## 3 Numerical Simulations and Comparisons
### Numerical Results  
In all simulated cases, the \(K\) values were obtained from the ratios of numerical horizontal and vertical stresses. Figure 3 shows the simulated variations of \(K\) along the VCL of backfill openings, considering different values for opening width \(B\) (Figure 3a; Cases 1), Young's modulus \(E\) (Figure 3b; Cases 2), friction angle \(\varphi^{\prime}\) (Figure 3c; Cases 3), and Poisson's ratios \(\
u\) of the fills (Figure 3d; Cases 4).  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Cases** & **Figure Reference** & \(B\) **(m)** & \(E\) **(MPa)** & **Peak \(\varphi^{\prime}\) (\({}^{\circ}\))** & \(\
u\) **(-)** \\ \hline
1 & Figure 3a & 4 to 20 & 300 & 30 & 0.2 \\
2 & Figure 3b & 8 & 10 to 1000 & 30 & 0.2 \\
3 & Figure 3c & 8 & 300 & 10 to 45 & 0.2 \\
4 & Figure 3d & 8 & 300 & 30 & 0.001 to 0.499 \\
5 \({}^{\ast}\) & Figure 4 & 8 & 300 & 10\({}^{\circ}\) to 40\({}^{\circ}\) & 0.452 to 0.263 \\ \hline \end{tabular} \({}^{\ast}\) The values of \(\varphi^{\prime}\) and \(\
u\) are related through Equation (7).  
\end{table}
Table 1: Geometric and mechanical characteristics of the numerical models simulating the backfilled slopes.  
For Cases 1 (Figure 3a), it is seen that the stress ratio \(K\) in the fill along the VCL appears almost unchanged with the variation of the width of opening \(B\) (ranging between 4 and 20 m). As expected, the \(K\) values near the center line of the backfill opening are close to Rankine's active coefficient \(K_{\mathrm{a}}\), for \(B\) varying between 4 and 20 m (except near the fill surface).  
For Cases 2, Figure 3b illustrates that \(K\) along the VCL remains almost constant as the fill modulus \(E\) changes between 10 MPa and 1 GPa. The stress state near the opening center is known to be almost insensitive to the fill modulus, which also has little effect on the \(K\) value (for a range of \(E\) varying from 10 MPa to 1 GPa). The \(K\) values obtained from the numerical modeling are close to \(K_{\mathrm{a}}\), which is consistent with the respective values of the fill Poisson's ratio and internal friction angle (as predicted above from Equation (6)).  
Figure 3c indicates that as the value of internal friction angle \(\varphi\)' changes from \(10^{\circ}\) to \(35^{\circ}\) (Cases 3), the \(K\) values near the backfill opening center tend to decrease, but remain fairly around \(K_{\mathrm{a}}\), except near the fill top. As the values of \(\varphi\)' increase to \(40^{\circ}\) and \(45^{\circ}\), the ratio \(K\) continues to decrease slightly and then stays close to a constant value given by \((K_{0})_{\
u}\) (with \(\
u\) = 0.2). These numerical results are also consistent with the theoretical analyses presented above.  
Figure 3d shows that the \(K\) values near the backfilled opening center keep almost unchanged (near \(K_{\mathrm{a}}\)), when Poisson's ratio \(\
u\) varies between 0.001 and 0.2, except near the top surface of the backfilled opening (Cases 4). When the value of Poisson's ratio \(\
u\geq 0.3\), the \(K\) value at the center of vertical openings is seen to be a function of Poisson's ratio, with values near \((K_{0})_{\
u}\) (Equation (4)), except near the top surface and bottom of the backfill. Figure 3d also illustrates that the \(K\) value approaches unity as the value of \(\
u\) goes close to 0.5 (i.e., 0.499), indicating a hydrostatic (isotropic) state. Again, these numerical results agree with the theoretical analyses presented above.  
Figure 4 illustrates that, for Cases 5 (\(\
u\) and \(\varphi\)' related through Equation (7)), the simulated \(K\) values along the VCL are reduced when \(\varphi\)' is increased (and \(\
u\)  
Figure 3: Distribution of \(K\) along the VCL of backfilled openings as a function of: (**a**) \(B\) (Cases 1); (**b**) \(E\) (Cases 2); (**c**) \(\varphi\)’ (Cases 3); and (**d**) \(\
u\) (Cases 4) (see Table 1 for details).  
ranging between 0.452 and 0.263). These simulation results correspond well with those predicted by the theoretical analyses presented above.