1885 σ= E(1+ν ) + Eν (1+ν )(1-2ν ) tr (ε ) I (4 where E is the Young’s modulus, ν is the Poisson’s ratio, tr(ε) represents the volumetric strain, and I is the identity tensor. For the purposes of numerical analysis, the kimberlite deposit was designed as a cylindrical body with a diameter of 137.5 m and a vertical extent of 400 m. This ore body was positioned at the center of a larger granite block measuring 400 m by 400 m in plan and 450 m in depth, thereby creating a symmetric host rock environment around the modeled deposit, as illustrated in Figure 111.Within this configuration, two vertical shafts were introduced. The first shaft was excavated through the central axis of the kimberlite body, while the second was located adjacent to it, enabling the study of excavation interactions and the evaluation of stability when applying the VCM. Material parameters for both the ore and the host rock were assigned based on specifications provided by the industrial collaborator, with details summarized in Table 42. The backfill was modeled according to the properties of the engineered zero‑cement mixture. Boundary conditions were imposed on the granite domain to replicate realistic constraints: the lateral faces were restricted with roller supports, allowing vertical displacement (z-direction) but preventing horizontal movement (x and y-directions). The bottom boundary was held fixed and the upper boundary was left unconstrained, permitting deformation in all directions. This setup ensured that the simulated mechanical response of the system reflected plausible excavation conditions. In addition, gravitational loading was incorporated as a distributed body force to account for the self‑weight of the materials. Table 42 – Material properties used for the numerical model. Density (kg/m³) Young's modulus (GPa) Poisson's ratio Kimberlite 2673 36 0.18 Granite 2551 63 0.27 Backfill 2090 0.015 0.25 To simulate the excavation and subsequent backfilling sequence, the computational model was adjusted by removing the drilled portion to represent the newly created shaft. Once excavation was complete, the cavity was reintroduced into the geometry using elements that mimicked frozen backfill, with its designated material properties replacing those of the original kimberlite. To ensure the robustness of the numerical results, a mesh refinement study was undertaken. The finite element discretization was progressively refined across seven levels, employing tetrahedral elements. Mesh sizes ranged from a coarse grid of 3,720 elements to a highly refined configuration of 858,138 elements. As the number of meshes increased, the maximum von Mises stress rose steadily from 5.156 MPa to 15.182 MPa, indicating convergence toward a stable solution. The difference in peak stress between the two finest meshes (517,057 and 858,138 elements) was less than 2%, demonstrating that further refinement would not meaningfully affect the outcome. Consequently, the mesh with 517,057 elements was selected for the final simulations, striking an optimal balance between computational efficiency and numerical accuracy.
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