Rockfall Analysis of Old Limestone Quarry Walls—A Case Study

Abstract

This article presents the results of a rockfall analysis conducted for the limestone walls of a former quarry that is now used as an urban park. The performed simulations (2D statistical analysis using Rigid Body Impact Mechanics—RBIM and Discrete Element Modelling—DEM) enabled the determination of the maximum displacement range during the ballistic phase and the maximum rebound height at the slope base, which facilitated the delineation of a safe land-use zone. A hazard zone was also identified, within which public access must be strictly prohibited due to the risk posed by flying debris. Based on slope stability assessments (safety factor values and rockfall trajectories), recommendations were formulated for slope reinforcement measures and appropriate management actions for designated sections to ensure safe operation of the site. Three mitigation strategies were proposed: (1) no protective measures, (2) no structural reinforcements but with installation of a rockfall barrier, and (3) full-scale stabilisation to allow unrestricted access to the quarry walls. The first option—leaving slopes unsecured with only designated safety buffers—is not recommended.

1. Introduction

Rockfall is a common phenomenon that occurs on steep rock slopes, resulting from the sudden detachment of material, such as rocks and boulders, which move vertically or at steep angles along pre-existing discontinuities [1,2,3]. This process is highly dynamic and occurs at high velocities [4,5]. Detached rock masses predominantly move by free fall, bouncing, and rolling before coming to rest at a certain distance from the base of the slope.

Due to their abrupt and avalanche-like nature and their potential to mobilise large rock volumes, rockfalls pose a significant hazard. They can cause substantial damage to residential structures and infrastructure [6,7,8,9,10,11,12,13,14,15], as well as result in human casualties [10,15,16,17,18,19]. Therefore, appropriate land-use planning and the implementation of protective measures are essential to safeguard people, buildings, and critical infrastructure from rockfall-related threats. In rockfall-prone areas, warning signs are often installed, and slopes are secured using protective systems. These include steel wire meshes, barriers, rockfall curtains, and specialised retaining structures [7,20,21,22,23,24,25,26].

Before implementing stabilisation measures, a detailed stability analysis of the rock blocks must be conducted. This step is crucial for effective risk management, early detection of rockfall events, mitigation of their adverse impacts, and selection of appropriate protective systems [9,20,23,27,28,29].

Rockfall analysis is fundamental to understanding and reducing the risks associated with slope instabilities. It should include both qualitative assessment—based on field observations to evaluate visual indicators (e.g., jointing, bedding dip, vegetation presence), geomechanical classification (e.g., GSI, RMR, Q-system), and identification of potential failure mechanisms (kinematic analysis) [10,30,31,32,33,34,35]—and quantitative analysis, such as numerical modeling and probabilistic/risk assessments, which enable simulation of rock mass behavior under various conditions [7,32,35,36,37,38,39].

Based on stability evaluations and dynamic modelling of rock block detachment, trajectory, and impact, it is possible to delineate the expected rockfall dispersion area and identify locations requiring protective reinforcement. This issue is particularly significant when the site is used as a public urban park, where large gatherings may occur, further amplifying the need for reliable safety measures.

This article presents a case study that assesses rockfall hazards in the area of a former limestone quarry, which is now repurposed as an urban park. The analysis of rockfall was performed using a 2D statistical simulation tool incorporating Rigid Body Impact Mechanics (RBIM) and Discrete Element Modelling (DEM). Based on the analysis results, alternative recommendations for slope stabilisation were developed, considering different technical intervention scenarios to ensure the safe use of the former excavation site, now known as Wojciech Bednarski Park. To delineate areas at risk of instability—particularly those exposed to potential block detachment—and to propose suitable mitigation strategies, a combination of qualitative methods (e.g., characterisation of joint sets, kinematic analysis) and quantitative approaches (e.g., finite element stability analysis, rockfall modelling) was applied. Particular emphasis was placed on evaluating the effectiveness of proposed measures in the context of public safety, while maintaining open access to the site. The study contributes to the advancement of practical applications of rockfall kinematics modelling in urbanised environments by offering an approach that integrates field data with numerical analysis. These analyses formed part of a detailed geological, geotechnical, and geomechanical assessment of the quarry walls, conducted in support of construction works associated with the park’s revitalisation project [40].

2. Objective of the Analysis and Study Area Location

The revitalisation of Wojciech Bednarski Park in Kraków, located within the boundaries of a former limestone quarry (Figure 1), necessitated a geotechnical hazard assessment related to potential rock block detachment from quarry walls. The objective of this study was to identify critical sections that require protection and to define the minimum scope of engineering measures necessary to ensure the safe use of the park area. Additionally, local community stakeholders expressed a strong preference for minimising structural reinforcement and preserving the natural appearance of the rock slopes. To assess the extent of necessary stabilisation measures, an inventory of the rock slopes and outcrops was conducted, including an evaluation of weathering intensity, discontinuity mapping, an inventory of detached blocks (impact range), and subdivision of sections based on rock mass quality. For selected sectors, slope stability analysis [41] and rockfall hazard evaluation were performed. The results of these assessments served as the basis for formulating alternative recommendations for slope stabilisation and risk management in the context of future land use planning.

The results of slope stability analyses, conducted for eight designated sectors (Figure 2), are presented in [41]. These sectors were defined based on slope height, the presence of inclined surfaces covered with loose weathered material, the occurrence of jointing, the presence of rock fragments weakly bonded to the rock face, and the identification of unstable rock blocks. A detailed characterisation of the mapped discontinuity systems is provided in reference [41].

The characteristics of the analysed rock mass, including a dense network of discontinuities, and the anthropogenic modification of the rock face due to former quarrying activities, have led to the exposure of discontinuity surfaces. This, in turn, promotes weathering and detachment processes, resulting in both shallow block falls and deeper structurally controlled slides. To ensure the safe use of the site, it is necessary to delineate safety buffer zones separating visitors from hazardous areas. For this purpose, a rockfall analysis was conducted.

3. Geology

In Bednarski Park, the slope bordering the park consists of a rock face and an overlying inclined slope. The rock face is composed of bedded Upper Jurassic limestones with chert nodules. The inclined slope above the rock face often contains rocky ledges and is covered by a layer of loose weathered material. Loose blocks and limestone debris are present on the surface of this weathered layer (Figure 3). The slope height ranges from approximately 1 m to 16.6 m, with inclinations typically varying between 44° and 70°. Locally, the rock face is vertical, occasionally featuring rock overhangs. At the top of the limestone sequence lies a weathered layer (clayey rubble) with a thickness of up to approximately 3.0 m, overlain by Quaternary cohesive soils (sandy clays and clayey sands) in a stiff-plastic consistency state.

The slope of the park (former quarry) is predominantly composed of fractured, moderately strong to weakly cemented Upper Jurassic limestones (engineering categories IV and V [46]). Throughout the entire slope, weakly and very weakly bonded rock blocks (with volumes up to several cubic meters) are present, along with numerous limestone fragments of varying size. Many of these blocks have detached from the parent rock and are held in place solely by the vegetation—trees and shrubs—growing on the slope. The detachment of rock fragments is further promoted by root systems penetrating fractures and joints filled with soil, as well as by frost weathering affecting the near-surface portions of the rock face.

The slopes of Bednarski Park are affected by mass movement processes, including both rockfalls and creeping of the weathered cover material. Numerous locations along the rock face exhibit fresh traces of detached rock blocks or limestone debris. This material accumulates at the base of the slope, forming a talus cone that extends the entire length of the slope (approximately 630 m). The size of detached rock blocks can reach up to 1 m (e.g., 1.0 × 0.49 × 0.36 m), although most commonly the blocks do not exceed approximately 0.3 m in size (with a mass up to 8 kg). Isolated or clustered rock fragments are occasionally present, and their rolling distance from the slope base may reach several meters. Figure 2 shows the extent of talus cones at the base of the rock face and the maximum (field-observed) reach of falling rock fragments.

A key factor contributing to rock detachment from the rock face in the park is the presence of a dense network of discontinuities. Particularly unfavourable are fractures oriented parallel or nearly parallel to the slope surface, with spacings of only several tens of centimetres. Often, the rock face itself is inclined at an angle consistent with these discontinuities, increasing the likelihood of both relatively shallow block falls and deeper structurally controlled slides. Furthermore, these fractures are frequently exploited by tree and shrub roots, which expand them and facilitate the detachment of blocks and debris.

The risk of rock fragment detachment from the rock face, or the downslope movement of debris and rock blocks from the upper inclined slope, increases during adverse weather conditions such as intense and/or prolonged rainfall or snowmelt. Such mass movement events are particularly likely in sectors where the upper slope is wide or bowl-shaped, which promotes the accumulation of water. Short-duration, high-intensity rainfall events (cloudbursts) pose a significant threat to the stability of the slope-covering debris and unstable rock blocks within the rock face.

4. Description of the Computational Method and Adopted Assumptions

The rockfall analysis was conducted using a 2D statistical analysis program (RocFall2 by Rocsciese, version 8.0.2.5) designed to assess rockfall hazard from slopes. This software integrates rigid body impact mechanics (RBIM) and discrete element modelling (DEM). It allows for the definition of rock blocks with various shapes, sizes, and densities, as well as the initial conditions of falling blocks, including vertical and horizontal velocities, angular velocities, and initial rotations. Additionally, it requires the specification of several key input parameters, such as the normal coefficient of restitution, tangential coefficient of restitution, dynamic friction, and rolling resistance—each of which is dependent on the material properties of the slope surface.

To assess the mechanisms of potential rockfall initiation, comprehensive mapping of the quarry wall surfaces was conducted. The geological discontinuities within the rock mass were classified into five joint sets (A–E), with their stereographic projections also presented in [41]. In the DEM model, discontinuities were indirectly represented through the spatial distribution of particles and the parameters of contact bonds, whose failure reflected potential planes of weakness. A simplified statistical distribution of orientation and material parameters was assumed for the particles, corresponding to the average properties of the identified discontinuity sets.

Three slope-building materials were considered in the calculations:

• weathered debris/talus, modelled as loose rock debris,
• limestone, modelled as bedrock,
• clay with stones, crushed rock, and soil, modelled as soil material.

In some cross-sections, rock debris was introduced between the soil and bedrock at the base of the slope.

For the materials mentioned above, the following parameters were implemented: normal coefficient of restitution (defined as the square root of the negative ratio of outgoing to incoming energy), tangential coefficient of restitution (introduced as artificial damping), dynamic friction, and rolling resistance. The values for these parameters were derived from the literature, including refs. [47,48] for rock debris, refs. [47,48,49,50] for bedrock, and refs. [47,48,51] for soil.

Moreover, a normal distribution (±one standard deviation) was applied to each parameter. The adopted material parameters for the slope surface are summarised in

Table 1. Material parameters of the slope geology.

Type of Material	Normal Coefficient of Restitution	Tangential Coefficient of Restitution	Dynamic Friction	Rolling Resistance
Weathered Material/Talus	0.32 ± 0.04	0.82 ± 0.04	0.557 ± 0.054	0.65 ± 0.017
Limestone	0.315 ± 0.064	0.712 ± 0.116	0.576 ± 0.13	0.40 ± 0.083
Clay with Stones, Crushed Rock, Soil	0.30 ± 0.00666667	0.815 ± 0.005	0.56 ± 0.0867	0.815 ± 0.068

.

Rock and weathered blocks were modelled by assigning appropriate shapes, sizes, and bulk densities. For the weathered material, a density of 1.85 g/cm³ was adopted in the geotechnical report [52]. Four block shapes were analysed: spherical, triangular, square, and rhomboid. For limestone, a density of 2.56 g/cm³ was used (based on archival documentation [52,53]), and three block shapes were considered: triangular, square, and rhomboid. The dimensions of weathered fragments and rock blocks were determined based on the results of conducted geological investigations [40].

The adopted sizes of rock and weathered blocks are presented in

Table 2. Material parameters of the slope geometry.

Sector	Rock Block Size [m]	Weathered Fragment Size [m]	Remarks
1	0.05; 0.10; 0.15; 0.50; 1.00	0.03; 0.10; 0.15	Numerous irregular clusters of rock fragments (up to several centimetres) are present on the rock surface. In the upper and middle parts—blocks up to approx. 1 m.
2	0.05; 0.10; 0.15; 0.50; 1.00	0.03; 0.10; 0.15; 0.20; 0.30	At the upper edge of the debris zone.
3	0.05; 0.10; 0.15; 0.50; 1.00	0.03; 0.10; 0.15; 0.20	Localised clusters of rock fragments (up to several centimetres) are poorly bonded to the rock. Isolated unstable rock blocks (up to approx. 1 m) in the middle and upper parts of the rock wall. Individual rock blocks up to approx. 0.30 m in the weathered zone at the top.
4	0.05; 0.10; 0.15; 0.50; 1.00; 1.50	0.03; 0.10; 0.20; 0.50; 0.80	The rock wall contains blocks up to approximately 1 m (most commonly 0.5 m).
5	0.05; 0.10; 0.15; 0.20; 0.40; 1.00	0.03; 0.10; 0.20; 0.50	Localised clusters of rock fragments up to several centimetres in size.
6	0.05; 0.10; 0.15; 0.20; 0.30; 0.40; 0.50	0.03; 0.10; 0.20; 0.50	Creeping of weathered material is observed in many locations.
7	0.01; 0.10; 0.15; 0.20; 0.30	0.05; 0.10; 0.20; 0.50	Separation of rock blocks up to approx. 1.5 m (most commonly 0.5 m).
8	0.01; 0.03; 0.05; 0.10; 0.15; 0.20; 0.30	0.03; 0.10; 0.20; 0.50	Numerous unstable blocks are present in this sector.

.

The potential detachment of rock fragments was analysed for the upper and lower parts of the weathered layer, as well as for the top, middle, and bottom sections of the rock face. An initial velocity of zero was assumed. For each designated block (defined by specific size and shape), fifty possible flight paths were simulated.

Calculations were performed for eight designated sectors, using cross-sections that had been previously subjected to slope stability analyses [41]. Additionally, in sectors 3 and 4, a triangular-section buttress composed of weathered material (with properties defined in Table 1) was modelled. Two buttress heights were considered: 0.7 m and 1.0 m (Figure 4).

The purpose of the buttress is to isolate the zone where rock and/or weathered fragments reach the highest rebound heights (~1.0–2.0 m) or remain in flight from the rock slope. The presence of the buttress also shortens the travel distance of detached rock and weathered fragments.

5. Simulation Results and Discussion

5.1. Model Verification and Validation

To ensure the reliability of the conducted analysis, fundamental principles of verification and validation (V&V) of the simulation model were applied. Model verification ensured the correctness of input data implementation and the proper functioning of the simulation algorithm. This process included trajectory control tests, behavioural analysis of blocks with known properties, and consistency checks between adopted material parameters and values reported in the literature. Model validation was carried out by comparing simulation outcomes with field observations. For each analysed sector, input data such as block size and shape (based on field mapping and geotechnical documentation), ground conditions (defined into eight sectors), material density, and contact parameters (derived from literature sources [47,48,49,50,51,52,53]) were compiled.

The analysis included more than 50 trajectories for each block size and shape, incorporating parameter variability through the application of a normal distribution for restitution coefficients, dynamic friction, and rolling resistance. This parameterisation approach enabled a statistical representation of terrain condition variability and reduced the risk of overestimation or underestimation of hazard levels.

Simulation results, including maximum displacement range, rebound height, and flight zone extent, were compared with actual field observations. It was found that the predicted displacement ranges aligned with the locations of accumulated rock debris and weathered material, and the maximum simulated values corresponded to the identified hazard zones.

It is important to note that the rockfall analysis was conducted using a two-dimensional (2D) model. This decision was based on the slope geometry identified through LIDAR laser scanning, which enabled the precise reconstruction of the slope face topography and the adjacent terrain configuration.

The quarry wall’s horizontal alignment is relatively straight and uniform, making it possible to accurately represent key block trajectories in vertical cross-sections without the need for a complete three-dimensional (3D) model. Moreover, available field observations (distribution of rock fragments and displacement traces) confirmed that the dominant direction of block movement occurs within the 2D cross-sectional plane.

In conclusion, the performed verification procedures and consistency between simulation results and field data indicate that the model is sufficiently accurate for rockfall hazard assessment in the analysed area.

5.2. Analysis of the Range of Detached Rock and/or Weathered Fragments

The results for Sector 1 (Figure 2) are presented in

Table 3. Range of detached rock/weathered fragments—Sectors 1–3.

Block and Weathered Fragment Size [m]	Maximum Distance Reached by Detached Block/Fragment [m]	Detachment Location for Maximum Distance	Maximum Airborne Distance from Base of Rock Face (Excluding Rolling Phase) [m]	Maximum Rebound Height at Toe of Slope [m]
Sector 1
0.03–0.05	3.16	Upper edge of the weathered slope	-	0.32
0.10–0.15	2.77	Top of the upper rock face	-	0.42
0.50–1.00	0.98	Top of the lower rock face	-	-
Sector 2
0.03–0.05	7.69	Upper edge of the weathered slope	-	0.44
0.10–0.20	6.92	Lower edge of the weathered slope	-	0.54
0.30–0.50	5.79	Upper edge of the weathered slope	-	0.67
1.00	1.39	Top of the rock face	-	-
Sector 3
0.03–0.05	8.36	Upper edge of the weathered slope	0.72	0.88
0.10–0.20	7.50	Lower edge of the weathered slope	1.18	1.10
0.50	5.46	Top of the rock face	0.90	0.98
1.00	3.63	Top of the rock face	-	-

and illustrated through the migration path diagrams of detached blocks and rock fragments (Figure 5).

The most significant migration distance in Sector 1 was observed for rock/weathered fragments measuring 0.03–0.05 m, reaching a maximum distance of approximately 3.16 m from the slope. Both 0.03–0.05 m and 0.10–0.15 m fragments, when detached from the top of the weathered slope, traverse the rock ledge and reach the base of the lower rock face.

Blocks measuring 0.50–1.00 m, falling from the top of the upper rock wall, are stopped by the ledge located between the rock walls, with a maximum rebound height on the ledge of 0.40 m.

The analysis results for Sector 2 (Figure 2) are presented in Table 3 and illustrated through block and fragment migration path diagrams (Figure 6).

In this sector, the most incredible migration range was again recorded for fragments sized 0.03–0.05 m, detaching from the top of the weathered slope and reaching a maximum distance of ~7.69 m from the hill, with a maximum rebound height of 0.44 m. Larger fragments, measuring 0.10–0.20 m and 0.30–0.50 m, reached maximum distances of ~6.92 m and ~5.79 m, with corresponding rebound heights of 0.54 m and 0.67 m, respectively. One-meter rock blocks came to rest immediately after reaching the base of the rock wall. The results obtained for Sector 3 (Figure 2) are included in Table 3 and illustrated in the block and fragment migration path diagrams (Figure 7 and Figure 8).

In Sector 3, the smallest rock/weathered fragments (0.03–0.05 m) exhibited the longest migration distance, reaching approximately 8.36 m, with a maximum rebound height of 0.88 m. Fragments of larger sizes, such as 0.10–0.20 m and 0.50 m, reached maximum distances of ~7.50 m and ~5.46 m, respectively, and attained rebound heights of 1.10 m and 0.98 m. In contrast, 1.00 m rock blocks achieved a maximum distance of ~3.63 m. Additionally, a triangular-shaped buttress composed of weathered material (with parameters as defined in Table 1) was modelled in Sector 3, with heights of 0.7 m and 1.0 m and placed at distances of 3.0 m and 4.0 m from the rock wall. The obtained results are illustrated in Figure 9 and Figure 10.

The introduced buttress in Sector 3 effectively intercepted the majority of detached rock blocks and delineated the zone where rock/weathered fragments experienced the highest rebound (~1.0 m). At a 3.0 m distance, a 1.0 m high buttress proved most effective, as a slightly lower structure (0.70 m) presented a higher risk of fragments rebounding to hazardous heights and distances. Positioning the buttress (both 0.7 m and 1.0 m high) at 4.0 m from the wall nearly eliminated high rebounding of rock/weathered fragments.

The analysis results for Sector 4 (Figure 2) are presented in

Table 4. Range of detached rock/weathered fragments—Sectors 4–6.

Block and Weathered Fragment Size [m]	Maximum Distance Reached by Detached Block/Fragment [m]	Detachment Location for Maximum Distance	Maximum Airborne Distance from Base of Rock Face (Excluding Rolling Phase) [m]	Maximum Rebound Height at Toe of Slope [m]
Sector 4
0.03–0.05	9.89	Top of the weathered slope	1.22	1.33
0.10–0.20	9.99	Top of the weathered slope	2.41	1.30
0.50–0.80	7.79	Top of the weathered slope	1.50	1.36
1.00–1.50	4.63	Top of the rock face	-	-
Sector 5
0.03–0.05	8.61	op of the weathered slope	2.13	1.61
0.10–0.20	8.20	Top of the weathered slope	2.32	1.97
0.40–0.50	6.10	Top of the rock face	-	1.11
1.00	2.78	Top of the rock face	-	1.43
Sector 6
0.03–0.05	7.54	Top of the weathered slope	-	0.55
0.10–0.20	9.59	Top of the rock face	-	0.89
0.30–0.50	6.74	Top of the rock face	-	1.06

and illustrated through block and fragment migration path diagrams (Figure 11).

The most significant migration distance in Sector 4 was exhibited by rock/weathered fragments sized 0.10–0.20 m, which reached approximately 9.99 m from the slope. Similarly high distances (~9.89 m) were observed for the smallest fragments (0.03–0.05 m). Blocks measuring 0.50–0.80 m achieved a maximum distance of ~7.79 m, while 1.0–1.5 m blocks reached up to ~4.63 m. All aforementioned fragment sizes attained their maximum migration distance when falling from the top of the weathered slope. The maximum rebound heights ranged between 1.30 m and 1.36 m. Additionally, as in Sector 3, a triangular berm composed of weathered material (properties listed in Table 1) was modelled at a height of 0.7 m and 1.0 m, positioned 4.0 m from the rock face; the results are shown in Figure 12.

The implemented 1.0 m high berm at the base of the slope in Sector 4 effectively stops most detached rock blocks and separates the zone where rock/weathered fragments exhibit the highest rebound (~1.0 m). In contrast, a 0.70 m high berm may still allow hazardous high rebounds of rock/weathered fragments.

The analysis results for Sector 5 (Figure 2) are presented in Table 4 and illustrated through block and fragment migration path diagrams (Figure 13).

The most significant migration distance in Sector 5 was recorded for 0.03–0.05 m fragments detached from the top of the weathered slope, reaching a maximum distance of ~8.61 m and a rebound height of 1.61 m. Larger fragments, such as 0.10–0.20 m and 0.40–0.50 m, reached maximum distances of ~8.20 m and ~6.10 m, with corresponding maximum rebound heights of 1.97 m (2.32 m in flight from the bottom of the rock face) and 1.11 m, respectively. One-meter rock blocks stopped at the base of the rock wall at a distance of approximately 2.78 m.

The results for Sector 6 (Figure 2) are presented in Table 4 and shown in the block and fragment migration path diagrams (Figure 14).

The most significant migration distance in Sector 6, approximately 9.59 m, was observed for rock/weathered fragments measuring 0.10–0.20 m in diameter, detached from the upper part of the rock face, with a maximum rebound height of approximately 0.89 m.

The smallest fragments (0.03–0.05 m) reached a maximum distance of ~7.54 m and a rebound height of 0.55 m.

Rock/weathered blocks sized 0.30–0.50 m achieved a maximum distance of ~6.74 m from the slope base and a rebound height of ~1.06 m.

The analysis results for Sector 7 (Figure 2) are presented in

Table 5. Range of detached rock/weathered fragments—Sectors 7–8.

Block and Weathered Fragment Size [m]	Maximum Migration Distance of Detached Block/Fragment [m]	Detachment Location for Maximum Distance	Maximum Airborne Distance Measured from Base of Rock Wall [m]	Maximum Rebound Height (at Slope Base) [m]
Sector 7
0.01–0.05	3.90	Top of weathered/debris slope	0.69	0.42
0.10–0.20	4.95	Top of weathered/debris slope	0.96	0.60
0.30–0.50	2.86	Top of rock face	0.86	-
Sector 8
0.01–0.05	4.66	Top of weathered/debris slope	0.36	0.53
0.10–0.20	4.32	Top of weathered/debris slope	0.70	1.00
0.30–0.50	2.93	Top of rock face	-	-

and illustrated through migration path diagrams of falling blocks and fragments (Figure 15).

The most significant migration distance in Sector 7 was observed for rock/weathered fragments or debris measuring 0.10–0.20 m, reaching a maximum distance of approximately 4.95 m from the slope (top of the weathered/debris slope) and a maximum rebound height of ~0.60 m (with an airborne distance of 0.96 m).

Fragments of the smallest size (0.01–0.05 m) travelled a ~1.0 m shorter distance, reaching 3.90 m (maximum rebound height ~0.42 m, maximum airborne distance 0.69 m).

Blocks measuring 0.30–0.50 m detached from the top of the rock face were stopped after reaching a maximum distance of 2.86 m.

The results for Sector 8 (Figure 2) are summarised in Table 5 and illustrated in the migration path diagrams of detached blocks and fragments (Figure 16 and Figure 17).

In Sector 8, the most significant migration distance was achieved by the smallest rock/weathered fragments (0.01–0.05 m), reaching approximately 4.66 m with a maximum rebound height of 0.53 m.

Larger fragments, sized 0.10–0.20 m and 0.30–0.50 m, reached distances of ~4.32 m (with a maximum rebound height of 1.00 m) and ~2.93 m, respectively.

5.3. Recommendations for Slope Stabilisation and Actions Regarding the Designated Sections for Safe Site Use

Based on the stability calculations presented in [41] and the analysis of rockfall reach results,

Table 6. Summary of slope stability calculation results.

Sector	Factor of Safety FS	Maximum Distance Reached by the Detached Rock Block/Weathered Fragment [m]/During the Flight Phase [m]
1	5.18	3.16–0.98/-
2	2.99	7.69–1.39/-
3	1.41 (within the weathered zone) 5.51 (entire slope)	8.36–3.63/0.72–1.18
4	<<1.0	9.89–4.63/1.22–2.41
5	1.17 (weathered material + rock blocks)	8.61–2.78/1.33–2.32
6	<<1.0	9.59–6.74/-
7	1.20 weathered material 1.60 (debris) 1	4.95–2.86/0.69–0.96
8	2.84 (debris) 2.18 (weathered material)	4.66–2.93/0.36–0.70
	FS > 1.5 condition met;
	FS > 1.5 condition not met;
	FS > 1.5 condition not met; slope with limited stability margin;
	FS > 1.5 condition not met; unstable slope.

¹ For debris, FS > 1.5 condition is met.

provides a summary of slope stability assessments for individual sectors. A minimum safety factor of FS > 1.5 was adopted as the required safety threshold. In the case of allowing mass events to be organised in the Park, the required factor of safety level should be increased (a minimum of FS > 2.0 is recommended), and the corresponding mitigation measures should be adjusted accordingly.

If no protective measures are implemented in Bednarski Park, it will be necessary to designate safety zones or buffer barriers to separate users from hazardous areas, as follows:

• Sector 1—3.16 m;
• Sector 2—7.69 m;
• Sector 3—8.36 m;
• Sector 4—9.89 m;
• Sector 5—8.61 m;
• Sector 6—9.59 m;
• Sector 7—4.95 m;
• Sector 8—4.66 m.

For sectors 4 and 6, the calculations indicate the presence of instability conditions [41]. Failure to implement appropriate stabilisation measures may lead to progressive failure processes and substantial damage in the upper and middle parts of the slope. The extent of the safety zone may be reduced by installing a barrier designed to minimise the rolling of rock blocks.

Calculations for sector 3, aimed at minimising the safety zone, suggest that it would be sufficient to construct a barrier with a minimum height of 0.7 m located 4 m from the quarry wall, or with a minimum height of 1.0 m positioned 3 m from the wall. For sector 4, minimising the safety zone would require a barrier of at least 1.0 m in height installed 4 m from the wall.

Introducing a barrier in the remaining sectors necessitates additional verification calculations. Moreover, a review and supplementation of the existing geotechnical and engineering–geological investigations are recommended.

To ensure full access to the quarry walls, it is essential to remove vegetation that is contributing to wall degradation, scale off weathered and loose rock blocks, and implement reinforcement measures as specified below:

• Sector 3—Installation of steel mesh and anchoring of the upper slope section (within the weathered zone). After rockfall removal and vegetation clearance, it will be possible to subdivide the area into two zones: Figure 18: Sector 3a (without protection, with delineated safety zone—black ellipse) and Sector 3b (with protective measures—red ellipse);

• Sector 4—installation of steel mesh and anchoring of the upper and middle sections of the slope;
• Sector 5—installation of steel mesh and anchoring of the upper slope section (within the weathered zone), and locally the middle section where a dense joint network is present;
• Sector 6—installation of steel mesh and anchoring of the upper and middle sections of the slope;
• Sector 7—installation of steel mesh and anchoring of the upper section of the slope;
• Sectors 1, 2, and 8—delineation of a safety buffer zone with a minimum width of 1.5 m (Sectors 1 and 2) and 3.0 m (Sector 8), e.g., by planting dense vegetation to discourage entry.

6. Conclusions

The adaptation of the former limestone quarry into an urban park, while preserving its shaped rock slopes and visible traces of past mining activity, necessitates ensuring user safety. This study presents an applied approach to assessing the stability conditions of rock blocks, aimed at defining the scope of necessary reinforcement measures while minimising interference with the natural fabric of the historical quarry site.

The analysed rock mass is highly fractured and subject to weathering processes, which contribute to the detachment of rock and weathered fragments.

Based on the literature review, field investigations, and stability analyses, the following conclusions can be drawn:

• Given the slope geometry and the practical objective of the study (risk assessment in the context of land-use planning), the use of 2D modelling was deemed sufficient. The 2D analysis, when conducted with conservative parameter values, enables a safe estimation of the displacement ranges of the rock mass.
• The choice of 2D analysis was a deliberate compromise between model realism and its engineering applicability for risk evaluation and safety zone design. In the future, the use of three-dimensional (3D) models may be justified for detailed design analyses or for planning new forms of land development.
• The numerical simulations of the displacement range of rock and weathered blocks enabled the formulation of the following conclusions:
  • Sectors 1, 7, and 8 exhibit the shortest maximum travel distances for detached rock and weathered fragments, ranging between 3.16 and 4.95 m;
  • Sectors 4 and 6 demonstrate the most extended migration distances, between 9.59–9.99 m, consistent with field survey data (maximum mapped reach ~9.38 m);
  • Sectors 2, 3, and 5 show intermediate travel distances, between 7.69 and 8.61 m;
  • The most hazardous rebound height was observed in Sector 5, reaching 1.97 m for 0.10–0.20 m blocks. Slightly lower values were found in Sectors 4 (1.36 m for 0.50–0.80 m blocks), 3 (1.10 m for 0.10–0.20 m blocks), and 6 (1.06 m for 0.30–0.50 m blocks);
  • The greatest aerial flight distances, measured from the toe of the rock wall (excluding rolling phase), were observed in Sectors 4 and 5: 2.41 m and 2.32 m, respectively (0.10–0.20 m blocks); in Sectors 3 and 7, these values approached 1.0 m;
  • The implemented berm in Sector 3 effectively arrests most of the detached blocks and shields the rebound zone (~1.0 m). At a 3.0 m offset, a 1.0 m high berm is more effective than a 0.7 m berm, which may still allow hazardous rebounds. A berm placed 4.0 m from the wall, regardless of height (0.7 m or 1.0 m), nearly eliminates high rebounds;
  • In Sector 4, a 1.0 m high berm successfully intercepts most fragments and blocks the zone of highest rebounds (~1.2–2.4 m). A 0.7 m high berm at 4.0 m offset may still permit unsafe rebound trajectories;

These results confirm that simulation-based methods can effectively support decision-making processes in geotechnical engineering.

4.

The outcomes of the numerical analyses (combining RBIM and DEM approaches) have direct applications in engineering practice, particularly for assessing hazards associated with rock face instability and for planning appropriate protective measures.

5.

The conducted simulations enable a quantitative assessment of the potential displacement range of detached rock fragments and weathered material, providing a basis for defining safety zones and selecting suitable protective measures. Recommendations concerning safety zones and possible protective actions are outlined below:

• Within the flight zone (understood as the maximum distance a detached block or weathered fragment can travel during the airborne phase—excluding post-impact rolling at the slope base), human access must be strictly prohibited. Any scaling operations should only be carried out using appropriate protective measures;
• Delineation of safety zones to prevent unauthorised access to hazardous areas;
• Installation of barriers (e.g., buttresses) to limit the rolling of rock fragments;
• Continuous monitoring and visual inspection of rock faces to assess their stability;
• In cases where full access to quarry walls is required, adequate reinforcement (e.g., wire mesh and anchoring) should be implemented, combined with systematic technical inspections of the slopes;
• The final scope of required stabilisation and reinforcement should be defined following the removal of vegetation that destabilises the rock mass and after scaling of weathered and loose rock material;
• For Sectors 4 and 6, the conducted calculations indicate the presence of instability conditions. A lack of stabilisation measures may lead to the development of destructive processes, resulting in significant damage to the upper and middle parts of the slope.

6.

The presented approach—integrating field data with numerical analysis—contributes to the advancement of practical applications of rockfall kinematics modelling in urbanised environments.

The conducted analyses and formulated recommendations were implemented as part of the revitalisation program for Wojciech Bednarski Park in Kraków. The proposed measures provided a technical basis for securing quarry slopes and defining exclusion zones where human presence is prohibited due to life-threatening hazard levels.