## 4 Discussion  
In soil mechanics, the stress state behind a retaining wall is commonly associated with its movement [33]. The soil is initially assumed to be at rest before introducing a (stiff and very thin) retaining wall. An immobile wall is thus sometimes regarded as a necessary and sufficient condition to maintain an at-rest state in the soil. However, it has been shown above that the stress ratio \(K\) (and corresponding state) in backfilled openings is not solely associated with the wall movement (see also simulation results from [PERSON] et al. [24] and [PERSON] et al. [28,29]). In such openings, yielding of the fill may occur even if the walls are immobile. The theoretical analyses show that the state of stress of backfill at the center of openings also depends on the respective values of \(\varphi\)' and \(\
u\). This has been confirmed by the numerical results presented above. The theoretical and numerical analyses presented above would benefit from further validation, based on specific experimental data that comprise all the required parameters, including the vertical and horizontal normal stresses and backfill properties (\(\varphi\)' and \(\
u\)). To the authors' knowledge, such a complete data set does not yet exist. A few results, such as the in-situ measurements performed by [PERSON] et al. [45], nonetheless indicate that the \(K\) value may approach \(K_{\text{a}}\) or \(K_{0}\), depending on the local conditions in the backfilled stop. These results appear to follow the trends identified by the theoretical and numerical results shown above, particularly for a relatively large \(\varphi\)' value (=37\({}^{\circ}\)), as shown for Cases 3 in Figure 3c. However, a more complete assessment would require additional testing results to specifically determine the backfill Poisson's ratio. In addition, the complex conditions in the field may influence the stress state and related results due to additional factors not considered here (i.e., complex geometry, water pressures, cementation and curing, etc.). All these indicate that more effort should be made to determine the backfill Poisson's ratio, despite the difficulties raised by accurate measurements. Ongoing study is also being performed by the authors to help obtain the value of \(\
u\) from indirect means. The numerical analyses performed here consider dry cohesionless filling materials. Previous studies have shown that the generation and dissipation of pore water pressure, cohesion, and cement curing could affect the state of stress developed in backfilled mine stops [25,43,45]. It has also been shown that a nonzero dilation angle would alter the simulated stress state in backfill openings [17,25,40]. Additional work is urged to assess the impact of these additional aspects. The simulations conducted here assume constant values for Poisson's ratio and Young's modulus of the backfill. In practice, these parameters can vary in the opening due to the changes in falling height or loading associated with the addition of fill layers. It would thus be useful to assess the effect of a spatial variation of the fill properties using numerical simulations and experimental tests. Figure 4: Distribution of \(K\) along the VCL of backfilled openings as a function of \(\varphi\)’ and \(\
u\) (Cases 5 in Table 1). In order to analyze the effect of parameters of interest, the parameter of interest was treated as a variable while other parameters were kept constant (see Cases 1-4 in Table 1). The values of \(\
u\) and \(\varphi^{\prime}\) were varied simultaneously in Cases 5. Additional work is required to check the validity of the simulated outcomes when two or more parameters are varied at the same time. The results presented above are based on plane strain conditions, thus corresponding to openings with very long third dimensions. It is known that the magnitude of arching may be under-predicted by plane strain models. Additional work is needed to evaluate the effect of the third dimension, to complement the recent work of [PERSON] [43].