A risk-based pillar design approach combining stochastic continuous and discontinuous modeling in an underground stone mine

This study introduces a comprehensive risk-based pillar design methodology that integrates LiDAR and photogrammetric surveys with stochastic discrete and continuous numerical modeling. The framework quantifies pillar failure probability by combining strength and stress variability through the reliability method. Stochastic discrete element modeling (DEM) revealed that variability in intact rock strength plays a more significant role in pillar strength uncertainty than the spatial variability of discontinuities. A finite volume model (FVM), using the point estimate method, was applied to assess stress variability across a real mine layout, accounting for changes in rock elastic properties. Field validation using laser scanning and drone surveys confirmed strong agreement between simulated and observed pillar behavior, demonstrating the method’s applicability for site-specific ground control design. Introduction Pillar collapse is a high-consequence, low-frequency event in underground mining. Given the significant risk associated with these failures, design approaches that incorporate variability and uncertainty should be favored over traditional deterministic methodologies. This work presents a risk-based pillar design framework that enables the estimation of probability of failure by integrating stochastic modeling of pillar strength and stress estimation. The methodology was implemented in a dipping underground limestone room-and-pillar operation and validated using laser scanning and photogrammetric survey data. Fig. 1 Proposed workflow for the integration of laser scanning and stochastic modeling in the case study mine. Methodology Pillar strength was estimated using the bonded block method (BBM) in 3DEC, combined with filtered discrete fracture networks (DFNs) derived from site-specific discontinuity mapping. Models were calibrated against the National Institute for Occupational Safety and Health (NIOSH) empirical pillar strength formula using intact rock parameters from laboratory testing. Simulations were conducted for three pillar geometries — width/height (W/H) ratios of 0.5, 0.8 and 1.0 — with 15 stochastic realizations per case. A parallel pillar stress analysis was performed using a finite volume modeling (FVM) approach in 3DEC. The model accounted for the variability in rock mass deformation properties and three horizontal-to-vertical stress ratio scenarios (K0 = 0.5, 1.0 and 1.5). The point estimate method was used to evaluate the influence of input variability on pillar stress distribution. Stochastic strength and stress outputs were then combined using the reliability method. Pillar probability of failure (Pf) and factor of safety (FoS) were calculated using the limit state function defined as the difference between pillar strength and stress. Results The stochastic DEM simulations demonstrated that discontinuities reduced pillar strength slightly, with fractured pillars achieving 95 to 98 percent of the strength of intact models. However, the variability introduced by discontinuities was low, with coefficients of variation (COV) between 1.3 and 1.6 percent, compared to 13.3 percent COV from intact uniaxial compressive strength (UCS) variability. The average fractured pillar strength for 0.8 W/H geometry was 34.6 MPa. Stochastic stress modeling revealed stress variability across the mine geometry. Pillars at greater depths and beneath topographic highs experienced higher average vertical stresses. Three stress scenarios yielded different maximum stress values, with scenario 3 (K0 = 1.5) reaching up to 23.4 MPa. By applying the reliability method, the probability of failure for each pillar was estimated. Under the highest stress scenario, the maximum Pf observed was 0.83 percent, with 17 of 72 pillars showing a FoS below the standard 1.8. A negative logarithmic relationship between FoS and Pf was observed. Based on this trend, the study proposes a 0.05 percent Pf threshold as a potential design criterion aligned with the conventional 1.8 FoS recommended by NIOSH. Model validation One of the pillars within the case study mine (CSM) was selected for a validation exercise using terrestrial LiDAR and drone-based photogrammetric surveys. The scanned pillar was modeled using the same BBM + DFN approach under simulated in situ loading conditions representative of the mine. The model predicted maximum displacements on the order of 2.99 mm, with higher deformations occurring at corners where intersecting discontinuities were present. These locations correlated with areas of observed scaled loose rock in the scanned point cloud. Discussion The proposed methodology incorporates geomechanical variability into both the strength and stress estimation components of pillar design. While discontinuities do influence strength, their contribution to variability was limited compared to the uncertainty introduced by intact rock UCS. This observation is consistent with earlier research indicating that DFN size filtering and rock bridge continuity can reduce conservatism in strength predictions. The use of the reliability method enabled to quantify pillar failure risk through estimating the probability of failure. While this parameter is not currently part of stone pillar design standards in the United States, it provides a more complete representation of design uncertainty. In the absence of formal acceptance criteria, the derived relationship between FoS and Pf can serve as a basis for defining acceptable risk thresholds. By integrating this methodology into ground control management plans (GCMPs), mining operations can better document, monitor and respond to evolving geotechnical conditions. The framework is compatible with performance-based design and aligns with international best practices in risk-informed ground control. Fig. 2 Top view of the simplified CSM model colored by probability of failure for each stress scenario: (a) stress scenario 2: K0 = 0.5, (b) stress scenario 1: K0 = 1.0and (c) stress scenario 3: K0 = 1.5. Conclusion This work presents a validated risk-based pillar design approach that integrates stochastic discrete and continuous modeling with a reliability-based failure assessment. The methodology enables the estimation of failure probability and supports informed decision-making by addressing geotechnical uncertainty explicitly. Results from this study demonstrate that while discontinuities affect strength, intact rock variability is the dominant factor in uncertainty characterization. The framework can serve as a practical tool to improve the safety and reliability of pillar designs in underground stone mining. References A list of all references is available in the full paper.

A risk-based pillar design approach combining stochastic continuous and discontinuous modeling in an underground stone mine

Authored by: Juan J. Monsalve, Aman Soni, Richard Bishop, Jim Hazzard, Adrian Rodriguez-Marek, Cheng Chen and Nino Ripepi

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