FLAC3D-IMASS Modelling of Rock Mass Damage in Unsupported Underground Mining Excavations: A Safety Factor-Based Framework

Abstract

The implementation and application of a safety factor (SF)-based numerical framework in FLAC3D-IMASS (Itasca Model for Advanced Strain Softening) is presented for the evaluation of the short-term stability of unsupported underground excavations in sedimentary rock masses during pillar recovery in bord-and-pillar mining. The stability of underground openings during the initial hours post-excavation must be ensured, as they are not accessed thereafter; therefore, short-term stability assessment is essential. The framework was specifically calibrated to field observations and applied to a case study from an Australian bord-and-pillar mine, focusing on plunge and bellout configurations commonly used during the pillar extraction stage to enhance ore recovery. The modelling approach was integrated with rock mass degradation behavior under static loading conditions and was used to calculate three-dimensional distributions of SF to identify potential failure zones. The results demonstrate that the coal (CO) roof scenario generally maintains structural stability, while the impure coal (Cox) roof scenario is observed to exhibit significant instability, particularly at greater excavation advancement. Among the tested bellout geometries, 8.0 m spans were observed to provide improved performance due to shorter tunnel lengths that enhance confinement and reduce the volume of disturbed rock. Overall, the proposed SF framework effectively captures localized failure mechanisms and is demonstrated as a practical design tool for assessing the short-term stability of unsupported structures during critical stages of underground mining operations.

1. Introduction

The stability of unsupported underground excavations in pillar mining is considered a critical issue in mining engineering. One of the most prevalent scenarios involving unsupported openings arises during pillar extraction [1,2,3,4], where economic incentives often drive the decision to recover previously left-in-place pillars. While pillar recovery can significantly improve overall profitability through increased ore extraction and reduced dilution losses, it simultaneously introduces substantial geomechanical challenges, specifically pillar yielding, increased failure risk, roof instability, and complex stress redistribution associated with retreat sequencing and mining depth [5,6,7,8]. These challenges require careful consideration of safety design, extraction sequencing, and ground control strategies to ensure that pillar recovery operations remain both productive and sustainable.

A range of methods have been developed to assess pillar stability in underground mining, typically classified into empirical approaches [9,10,11,12,13], artificial intelligence (AI)-based models [14,15,16,17,18,19,20,21,22], and numerical simulations [5,23,24,25,26,27,28,29,30,31]. Recent comprehensive reviews by Zhang et al. (2024) [32] and Dzimunya et al. (2025) [33] offer valuable insights into the evolution and current practices of pillar stability prediction methods. Zhang et al. (2024) [32] conducted a systematic global review highlighting a paradigm shift toward hybrid data- and theory-driven pillar design (i.e., integrating AI with numerical modelling [18]) while Dzimunya et al. (2025) [33] emphasized practical challenges and recent advances through case studies, offering critical insights into the limitations of existing design strategies.

However, there exists currently no universally accepted approach for evaluating the stability of excavations during pillar extraction, particularly under conditions of minimal or no support. Recent studies demonstrate that the decision to recover a rock pillar in underground mining involves a complex interplay between geomechanical stability, operational efficiency, and long-term safety. In a comprehensive review, Zhang et al. (2024) [32] emphasized that, although pillar recovery can improve ore extraction and profitability, it also requires a rigorous site-specific assessment involving geological characterization, advanced monitoring, and risk-based design. In the absence of a standardized design methodology, practitioners are required to rely on a combination of empirical data, numerical modeling, and economic analysis to guide recovery strategies [32].

Babanouri et al. (2023) [5] numerically investigated four commonly applied pillar recovery methods in a room-and-pillar coal mine using FLAC3D, focusing primary on the geomechanical response during excavation sequences. Their analysis incorporated the shear strength reduction (SSR) technique to evaluate safety factor evolution, concluded that the split and fender and Christmas tree methods induced a more controlled reduction in pillar stability and delayed yield onset. Feng and Wantg (2020) [34] examined the simultaneous recovery of remnant coal pillars and an underlying ultra-close seam using longwall top coal caving (LTCC), revealing through physical and FLAC3D modeling that multi-seam interactions—particularly stress redistribution and pillar caveability—play a critical role in roof stability and controlled caving behavior. Other studies have focused on the numerical modeling of remnant pillar extraction [7,35]. While these investigations contribute valuable insights into recovery sequencing, they predominantly utilize conventional Mohr–Coulomb constitutive models or apply overly simplified softening responses. These approaches fail to capture the strain-softening behavior typical of roof lithologies in the post-peak regime. As a result, the models possess limited ability to simulate progressive failure mechanisms accurately, thereby reducing the reliability of stability predictions under unsupported or partially supported excavation scenarios.

Similarly, empirical approaches remain largely constrained to the geometry and strength of the pillar itself, with limited consideration of roof lithological conditions [36,37,38,39]. This narrow focus prevents a comprehensive assessment of underground stability when excavation sequences interact with complex geomechanical settings, such as bellout pillar extraction. Empirical methods provide fundamental interpretations of load distribution mechanisms, but they are limited in their ability to incorporate site-specific geological variability. Specifically, they often assume uniform and idealized geological conditions, neglecting weak carbonaceous layers, partings within the coal seam, and variations in roof and floor strength, which can compromise stability and lead to overestimation of pillar safety. Furthermore, empirical formulas are generally derived from historical datasets, limiting their applicability to conditions similar to those in which they were developed. As a result, they are often unreliable for novel pillar shapes, unusual seam thicknesses, variable mining depths, or extraction sequences, such as bellout pillar recovery. Finally, empirical methods oversimplify pillar failure mechanisms by reducing them to a single safety factor and ignoring progressive failure phenomena such as spalling, slabbing, and stress redistribution, which are critical in assessing real-world pillar behavior. These limitations highlight the need for more sophisticated approaches that can incorporate complex geometrical configurations, site-specific geological variability, dynamic loading, and progressive failure processes.

AI-based models [17,40,41,42] have also emerged as promising tools; however, they are generally limited to input parameters describing pillar geometry and rock mass strength, and seldom incorporate the effects of recovery sequences or complex geometrical configurations such as bellout excavations. These limitations underscore the need for a framework that explicitly accounts for excavation geometry, roof–pillar interactions, and short-term stability during pillar extraction.

Zvarivadza et al. [43] proposed a hybrid pillar stress analysis framework integrating numerical modelling (FEM and DEM), machine learning (GBM, XGBoost), and geostatistics (kriging) to predict pillar stress distributions in structurally complex hardrock environments. The study successfully addresses the limitations of traditional empirical methods by capturing geological heterogeneity and providing high-resolution, data-driven insights into stress redistribution. While highly valuable for hardrock pillar stress analysis, this hybrid framework primarily focuses on stress prediction rather than operational stability during pillar recovery, particularly for irregular or complex geometrical configurations such as bellouts or split pillars. Additionally, the machine learning component requires extensive training datasets, limiting applicability in scenarios with limited site-specific data, and the framework does not provide direct, stage-wise safety factors for short-term extraction sequences.

Among the current numerical tools available, FLAC3D [44] has been proven effective in simulating pillar stability [5,45,46,47], ground support performance [48,49,50], and caving processes [51,52,53]. A key advantage of FLAC3D is its implementation of the Itasca Model for Advanced Strain Softening (IMASS) [44,54], a robust constitutive model that incorporates two distinct residual strength envelopes: the first governs post-peak strength mobilization following initial plastic deformation, while the second captures the ultimate residual strength of the rock mass under large deformations. This dual-stage softening framework enables a realistic representation of friction mobilization at low confining pressures during the onset of rock mass fracturing [54]. IMASS also introduces a damage indicator termed strength loss (sloss), which, when calibrated against field observations and known failure or stable cases, can serve as a powerful tool for assessing the serviceability and stability of rock structures [44]. Several studies in mining geotechnical assessment have successfully employed IMASS to simulate the fracturing processes of rock masses in various underground structures [51,55,56,57].

The aim of this study is to develop a methodology that defines a safety factor (SF) as an exponential function of sloss, using a calibration scheme based on historical field performance. This study is built upon our previous work in numerical modeling for mining applications. While our 2024 publication [51] focused on simulating the large-scale fracturing response of overburden during longwall mining, the present work introduces a distinct SF-based framework for evaluating the short-term stability of specific unsupported geometries—such as bellout and plunge configurations—commonly used in bord-and-pillar extraction projects. In particular, it addresses the critical pillar recovery stage, during which roof and rib stability must be ensured for several hours after pillar extraction. The framework provides a practical tool for optimizing excavation layout and geometrical configurations, which is a fundamentally different objective from the previous longwall caving study. The current study also draws on earlier empirical and numerical investigations (e.g., [5,32,34]) by providing a direct comparative analysis of CO and COx roof conditions under varying excavation geometries, including both plunge and bellout scenarios. The framework integrates evidence from back-analysis of field observations with numerical modelling of unsupported underground excavations, thereby providing a more robust stability assessment tool compared with conventional empirical approaches.

By expressing roof and rib damage in terms of a calibrated SF indicator, the post-processing of numerical results is rendered more straightforward, interpretable, and practical for geotechnical decision-making. This approach enhances geotechnical assessments by introducing a field-based damage threshold, particularly suitable for the evaluation of unsupported underground excavations during pillar extraction under varying geometrical configurations.

2. Development, Implementation and Back-Analysis of the Safety Factor Approach

2.1. Framework for Damage-Based Safety Factor

The safety factor (SF) approach is implemented by calibrating the strength loss (sloss) parameter in IMASS [54] of a numerical model designed to back-calculate field observations from an Australian bord-and-pillar mining operation. When sloss is calibrated against field data in terms of observed damage, it becomes a powerful predictive tool for assessing the serviceability and stability of underground structures [44].

As discussed by Ghazvinian et al. (2020) [54], sloss is an indicator of damage within the IMASS constitutive model, representing the degree of softening or weakening of the rock mass. Figure 1 schematically illustrates sloss at different damage and degradation levels within a rock mass. As depicted in the stress–strain relationship graph in Figure 1:

• Point A corresponds to the elastic stage of rock behaviour, where the material exhibits no damage, and stress is proportional to strain.
• Point B represents the peak strength, where the rock mass achieves its maximum load-bearing capacity.
• Point C marks the strain-softening stage after the peak, where stress decreases as strain progresses, indicating progressive damage.
• Point D indicates the post-failure stage with significant progressive rock fracturing (bulking), where the rock mass is fully caved, and unlined shafts or minimally supported drifts are considered unserviceable.

In the context of sloss, the range varies from −1 to +1. For cave mining simulations, only sloss values below 0.0 (Point “D”) are applicable.

In the IMASS constitutive model, jointing effects are implicitly captured through the Geological Strength Index (GSI), which integrates the influence of structure and joint condition on bulk rock mass behavior [44,54]. Explicit geometric modeling of individual joints (orientation, persistence, infill) is not required, as GSI was developed specifically to represent rock mass behavior without such complexity [58]. IMASS assumes that failure is governed by continuum strain-softening, suitable for massive to moderately jointed strata (GSI > 35 [58,59]). It is not intended for structurally controlled failures dominated by discrete joint sets or wedge mechanisms, where kinematic or discontinuum analyses are more appropriate [60]. The strain-softening behavior predicted by IMASS provides the basis for the sloss parameter, which alone may lead to subjective interpretation of stability. To overcome this limitation, we introduce a novel mathematical mapping that systematically links sloss to a practical safety factor (SF) metric. The relationship is expressed by the following exponential function:

$$\mathrm{S}\mathrm{F}=1+\left[{\mathrm{S}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}-1\right]\exp \left[\mathsf{\alpha }(\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}-1)\right]$$

(1)

where $SF$ is the value of safety factor at a given state of FLAC3D zone, ${SF}^{max}$ is a maximum safety factor that can be selected through rigorous mapping, and $\alpha$ is a coefficient back-calculated from field observations to establish the correspondence between sloss and SF. An sloss threshold is defined from field evidence such that values above this threshold correspond to stable unsupported openings, while values below it indicate instability.

It should be noted that while the well-established IMASS model is employed, the innovation of the current study lies in translating model outputs such as strength loss (sloss) into a calibrated SF distribution, updated dynamically after each excavation stage. This workflow enables the generation of SF contour plots that provide a direct, practical assessment of short-term stability for unsupported excavations. In this sense, the study introduces a new application framework that bridges complex numerical modeling with operational decision-making, rather than a new numerical method.

Section 2.2 describes the implementation of this SF-based framework into FLAC3D, and Section 2.3 outlines the back-analysis procedure used to determine the sloss threshold and calculate the coefficient $\alpha$.

2.2. FLAC3D Implementation

The proposed safety factor (SF) approach was implemented in FLAC3D through a custom FISH function. Figure 2 illustrates the flowchart of the implementation process in FLAC3D. This function is automatically executed at each excavation stage, where the computed SF values are stored as extra FISH variables within the model, allowing visualization and post-processing for stability assessment and calibration. The calculated SF distribution can then be plotted to evaluate the progressive development of damage in the roof and ribs.

Figure 2 presents a step-by-step flowchart illustrating the practical implementation procedure of the proposed SF approach in FLAC3D. Once the model reaches initial equilibrium, a custom function is executed to compute the SF for all zones governed by the IMASS constitutive model, based on the sloss parameter, as defined in Equation (1). Zones not representing material (e.g., excavated regions) are excluded from this computation. The resulting SF values are stored in a user-defined extra variable (available in FLAC3D 9.0 software FISH codes) for subsequent analysis. The staged excavation is then performed sequentially, either by advancing the plunge or by excavating individual fingers in the bellout configuration. After each excavation stage, the SF computation is repeated to update the SF distribution throughout the model. Finally, the results are saved and post-processed to generate SF contour plots, providing a numerical visualization of potential failure zones and enabling a practical, applied assessment of short-term stability for unsupported underground excavations.

FLAC3D employs an explicit finite-difference/finite-volume formulation. A structured hexahedral mesh was adopted to accurately represent the excavation geometry while maintaining computational efficiency. This mesh type allows the model to capture stress redistribution and plastic deformation in the rock mass effectively. Mesh sensitivity was verified to ensure that the results were independent of the chosen zoning arrangement, thereby providing confidence in the reliability of the numerical outcomes. The primary assumptions of the numerical model reflect the FLAC3D explicit formulation and the selected constitutive laws. The rock mass is treated as a continuum with properties defined by the IMASS constitutive model. The numerical framework intentionally excludes long-term mechanisms (e.g., creep) as pillar recovery excavations are abandoned immediately post-mining and require stability for only a few hours—the critical window for safe equipment retreat. Long-term deformation is therefore irrelevant to the design objectives of this framework.

2.3. Back-Analysis Results in FLAC3D

The back-analysis presented in this study is based on observations from a real mining site. The selected site is representative of typical Australian bord-and-pillar operation in the Bowen Basin region of central Queensland, Australia. However, specific location details and associated maps are confidential and cannot be disclosed due to company policies. Field observations indicated that for coal roof plunges, the roof remains stable at an advance length of 15.0 m. In contrast, for stone roof plunges, failure initiates when the excavation advancement reaches approximately 5.0 m or beyond. Field observations further highlighted the importance of leaving a 0.5 m coal beam on the roof to enhance stability. When the coal beam is cut, exposing the carbonaceous siltstone roof (XT unit), a stable unsupported plunge cannot be achieved at a 15.0 m excavation advancement. In this case, stability is only maintained for an unsupported plunge at a reduced excavation depth of 5.0 m. Based on these field observations, a calibration scheme was established using two FLAC3D models: one with a coal roof and the other with a stone roof (XT unit). The lithological configurations of these models are illustrated in Figure 3.

The lateral boundaries were positioned at a distance of 12.0 m from the excavation boundaries. The top and bottom boundaries were located at approximately 7.0 m above the roof and below the floor, respectively. These dimensions were selected to be sufficiently large to ensure that excavation-induced stress changes do not influence the model boundaries, thereby simulating an infinite domain while maintaining computational efficiency. All simulations were performed on a high-performance workstation (Intel^® Xeon^® W-3275M, 28 cores/56 threads, 384 GB RAM, SSD storage, Santa Clara, CA, USA), with a typical model run requiring approximately 7 h.

The in situ stress boundary condition is applied on the models based on Tectonic Stress Factor (TSF) proposed by Nemcik et al. [61]. The field stress is assigned to the FLAC3D model as follows:

• The required lithological units are imported into the model (in kg/m³).
• Assign vertical (gravitational) stress with a rate of $\mathsf{\rho }\mathrm{g}·\mathrm{h}$

Where

ρ = density of lithological unit.

g = acceleration due to gravity (in m/s²).

h = depth of cover measured from surface topography (in m).

• The major and minor horizontal stresses are assigned using the following equation by assuming a combination of lithostatic loading (i.e., Poisson’s effect) and tectonic stress (i.e., TSF approach):

$${\mathsf{\sigma }}_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}}={\displaystyle \frac{\mathsf{\upsilon }}{1-\mathsf{\upsilon }}}{\mathsf{\sigma }}_{\mathrm{V}}+\mathrm{T}\mathrm{S}\mathrm{F}\times \mathrm{E}$$

(2)

where

σ_lateral = maximum (σ_H) or minimum (σ_h) principal horizontal stress (in MPa).

σ_V = vertical stress calculated as the gravitational load of the overburden strata (in MPa).

ν = Poisson’s ratio.

E = Young’s modulus of the strata (in GPa).

TSF = Tectonic Stress Factor.

A TSF of 0.45 was used to calculate the estimated maximum horizontal stress (σ_H). Similarly, the minimum horizontal stress (σ_h) was estimated using a TSF coefficient of 0.3. The TSF coefficients used to estimate the maximum and minimum horizontal stresses were derived from regional stress measurements reported for the Bowen Basin. The adopted coefficients are consistent with field stress data reported in coal mining projects in the region [61] and provide a reasonable approximation of the in situ horizontal stress regime. The average Young’s modulus of the strata was assumed to be 16.8 GPa, with a Poisson’s ratio of 0.23. The depth of cover was set to 160.0 m. While the Young’s modulus (E) is expressed in GPa, the TSF coefficient is calibrated empirically, ensuring that the resulting lateral stress is in MPa, consistent with field measurements.

It is important to note that the numerical modelling in this study is conducted using FLAC3D, which is formulated on an explicit finite-difference formulation (finite-volume approach) rather than the finite element method. In this scheme, numerical errors can result from discretization (truncation error) and computer precision (round-off error). Round-off errors are considered negligible due to double-precision arithmetic, while truncation errors are managed through adequate zoning and verified by monitoring model convergence using the built-in unbalanced force ratio criterion [44].

IMASS provides a user-defined multiplier for the critical plastic shear strain of the matrix material, which is calibrated based on site-specific conditions. The critical plastic shear strain multiplier (${\lambda }_{\mathrm{e}}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$) is set to a value less than 1.0 (but greater than 0.0) to represent a more brittle material response, or greater than 1.0 to simulate a more ductile behavior [44]. The IMASS parameters are calibrated to replicate this behavior in FLAC3D modelling. Intact rock properties are assigned based on laboratory data, as summarized in

Table 1. Material Properties Used for the Back-Analysis in FLAC3D-IMASS.

Lithology	Generalized Hoek-Brown Parameters
	UCSi	E	GSI	mi	${\mathit{\lambda }}_{\mathbf{e}}^{\mathbf{c}\mathbf{r}\mathbf{i}\mathbf{t}}$
	[MPa]	[GPa]	[-]	[-]	[-]
CO	9	5	50	8	0.75
XT	15	7	40	12	0.35
SS	45	17.7	65	12	0.75
ST	18	10.6	55	12	0.75

SS = Sandstone; ST = Siltstone; XT = Carbonaceous Siltstone; CO = Coal Seam.

, which lists the material properties used for the back-analysis. The physical and mechanical parameters listed in Table 1 are derived from a combined characterization program, incorporating laboratory testing (UCS and Young’s modulus) and borehole analyses (core logging and GSI estimation). The adopted values are further calibrated against a well-established database and long-term project experience in the Bowen Basin, ensuring site-representative input parameters.

Figure 4 and Figure 5 illustrate the sloss and safety factor contours resulting from back-analysis, respectively. The back-analysis results show that, for the coal roof scenario, the sloss value for the immediate roof exceeds 0.6. A section view of the coal roof scenario indicates that the coal beam is maintained stable, with an average $sloss$ value of 0.8 or higher. Additionally, a narrow shear band is formed in the XT unit with an average sloss of 0.3, which is effectively supported by the coal beam, as indicated by the high $sloss$ values in the coal beam unit. In contrast, when the excavation extends into the XT unit, the sloss value for a 15.0 m excavation advancement decreases significantly to around 0.1, indicating severe damage to the immediate roof. However, for a 5.0 m excavation advance, the stone roof scenario remains serviceable under unsupported conditions.

These numerical observations, derived from the back-analysis, indicate that the material properties incorporated in the model reproduce accurately the field observations for both stable and unstable conditions. The back-analysis showed that, in the coal roof scenario, the unsupported roof is maintained stable after a 15.0 m excavation advance. The sloss value corresponding to this stable condition, validated against field observations, is equal to 0.6. This is therefore adopted as the threshold for calculating the parameter α in Equation (1). Although sloss theoretically ranges from −1.0 to +1.0, the present framework is limited only the range 0.0–1.0. In this interval, fractured rock still exhibits interlocking and frictional resistance, allowing unsupported excavations to be marginally serviceable in the short-term (e.g., a few hours post-excavation). By contrast, values below sloss = 0.0 represent conditions where interlocking is lost and bulking dominates, rendering openings unserviceable [54]. Since the framework is explicitly designed for short-term stability assessments of unsupported excavations, the operational range of sloss is therefore constrained to 0.0–1.0.

In this study, calibration relied on visual field observations because instrumented monitoring is neither feasible nor necessary in temporary, unsupported excavations abandoned within hours—consistent with pillar recovery practice. The focus remains on capturing observable stability thresholds sufficient for calibrating relative trends that guide immediate engineering decisions. This approach is consistent with established practices in mining geomechanics, where visual or photographic evidence has been widely used to back-analyze underground excavation behavior when instrumentation is infeasible [62,63,64,65].

In practice, CO roof conditions are maintained stable for several hours post-excavation, whereas XT roofs showed rapid instability. This stability contrast is used in a back-analysis procedure in FLAC3D to determine the threshold sloss value of 0.6, which corresponds to approximately 40% rock mass damage according to (1 − sloss) × 100. This calibration forms the basis for linking field observations to the SF framework.

The parameter α in Equation (1) is uniquely determined by the selected values of SF^t, SF^max, and the threshold sloss value. For this study, α = 4.74 is obtained by setting SF^t = 1.3, SF^max = 3.0, and sloss = 0.6, which corresponds to the minimum stable condition observed in the field. In this sense, α functions only as a scaling coefficient that ensures the exponential curve passes through the calibration points, and its value changes automatically if the calibration parameters are altered. Therefore, no separate sensitivity analysis of α is required, as its robustness depends intrinsically on the selection of SF^t, SF^max, and sloss values.

The threshold safety factor (SF^t) is defined at 1.3, a conventional value commonly used in mining projects to indicate stability, with values below this indicating instability. In the present case study, this corresponds to a sloss value of 0.6 (≈40% rock mass damage). The proposed framework allows calibration against field observations, enabling adjustment of the sloss threshold for different lithological conditions. For example, in stronger siltstone or sandstone roofs (ST/SS), stability may be achieved at a higher sloss threshold (e.g., 0.7, ≈30% damage), which still corresponds to an SF^t of 1.3. Thus, the SF threshold can be adapted to site-specific conditions, with the minimum sloss value at which stability occurs regarded as the SF^t equivalent.

An upper bound of SF^max = 3.0 is defined, corresponding to a sloss value of 1.0 (i.e., intact rock mass without damage, Point A in Figure 1). Different SF^max values can be seen in Figure 5. The choice of 3.0 is made through trial-and-error calibration, ensuring practical contour generation within FLAC3D. Although higher values of SF^max (e.g., 4.0 or 5.0) can be defined, these do not represent more competent rock, since all such cases are associated with sloss = 1.0. Instead, higher SF^max values increases the steepness of the exponential curve and expand the interval between SF^t and SF^max, altering only the spacing of contour levels rather than the rock mass condition. Based on empirical testing, SF^max = 3.0 is adopted for consistency and clarity.

In this study, the relationship between SF and sloss is expressed using an exponential formulation (Equation (1)). The exponential form is selected because it meets the required boundary conditions (SF^max = 3.0 at sloss = 1.0 and SF^t = 1.3 at sloss = 0.6), is monotone and bounded over the operational range of sloss ∈ [0, 1], and preserves the resolution of critical SF interval near sloss of 0.6. The threshold condition uniquely defines the exponential decay constant, ensuring that no additional fitting parameters are needed. Compared with alternative formulations, the exponential mapping allows a smooth reduction in SF toward the calibrated threshold while avoiding oscillations or discontinuities that could arise from polynomial, logarithmic, or piecewise-linear functions.

The framework presented in this study is a novel application of existing numerical methods, rather than the development of a new constitutive model. While the well-established IMASS constitutive model within FLAC3D is used, the core contribution lies in the creation of a practical, SF-based framework for analyzing unsupported underground excavations. This framework includes a post-processing methodology that translates abstract model outputs, such as strength loss (sloss), into a quantifiable SF via a proposed exponential function. The approach offers a more intuitive and readily applicable design tool for engineers, bridging the gap between complex numerical simulations and practical stability assessment. The following sub-sections describe the framework’s development, implementation, and back-analysis.

The exponential function (Equation (1)) translates IMASS-derived sloss values into conventional safety factors, anchored by two field-based calibration points: SF^t = 1.3 at sloss = 0.6 (minimum stable condition) and SF^max = 3.0 at sloss = 1.0 (intact rock). This exponential formulation ensures a smooth, monotonic transition between unstable and stable conditions, preserves resolution near critical SF thresholds, and can be adapted for different lithologies or geometrical configurations. The scaling coefficient α is uniquely determined by these calibration points, making the function mathematically robust and practically relevant for short-term stability assessment of unsupported excavations.

After determining the parameter α in Equation (1), the SF is calculated for the FLAC3D models. Figure 6 shows the SF results for both coal and stone roof scenarios. As expected, the coal roof scenario exhibits an SF of 1.3 or higher, indicating a relatively stable condition for an unsupported roof at a 15.0 m excavation depth. This result is consistent with field observations, where the coal roof is maintained intact without the need for additional support.

In contrast, the stone roof scenario exhibits an SF below 1.3 for the same excavation depth, highlighting the instability of the unsupported roof at 15.0 m. However, at a reduced excavation advancement of 5.0 m—the maximum depth at which an unsupported plunge is maintained stable—a narrow, localized zone with an SF slightly below 1.3 is observed on the roof. This shows that while the 5.0 m excavation remains serviceable, it lies at the threshold of stability, and further excavation may result in failure without support.

Figure 6 illustrates the exponential relationship between SF and $\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}$. The exponential function provides a smooth transition from unstable to stable conditions, preventing the model from sudden changes in SF as the rock mass behavior evolves. This smooth progression ensures that the numerical assessment represents accurately the gradual nature of rock mass damage accumulation and stability recovery ($\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}$ = 0.6), providing a more reliable framework for design and analysis in rock engineering. By avoiding sudden shifts in SF, the function enhances the model performance, particularly in scenarios involving progressive, gradual failure of soft sedimentary rocks under varying excavation and support conditions.

These findings underscore the importance of advance length in determining roof stability and highlight the critical nature of material properties in influencing SF values. The calibration of the SF–sloss relationship is performed using a plunge excavation with a 15.0 m advance length at 160.0 m depth, for which field observations of roof stability exist. This back-analysis defines the correspondence between sloss values and the threshold safety factor (SF^t) under the specific lithological and mining conditions of the study site. Once calibrated, the same parameters (α, SF^t, SF^max, and sloss threshold) are applied to assess other excavation geometries, including bellouts and crosscuts or different depths of cover, provided that the rock mass domain is comparable. This approach is consistent with common practice in numerical modelling, where calibration against a well-documented case offers a robust basis for extending the framework to cases without direct observational data [66,67,68,69]. In cases where lithology, structural conditions, or rock mass strength vary significantly, the framework is recalibrated through the same back-analysis procedure to capture site-specific behavior.

Beyond roof composition, other factors affect the SF contour distribution, including the mechanical properties of the rock mass, such as the plastic shear strain multiplier, and varying geometrical configurations of the excavation. While a full sensitivity analysis lies outside the scope of this study, the current work assesses the influence of roof condition (CO vs. COx) across several excavation scenarios. A comprehensive parametric study considering excavation geometry, rock mass strength, and plastic shear strain multipliers will be conducted in future research.

It should be noted that the dynamic loads, such as machine-induced vibrations, are not modeled because pillar recovery excavations are mechanically mined (no blasting) and abandoned immediately post-mining, removing both the source and the need for vibration analysis. Stability is therefore evaluated under static conditions, consistent with the short-term, unsupported nature of these excavations. This approach aligns with established numerical modeling practices for board-and-pillar mining [46,70]. Applying the framework to excavations subjected to operational traffic or blasting would require additional dynamic analysis.

While the current framework is designed to evaluate immediate post-excavation stability, it assumes quasi-static input conditions and does not implement a dynamic database for real-time updates. This assumption is consistent with general practice in quasi-static numerical modelling studies [46,66,71,72,73,74,75]. As a result, the model does not normalize or filter time-dependent variations in rock mass properties, excavation sequences, or stress redistribution. Noise removal and data preprocessing must therefore be managed externally before applying the framework to continuously evolving datasets. Future work could address this limitation by integrating real-time monitoring data with automated noise filtering and normalization procedures [76], thereby enabling a dynamically updated assessment of short-term pillar stability.

The present study investigates the short-term stability of unsupported excavations, and no structural supports are included explicitly. However, the proposed SF-based framework is independent of support conditions and may be applied to supported excavations using FLAC3D’s built-in structural elements, allowing calibration with field monitoring data to determine stability thresholds. The framework is calibrated via back-analysis against field observations, minimizing input uncertainty and ensuring that the stability assessment reflects realistic short-term excavation behavior. It is intended for use within geologically consistent zones of the same mine or similar settings. Application to distinct geological conditions requires site-specific recalibration using local field observations—consistent with standard rock engineering practice [62,77].

3. Pillar Extraction Case Study

3.1. Overview of Plunge and Bellout Configurations in Board and Pillar Layout

For a more in-depth analysis of the proposed SF approach, a case study from an Australian mining operation was selected. The objective of project is to assess the short-term stability of different pillar extraction scenarios with various geometrical configurations as plunge and bellout. Figure 7 illustrates the schematic layout of a typical bord-and-pillar mining operation. The bellout excavations are designed to facilitate pillar recovery. The bellout is an underground mining structure comprising multiple excavation drives extending into the pillar to enable secondary ore extraction (Figure 7). This structure is typically left unsupported, and upon completion of excavation, equipment is relocated to the next pillar for continued recovery operations. The span of the bellout directly influences ore recovery, with larger spans generally increasing the amount of extractable ore.

While multi-parameter sensitivity analysis is valuable in site-specific studies, this work intentionally focuses on roof profile and excavation geometry—the dominant, controllable variables in short-term pillar recovery design. This targeted approach ensures the framework delivers immediately actionable stability guidance for field engineers during pillar recovery in board-and-pillar mining.

3.2. FLAC3D Model Setup for Bellout for Case Study Stability Assessment

Figure 8 illustrates the FLAC3D model created for the stability assessment of plunge and bellout. The numerical modelling of unsupported plunges aims to evaluate the following:

• Prediction of the maximum unsupported plunge depth under varying rock mass conditions.
• Evaluation of the effectiveness of clean coal beam in unsupported plunges.
• Assessment of maximum excavation advancement (5.0, 10.0, or 15.0 m) at which the stability of the unsupported roof is preserved.
• Assessment of the fracturing and damage responses of weak carbonaceous siltstone (XT) on the immediate plunge roof.

The numerical modelling of unsupported bellout evaluates:

• Prediction of the unsupported roof stability in bellouts of varying geometry.
• Evaluation of the effectiveness of coal beam in unsupported bellouts.
• Assessment of fracturing and damage response of weak XT on the immediate roof.

Based on empirical assessments and previous field experience, two bellout configurations are defined for pillar recovery. These configurations are illustrated in Figure 9. The initial roadways of the plunge and bellout structures are supported (as illustrated in Figure 8 and Figure 9), and in the FLAC3D model, an equivalent support pressure was applied to the roof and ribs. Although these supported areas are not the main focus of the stability assessment, they are included in the model for completeness.

The in situ stresses were assigned to the models using Equation (2). A TSF of 0.5 was used to estimate the maximum horizontal stress (σH), while the minimum horizontal stress (σh) was calculated using a TSF of 0.3. The average Young’s modulus of the strata was assumed to be 8.3 GPa, and the Poisson’s ratio was taken as 0.2. The depth of cover for all models was specified as 150.0 m.

One of the key factors influencing the stability of underground excavations in stratified lithologies is the thickness and mechanical strength of the immediate roof strata. In this study, two distinct roof scenarios were modelled in FLAC3D: one with COx serving as the immediate roof, and the other with a 0.5 m layer of CO retained as the immediate roof material. Figure 10 illustrates the CO and COx roof scenarios considered in FLAC3D modelling. The stratified lithological sequences used in this study were obtained from borehole core logging data, which represents the standard practice for lithological characterization in coal mining projects. The objective of the numerical assessment was to assess the extent to which the presence of the stronger CO layer enhances roof stability.

We emphasize that the focus of this work is on the applied approach—the safety factor framework—rather than reproducing every geological detail of the site. Incorporating excessive complexity into numerical models can obscure the dominant mechanisms and introduce additional uncertainty without improving practical insight. As highlighted by Cundall [78], simplified, physically representative models often provide more reliable understanding of key behaviors compared to overly detailed configurations. This approach has been widely adopted in numerical modelling of underground mining excavations in stratified lithologies [46,66,71,72,73,74,75].

It should be noted that while the geological configuration in this study is intentionally simplified—representing stratified lithologies as horizontally uniform layers—this methodological choice does not compromise the validity of the comparative stability analysis. The primary objective of this framework is to capture relative trends in stability response under varying design parameters (e.g., CO vs. COx roof profiles, plunge excavation lengths, and bellout spans), rather than to replicate site-specific geological heterogeneity. This approach aligns with established practices in numerical modeling of stratified underground excavations, where controlled simplifications are routinely employed to isolate the influence of key geometric or design variables [46,47,75,79]. For instance, Napa-García et al. [46] held geological variability constant to identify optimal pillar dimensions, while Jin et al. [47] examined roadway layout effects under simplified stress conditions—both demonstrating that such focused comparisons yield actionable, design-relevant insights despite idealized inputs. By adopting a similar strategy, this study provides a clear diagnostic tool for understanding how roof geometry and excavation sequencing govern stability trends.

Table 2. Material Properties Used for the Case Study Stability Assessment in FLAC3D-IMASS.

Lithology	Generalized Hoek-Brown Parameters
	UCSi	E	GSI	mi	${\mathit{\lambda }}_{\mathbf{e}}^{\mathbf{c}\mathbf{r}\mathbf{i}\mathbf{t}}$
	[MPa]	[GPa]	[-]	[-]	[-]
ST	19	5.7	60	12	0.75
XT	13	4	40	8	0.35
COx	7.8	3.5	45	8	0.35
CO	7	2.5	50	10	0.75
SS	24	7.2	65	17	0.75

SS = Sandstone; ST = Siltstone; XT = Carbonaceous Siltstone; CO = Coal Seam; COx = Impure Coal.

presents the material properties derived from rock mass characterization for the case study. Based on back-analysis results, the critical plastic shear strain multiplier was assigned a value of 0.35 for both COx and XT lithologies.

3.3. Stability Assessment of Plunge Using the SF Approach in FLAC3D-IMASS

The numerical results of the plunge stability assessment, based on SF values, for both CO and COx roof scenarios are presented in Figure 11 and Figure 12. The SF framework was used to evaluate short-term stability immediately post-excavation (see Section 2.2 for modeling assumptions). To enable a more detailed analysis, critical sections along the plunge were selected, and corresponding sectional view plots were generated. The damage response of the models, represented by SF contours, was presented for the roof and rib regions. In each plot, the major and minor principal stresses (${\sigma }_1$ and ${\sigma }_3$) were shown. The major principal stress (${\sigma }_1$) was oriented at an angle of 30 degrees relative to the North.

The numerical simulation results indicated that at an excavation advancement of 5.0 m, the scenario with a CO immediate roof effectively maintained the plunge stability. In contrast, the corresponding scenario with a COx roof experienced roof instability, particularly on the left side of the tunnel, primarily influenced by the orientation of the in situ stresses. The analysis further showed that increasing the excavation advancement to 10.0 m and 15.0 m caused unstable conditions even for the CO roof case, as evidenced by the distribution of SF contours. The extent of unstable zones (defined as areas with SF < 1.3) increased significantly with greater excavation depths. This effect was even more pronounced in the COx roof scenario, where a more widespread area of low SF values appeared. The most severe roof damage occurred in the COx case at the 15.0 m excavation depth, highlighting the sensitivity of this lithology to excavation-induced stress redistribution.

These findings underscore the critical role of the immediate roof composition in maintaining excavation stability. The presence of a 0.5 m CO beam in the roof offered a substantial enhancement in structural integrity compared to the weaker COx alternative. The CO beam functioned as a competent bridging layer that redistributed stresses and mitigated the propagation of shear damage into the surrounding rock mass. Although deeper excavations became unstable even with the CO roof, the results clearly demonstrate that for 5.0 m excavation advancement the stability remained acceptable under unsupported condition. The numerical assessment also indicted that preserving a thin layer of stronger material significantly delay the onset of instability and limited the extent of failure zones. This highlights the engineering advantage of selectively retaining stronger lithological units in the roof during excavation planning.

To facilitate a more detailed evaluation of roof stability of unsupported plunges, SF histograms were generated for both the CO and COx roof scenarios. Sectional views of the numerical models (Figure 11 and Figure 12) indicated that the damage zone did not extend into the underlying ST unit. Therefore, the SF distribution was assessed specifically within the affected lithological units (CO, COx, and XT), which comprised the primary zones of stress-induced damage. Based on this assessment, SF histograms were produced to quantify and compare the extent and severity of roof instability under the two different roof conditions.

Figure 13 and Figure 14 present the comparative SF histograms generated for roof stability assessment. The horizontal axis displays the safety factor, and the vertical axis displays the frequencies for specific SF. The threshold SF is indicated by a dashed line; the unstable zones appear in red colour while the stable zones appear in green. The histograms were produced by quantifying all zones within the fractured regions across the full extent of the three-dimensional model. In the CO roof scenario, measurements were taken in the CO, COx, and XT units, as fracturing predominantly occurs within these layers. For the COx roof scenario, measurements are focused on the COx and XT units, as fracturing primarily occurs within these weaker layers.

The histogram results showed that for the 5.0 m excavation advancement in the CO roof scenario, all calculated SF exceeded the critical threshold value of 1.3, suggesting a stable roof condition throughout the modelled zone. This outcome highlights the effectiveness of retaining a 0.5 m CO beam as the immediate roof in enhancing structural integrity. In contrast, under identical excavation conditions, the COx roof scenario showed a significant proportion of zones with SF values below 1.3, indicating localized instability and insufficient strength of the immediate roof to support the unsupported excavation.

As the excavation advancement increases to 10.0 m and 15.0 m, both roof scenarios displayed a growing presence of unstable zones (SF < 1.3), reflecting a progressive reduction in overall stability of unsupported structures. However, the extent and severity of instability were more pronounced in the COx roof condition, particularly at the 15.0 m excavation depth, where a widespread distribution of low SF values appeared. This comparison clearly showed the critical role of the retained CO layer in improving roof stability when unsupported excavation was performed in the operation. The CO beam functioned as a structural reinforcement, redistributing stresses and limiting the extent of failure zones, whereas its absence (in the COx scenario) resulted in a broader and deeper progression of instability, especially under larger excavation advancements. These results demonstrated the practical benefit of preserving a competent immediate roof layer to enhance excavation safety and minimize roof failures under unsupported conditions.

3.4. Stability Assessment of Bellout Structures Using the SF Approach in FLAC3D-IMASS

The stability of bellout underground structures was evaluated using FLAC3D, the proposed geometrical configurations of the bellout designs (Figure 9) imported into the software environment. The stability of each unsupported bellout structure was assessed under two distinct immediate roof conditions: coal (CO) and impure coal (COx). The predicted safety factor (SF) contours, generated using the proposed SF-based assessment framework implemented in FLAC3D, are shown in Figure 15, Figure 16, Figure 17 and Figure 18 for all modelling scenarios. The SF framework was used to evaluate short-term stability immediately post-excavation (see Section 2.2 for modeling assumptions). To enable detailed interpretation, each bellout configuration included corresponding sectional views through key locations of the models.

For bellouts with 6.0 m and 8.0 m spans, the plan views showed that in the COx roof scenario, unstable zones (SF < 1.3) developed primarily along the excavation periphery on the roof, indicating that stress orientation played a dominant role in inducing instability in the weaker COx roof. In contrast, the CO roof scenario exhibited improved stability, with the majority of zones, particularly in the immediate roof, demonstrated SF values above the 1.3 threshold, indicating a stable condition. Section views revealed that for the CO scenario, section A–A′ displayed minor localized instability within the XT unit, again likely caused by stress orientation. However, sections B–B′ and C–C′ showed uniformly stable zones with SF > 1.3. On the other hand, in the COx roof scenario, all three sections exhibited small but widespread zones with SF < 1.3, primarily concentrated near roof corners, which are typical locations for stress concentration.

When the bellout span increased to 8.0 m, the CO roof scenario continued to demonstrate similar stability characteristics as the 6.0 m span case, with slightly fewer zones below the SF threshold. This improvement was attributed to the slightly shorter tunnel length in the 8.0 m span model (~7.5 m) compared to the 9.0 m length in the 6.0 m span model, which enhanced confinement and reduced the influence of excavation-induced stress redistribution. For the COx roof scenario at 8.0 m span, the general pattern of instability remained similar to that of the 6.0 m span, with unstable zones still concentrated at the excavation corners on the roof. However, a marginal reduction in the number of zones with SF < 1.3 was observed, again likely due to the shorter tunnel length reducing the disturbed zone extent.

The comparative analysis of SF distributions across varying span lengths and roof conditions highlighted the decisive influence of the immediate roof’s geomechanical properties on the overall stability of bellout structures. The CO roof scenario consistently outperformed the COx roof case, showing fewer unstable zones and higher resilience. The reduction in instability for the 8.0 m span models, regardless of roof type, emphasized the importance of excavation geometry, particularly tunnel length, in controlling stress redistribution and maintaining roof integrity. The results demonstrate the critical role of preserving a competent beam of coal in the roof (CO condition), which acts as a stabilizing arch that helps confine stresses and limit failure propagation. These findings suggest that in similar geological settings, engineering designs should prioritize the retention of a strong immediate roof layer and consider optimizing tunnel geometry to enhance stability and reduce the risk of roof falls during short-term unsupported excavations.

It should be noted that the SF did not account for any time-dependent (creep) behaviour of the rock mass. The analysis was limited to the immediate response of the jointed rock mass and considers only its elastic and plastic deformability under static loading conditions. Accordingly, the model represented the short-term stability condition of the rock mass, typically within a few hours post-excavation, and does not capture delayed failure mechanisms associated with long-term creep or stress redistribution over time. To assess time-dependent roof damage of the unsupported excavation, creep constitutive modelling should be incorporated into the simulations.

Figure 19 and Figure 20 present the SF histograms for the bellout models. These histograms were obtained by quantifying all zones within the fractured regions across the full three-dimensional domain of the numerical model. In the CO roof scenario, the analysis induced the CO, COx, and XT units, as fracturing predominantly occurred within these lithological layers. For the COx roof scenario, the SF distribution is assessed within the COx and XT units, which represented as the primary zones of instability due to their lower mechanical strength.

The histogram results reveal that both the 6.0 m and 8.0 m span configurations contained a number of zones with SF values below the critical threshold (SF < 1.3), indicating localized failure zones. These findings were consistent with the section views presented in Figure 15, Figure 16, Figure 17 and Figure 18. Notably, the 8.0 m span configurations exhibited fewer zones with SF < 1.3, which attributed to the reduced tunnel length in these models and the associated redistribution of stress.

The comparison between the SF histograms of different span configurations highlights valuable insight into the role of tunnel geometry and immediate roof composition in controlling structural stability. Although increasing the bellout span from 6.0 m to 8.0 m might intuitively suggest increased instability, the concurrent reduction in tunnel length produced a more favorable stress distribution, resulting in a net improvement in stability. Specifically, the CO roof scenario demonstrated a significantly more robust performance in both span cases, with a notably lower frequency of zones falling below the stability threshold. This confirms the stabilizing effect of the 0.5 m CO beam retained as the immediate roof. In contrast, the COx roof scenario consistently exhibited more widespread failure-prone zones, particularly concentrated in the roof corners where stress concentrations were highest.

Although geometry-sensitive stability indicators are not explicitly incorporated into the SF-based framework, excavation geometry inherently influences SF contours and guides operational decisions. For instance, in the CO roof scenario (roadway width = 6.0 m), a short excavation length of 5.0 m exhibits stable conditions (SF > 1.3), while increasing the excavation length to 10.0 m and 15.0 m produces unstable zones, with the 15.0 m case showing a pronounced extension of SF < 1.3 regions. These results provide geotechnical engineers with actionable insight for selecting safe excavation lengths in accordance with Triggered Action Response Plans (TARPs).

In bellout geometries under both CO and COx roof conditions, staged excavation has minimal effect on overall SF values, likely because individual bellout fingers are largely independent and stress redistribution is negligible. By contrast, plunge excavations require a detailed assessment of excavation sequences to identify the length associated with low-risk conditions. Overall, while the framework does not define explicit geometry-sensitive indicators, SF contours effectively reflect the influence of excavation geometry, enabling informed decisions on safe extraction sequences.

4. Technical Limitations

While the proposed framework provides a practical tool for assessing immediate post-excavation stability, several technical limitations should be acknowledged:

Quasi-static assumptions: The framework assumes quasi-static input conditions and does not incorporate a dynamic database for real-time updates. As a result, time-dependent variations in rock mass properties, excavation sequencing, or stress redistribution are not directly normalized or filtered. Data preprocessing and noise removal must therefore be managed externally. Future work could address this limitation by integrating real-time monitoring data with automated noise filtering and normalization procedures to enable dynamically updated stability assessments.

Simplified lithological representation: The applied modelling approach represents stratified lithologies as horizontally uniform layers. This simplification was chosen to balance geological realism with numerical robustness, ensuring that the framework remains interpretable and design-oriented. While more detailed lithological modelling may capture additional complexity, excessive detail can obscure dominant mechanisms and introduce uncertainty without improving insight. This methodological choice aligns with established practices in numerical modelling of stratified underground excavations [46,66,71,72,73,74,75].

Focus on applied framework: The primary objective of this study is to evaluate stability trends under varying excavation geometries and roof conditions, rather than replicate every site-specific geological heterogeneity. Controlled simplifications are therefore adopted to isolate the influence of key design parameters. Such an approach has been widely demonstrated to yield actionable design insights despite idealized inputs [46,47,75,79].

Exclusion of long-term deformation mechanisms: Time-dependent processes such as creep are not considered in this framework. Pillar recovery excavations are temporary, abandoned immediately after ore recovery, and require stability for only a few hours—the critical window for safe equipment retreat. Accordingly, the framework is suitable for quasi-static analysis of underground openings where long-term deformability is not a design consideration. Long-term deformation becomes relevant only in cases where time-dependent rock mass behaviour must be simulated, such as in salt mines [80] or other deep mining operations where long-term serviceability of the opening is required [81].

Calibration constraints: Calibration relied on visual field observations because instrumented monitoring is neither feasible nor necessary in temporary, unsupported excavations. This practice is consistent with pillar recovery operations, where instrumentation is not installed in areas abandoned shortly after mining. Visual and photographic evidence, widely used in mining geomechanics when instrumentation is impractical [62,63,64,65], was therefore adopted as the appropriate calibration method for this framework. Where monitoring and instrumentation data exist—for example, in tunnels and roadways used during active mine operations—the SF-based framework can be recalibrated using field-observed parameters, such as displacements or convergence, to support site-specific stability assessments.

5. Conclusions

This study introduced and implemented a safety factor (SF)-based framework within FLAC3D to evaluate the stability of unsupported underground structures in bord-and-pillar mining layouts. The numerical modelling considered various excavation geometries and roof conditions, with particular focus on plunge and bellout configurations under two distinct roof material scenarios: coal (CO) and impure coal (COx).

The results demonstrated that the SF framework effectively identifies zones of potential instability by capturing localised stress redistributions and roof responses. Following the calibration of the proposed safety factor framework against field observations, an Australian bord-and-pillar case study was selected to evaluate the stability of unsupported plunge and bellout excavations during the pillar recovery phase. In the plunge configuration, the inclusion of a 0.5 m coal beam (CO roof) was found to provide a marked enhancement in roof stability at low excavation advancement (5.0 m), maintaining SF values well above the critical threshold of 1.3. In contrast, the absence of this beam (i.e., COx roof) led to significant instability, particularly along the left side of the tunnel due to the orientation of the major in situ stress. As excavation advancement increased to 10.0 m and 15.0 m, both CO and COx scenarios exhibited progressively larger damage zones, with COx showing the most extensive and severe instability. This finding underscores the critical role of the intact coal beam in distributing loads and suppressing failure propagation in short-term unsupported excavations.

For bellout configurations, the SF contours and histograms revealed that the CO roof scenarios consistently outperformed COx in maintaining structural integrity, especially at the excavation corners and immediate roof. Even at larger spans (up to 8.0 m), the CO roof condition showed fewer zones with SF values below 1.3, indicating its effectiveness in supporting broader openings. This improved stability is attributed to the reduced tunnel length, which enhances structural confinement through principal stress redistribution and improves roof stability by limiting the volume of excavated rock mass. The histograms provided a quantitative measure of instability by counting the fractured zones in 3D, confirming that COx roofs consistently resulted in higher volumes of unstable zones due to weaker material properties and stress concentration effects at excavation corners. Future work will extend this framework to parametric studies of excavation geometry, rock mass strength, and plastic shear strain multipliers.

The numerical framework intentionally excludes long-term deformation because pillar recovery excavations are abandoned immediately post-mining and require stability only for a few hours—the critical window for safe equipment retreat. Once extraction is completed, the openings are no longer serviced, and no personnel or machinery remain. Consequently, the analyses focus solely on the immediate mechanical response of the rock mass under static loading, and delayed failure mechanisms, including progressive damage or stress redistribution, are irrelevant to the framework’s design objectives. Future studies could extend this approach to assess long-term performance for openings that remain in service throughout mine life by incorporating creep constitutive models.

Overall, the study confirms the reliability of the proposed SF framework in accurately identifying and quantifying localized failure zones, thereby supporting design optimization and short-term stability assessments in applications such as pillar extraction in underground mining operations. It further highlights that preserving a minimal 0.5 m coal beam along the roof can substantially enhance stability and reduce the risk of failure compared to cutting the beam entirely. These findings have direct implications for improving safety and design efficiency in unsupported mining operations by optimising excavation geometries and retaining strategic zones of stronger roof material.