## 3 Design and Computation of Orthogonal Simulation Experiment
### Simulation of Ground Subsidence Displacements and Deformations  
(1) Computation of Ground Displacements (Vertical Subsidence and Horizontal Displacement) \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Layers** & **Thickness (m)** & **Density** & **Young’s** & **Poisson’s** & **Cohesion** & **Tension** & **Friction** \\  & **(m)** & **(Kg-m\({}^{3}\))** & **Modulus (GPa)** & **Ratio** & **(MPa)** & **(MPa)** & **Angle (\({}^{*}\))** \\ \hline Soil layer & 17 & 1700 & 12 & 0.3 & 0.02 & 12 & 20 \\ Rock layer & 180 & 2410 & 21.25 & 0.245 & 1.7 & 21.85 & 32 \\ Roof & 5 & 2410 & 24.5 & 0.24 & 1.8 & 26.5 & 32 \\ Coal Seam & \({}^{*}\) & 1320 & 13 & 0.26 & 1 & 13.5 & 30 \\ Floor & 100 & 2410 & 24.5 & 0.24 & 1.8 & 26.5 & 32 \\ \hline \hline \end{tabular} Note \({}^{*}\): the thickness of the coal seam is defined in Table 2 and varying for different tests.  
\end{table} 
Table 3: The rock layers and their thickness and physical and mechanical characteristics in the simulation models. Figure 2: Model of Test 5 after excavation: (**a**) FLAC\({}^{3\text{D}}\) simulation model (overlying layers are elevated for illustration); (**b**) vertical profile of model with dimensioning and boundary conditions. The vertical subsidence and horizontal displacement at all model nodes are extracted after the excavations. The ground values (top surface of the model) are managed with the software Mathematica in table form. As shown in Figure 3, the vertical subsidence is plotted as iso-contours and the horizontal displacement is plotted as vectors. The horizontal displacement could also be plotted as iso-contours in the strike and dip directions of the coalface, which would require two separate figures. The criticality of the excavation is related to the size of the coalface, including the lengths in the dip and strike directions. Based on the experiment design, the minimum length in the strike direction was 600 m to ensure the excavation was always super-critical along this direction with a mining depth of 200 m. The simulated vertical subsidence data show that when the length in the dip direction is 250, 300, 350, 400 and 450 m, the vertical subsidence coefficients are 0.47, 0.53, 0.56, 0.56 and 0.56, respectively. Therefore, when the length in dip direction reaches 350 m, the excavation is considered to be critical in this direction, giving a vertical subsidence coefficient of 0.56. (2) Computation of Ground Deformations (Slope, Curvature and Horizontal Deformation)  
Ground damage is not only determined from the ground displacements, which are the direct result of the simulations, but also by the ground deformations, including the slope, curvature and horizontal deformation. These deformation factors can be derived from the ground displacements. The slope \(i(x)\) is the first derivative of the vertical subsidence \(W(x)\):  
\[i(x)=\frac{dW(x)}{dx} \tag{7}\]  
The curvature \(K(x)\) is the first derivative of the slope \(i(x)\) (second derivative of the vertical subsidence \(W(x)\)):  
\[K(x)=\frac{di(x)}{dx}=\frac{d^{2}W(x)}{dx^{2}} \tag{8}\]  
The horizontal deformation \(\varepsilon(x)\) is the first derivative of the horizontal displacement \(U(x)\):  
\[\varepsilon(x)=\frac{dU(x)}{dx} \tag{9}\]  
These three deformation factors can be represented as two orthogonal components, normally along the strike and dip directions of the coalface. Taking Test 5 as an example, the slope and horizontal deformation are plotted in Figure 4.  
Figure 3: Vertical subsidence iso-contours and horizontal displacement vectors for Test 5. (3) Verification of Simulated Results  
The vertical subsidence of the ground surface was measured 11 times during the excavation of the prototype coalface; the 11 th dataset (obtained after the subsidence was stopped) was used to verify the corrections in the numerical simulation results. A simulation test with the coalface parameters close to the prototype is employed for the comparisons. Test 3 was appropriate as its length in the dip direction, thickness and mining speed were 300 m, 6 m and 14 m/day.