[C_{\gamma}=\frac{16.2\text{ kN}/\text{m}^{3}}{25.2\text{ kN}/\text{m}^{3}}=9{:}14 \tag{2}\]  
where 25.2 kN/m\({}^{3}\) represents the bulk density of the prototype material, such as the in-situ rock at the mining site. The stress similarity ratio \(C_{\sigma}\) was then calculated by multiplying the weight similarity ratio \(C_{\gamma}\) by the geometric similarity ratio \(C_{1}\), ensuring that the stress distribution in the model accurately reflects that of the following prototype:  
\[C_{\sigma}=C_{\gamma}\times C_{1}=9{:}1400 \tag{3}\]  
Laboratory tests were conducted to determine the compressive strength, tensile strength, modulus of elasticity, and other mechanical properties of the materials used in the model. These properties helped refine the proportions of the materials, as shown in Table 1 and Figure 1. The ends of the model were fixed, ensuring that the boundary conditions accurately represented the operational conditions of the pipeline in the prototype. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Layer** & **Name** & **Thickness (m)** & \
\begin{tabular}{c} **Compressive** \\ **Strength (MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Key Layer** \\ **Discrimination** \\ \end{tabular} & **Component Ratio** \\ \hline
1 & Clay & 16.2 & 9 & Negative & 855 \\
2 & Fine Sandstone & 12.6 & 43.6 & Negative & 637 \\
3 & Medium Sandstone & 5.5 & 52.4 & Negative & 537 \\
4 & Breccia & 3.7 & 80.6 & Main Key Layer & 237 \\
5 & Mudstone & 10.2 & 27.2 & Negative & 746 \\
6 & Silstone & 7.9 & 57.6 & Negative & 537 \\
7 & Mudstone & 14.8 & 27.2 & Negative & 746 \\
8 & Coarse Sandstone & 4.4 & 54.7 & Negative & 537 \\
9 & Fine Sandstone & 9.6 & 43.6 & Negative & 637 \\
10 & Medium Sandstone & 11.1 & 70.2 & Primary Key Layer & 228 \\
11 & Sandy Mudstone & 5.7 & 37.7 & Negative & 646 \\
12 & Medium Sandstone & 8.2 & 52.4 & Negative & 537 \\
13 & Sandy Mudstone & 3.9 & 37.7 & Negative & 646 \\
14 & Fine Sandstone & 7 & 52.4 & Negative & 537 \\
15 & Sandy Mudstone & 2.5 & 37.7 & Negative & 646 \\
16 & 3\# Coal Stratum & 7 & 14 & Negative & 755 \\
17 & Floor & 20 & 42.6 & Negative & 637 \\ \hline \hline \end{tabular} Note: The component ratio number is \(\omega\) (river sand)/ \(\omega\) (calcium carbonate)/ \(\omega\) (gypsum). For instance, ratio 637 indicates that the mass fraction of river sand in the similar materials is 60%, while calcium carbonate and gypsum account for 30% and 70% of the remaining mass fraction of ingredients other than river sand, representing 12% and 28% of the total weight, respectively. The water content is 10% of the total mass of the materials.  
\end{table} 
Table 1: Rock parameters and similar material ratios [27]. PVC pipe was employed as a similar material for the pipeline, measuring 2500 cm in length, with an outer diameter of 4 cm and a wall thickness of 1.3 mm. The boundary conditions of the pipe significantly influence its mechanical properties; however, the test constraints make it challenging to simulate the boundary conditions of an infinitely long pipe.