# Numerical Simulation-Based Study on the Arching Effect in Subsequent Backfill
## 3 Numerical Simulation
The normal and shear stiffness values are calculated using Formula (2) provided in the FLAC3D manual:  
\[k_{\rm n}=k_{s}=10\times\max\left(\frac{l+4G/3}{\Delta z_{\rm min}}\right) \tag{2}\]  
\(\Delta z_{\rm min}\) represents the minimum normal grid size of the contact surface and takes the average or the smaller value of the grid sizes on both sides of the contact surface. The finer the grid and the greater the normal stiffness, the stronger the resistance to intrusion ability of the contact surface will be. Conversely, coarse meshes will lead to insufficient stiffness and may cause unrealistic contact phenomena. However, although a grid size that is too small can improve accuracy, it will significantly increase the computational load. J and G denote the bulk modulus and shear modulus of the backfill, respectively, which can be calculated from the backfill's elastic modulus (E) and the Poisson's ratio using Equations (3) and (4):  
\[J=\frac{E}{3(1-2\mu)} \tag{3}\]  
\[G=\frac{E}{2(1+\mu)} \tag{4}\]  
Applying the above equations, the normal and shear stiffness of the model were calculated as \(\rm k_{n}=k_{s}=12.7\ GPa/m\). In mining engineering, the internal friction angle of interface elements is typically set within the range of \(1/2\) to \(1\) times that of the backfill material. For this simulation, a value of \(\delta=25\) was selected. According to the model requirements, the contact surface unit of the surrounding rock-filling body should be inserted as shown in Figure 4.  
Analysis of Backfill Parameters and Interface Elements on Stress Distribution in Backfill Structures  
As shown in Figure 5, the comparative analysis of the numerical simulation results indicates that the introduction of the contact surface element has a significant impact on the stress distribution characteristics of the filling body. The FLAC3D numerical simulation results show that under both working conditions, with and without considering the contact surface element, the stress of the filling body presents a distribution law gradually increasing from the top to the bottom. This trend is consistent with the mechanical response of the filling body under the combined action of self-weight stress and surrounding rock  
Figure 4: Backfill-surrounding rock interface element model. constraints in actual engineering. It is worth noting that the stress values calculated by the model introducing the contact surface element are generally higher than those without considering the contact surface, and the stress gradient changes are more significant. This difference mainly stems from the fact that the contact surface unit can more accurately simulate complex mechanical phenomena such as the friction effect, relative slip, and possible separation behavior between the interface of the filling body and the surrounding rock, thereby more truly reflecting the stress distribution characteristics in actual engineering. In contrast, the model that does not consider the contact surface simplifies the interface interaction mechanism, and its calculation results show overly smooth stress distribution characteristics, making it difficult to reflect the stress distribution situation in engineering practice. As shown in Table 2, in order to investigate the arching effect and stress distribution patterns, this section adopts a controlled, variable approach to study the influence of various parameters--including the cohesion, adhesive force, elastic modulus, and Poisson's ratio of the backfill, as well as the internal friction angle and cohesion parameters at the backfill-surrounding rock interface--on the stress distribution characteristics of the backfill material.