Evaluation of maximum rockfall dimensions based on probabilistic assessment of the penetration of the sliding planes into the slope

There is intrinsic difficulty in the investigation of the largest volume of rockfalls that is expected in an area, which lies in the small number of large events, in registrable times. The maximum credible rockfall size has been associated with the properties of the rock mass discontinuities, as they delimit detachable rock blocks, and in particular with the penetration of those discontinuities that comprise rockfall sliding planes. In highly fractured rock masses, the evaluation of the penetration remains an issue. A probabilistic methodology is proposed, to measure the penetration of potential sliding planes into the interior of a rocky slope. The main hypothesis of the method is that the sliding plane persistence is interrupted along its two directions, at the intersection with two lateral discontinuity sets, as the latter displaces the former. Due to the displacement, the sliding planes are formed by quasi-planes that contain a maximum number of spacings of the intersecting joints, hence their size is restricted. The methodology requires as an input the spacing of the intersecting joint sets. Its application to a granodiorite slope confirms that for the study site, there is a maximum volume of rockfalls, excluding the possibility of large stepped failures and rupture of rock bridges. The maximum calculated persistence for the two existing sliding planes in the study site is, respectively, 28.0 m and 48.5 m. The maximum calculated sliding plane surfaces are, accordingly, 282.5 m2 and 289.3 m2. These results are compared against the observed scar dimensions at the study site, which have been retrieved alternatively, by processing a LiDAR point cloud. The results from the two alternative sources are similar, indicating that the methodology can be efficiently used to assess the sliding plane persistence and the expected maximum rockfall magnitude at the study site.Peer ReviewedPostprint (published version

Comparing rockfall scar volumes and kinematically detachable rock masses

Scenario-based risk assessment for rockfalls, requires assumptions for different scenarios of magnitude (volume). The magnitude of such instabilities is related to the properties of the jointed rock mass, with the characteristics of the existing unfavourably dipping joint sets playing a major role. The critical factors for the determination of the maximum credible rockfall volume in a study site, the Forat Negre in Andorra, are investigated. The results from two previous analyses for the rockfall size distribution at this site are discussed. The first analysis provides the observed size distribution of the rockfall scars, and it is an empirical evidence of past rockfalls. The second one, calculates the kinematically detachable rock masses, indicating hypothetical rockfalls that might occur in the future. The later gives a maximum rockfall volume, which is one order of magnitude higher, because the persistence of the basal planes is overestimated. The tension cracks and lateral planes interrupt systematically the basal planes, exerting a control over their persistence, and restricting the formation of extensive planes and large rockfall failures. Nonetheless, the formation of basal planes across more than one spacings of tension cracks is possible and small step-path failures have been observed too. Concluding, the key factor for the determination of the maximum credible volume at the study-site is the maximum realistic length of the basal planes, penetrating into the rock mass, their spacing, and, if applied, the contribution of the rock bridges to the overall rock mass resistance.Peer ReviewedPostprint (author's final draft

A fractal fragmentation model for rockfalls

The final publication is available at Springer via http://dx.doi.org/10.1007/s10346-016-0773-8The impact-induced rock mass fragmentation in a rockfall is analyzed by comparing the in situ block size distribution (IBSD) of the rock mass detached from the cliff face and the resultant rockfall block size distribution (RBSD) of the rockfall fragments on the slope. The analysis of several inventoried rockfall events suggests that the volumes of the rockfall fragments can be characterized by a power law distribution. We propose the application of a three-parameter rockfall fractal fragmentation model (RFFM) for the transformation of the IBSD into the RBSD. A discrete fracture network model is used to simulate the discontinuity pattern of the detached rock mass and to generate the IBSD. Each block of the IBSD of the detached rock mass is an initiator. A survival rate is included to express the proportion of the unbroken blocks after the impact on the ground surface. The model was calibrated using the volume distribution of a rockfall event in Vilanova de Banat in the Cadí Sierra, Eastern Pyrenees, Spain. The RBSD was obtained directly in the field, by measuring the rock block fragments deposited on the slope. The IBSD and the RBSD were fitted by exponential and power law functions, respectively. The results show that the proposed fractal model can successfully generate the RBSD from the IBSD and indicate the model parameter values for the case study.Peer ReviewedPostprint (author's final draft

Magnitude and frequency relations: are there geological constraints to the rockfall size?

The final publication is available at Springer via http://dx.doi.org/10.1007%2Fs10346-017-0910-zThere exists a transition between rockfalls, large rock mass failures, and rock avalanches. The magnitude and frequency relations (M/F) of the slope failure are increasingly used to assess the hazard level. The management of the rockfall risk requires the knowledge of the frequency of the events but also defining the worst case scenario, which is the one associated to the maximum expected (credible) rockfall event. The analysis of the volume distribution of the historical rockfall events in the slopes of the Solà d’Andorra during the last 50 years shows that they can be fitted to a power law. We argue that the extrapolation of the F-M relations far beyond the historical data is not appropriate in this case. Neither geomorphological evidences of past events nor the size of the potentially unstable rock masses identified in the slope support the occurrence of the large rockfall/rock avalanche volumes predicted by the power law. We have observed that the stability of the slope at the Solà is controlled by the presence of two sets of unfavorably dipping joints (F3, F5) that act as basal sliding planes of the detachable rock masses. The area of the basal sliding planes outcropping at the rockfall scars was measured with a terrestrial laser scanner. The distribution of the areas of the basal planes may be also fitted to a power law that shows a truncation for values bigger than 50 m2 and a maximum exposed surface of 200 m2. The analysis of the geological structure of the rock mass at the Solà d’Andorra makes us conclude that the size of the failures is controlled by the fracture pattern and that the maximum size of the failure is constrained. Two sets of steeply dipping faults (F1 and F7) interrupt the other joint sets and prevent the formation of continuous failure surfaces (F3 and F5). We conclude that due to the structural control, large slope failures in Andorra are not randomly distributed thus confirming the findings in other mountain ranges.Peer ReviewedPostprint (author's final draft

Methodology to evaluate rock slope stability under seismic conditions at Solà de Santa Coloma, Andorra

An analytical methodology is presented to evaluate rock slope stability under seismic conditions by considering the geomechanical and topographic properties of a slope. The objective is to locate potential rockfall source areas and evaluate their susceptibility in terms of probability of failure. For this purpose, the slope face of a study area is discretized into cells having homogenous aspect, slope angle, rock properties and joint set orientations. A pseudostatic limit equilibrium analysis is performed for each cell, whereby the destabilizing effect of an earthquake is represented by a horizontal force. The value of this force is calculated by linear interpolation between the peak horizontal ground acceleration PGA at the base and the top of the slope. The ground acceleration at the top of the slope is increased by 50% to account for topographic amplification. The uncertainty associated with the joint dip is taken into account using the Monte Carlo method. The proposed methodology was applied to a study site with moderate seismicity in Sol`a de Santa Coloma, located in the Principality of Andorra. The results of the analysis are consistent with the spatial distribution of historical rockfalls that have occurred since 1997. Moreover, the results indicate that for the studied area, 1) the most important factor controlling the rockfall susceptibility of the slope is water pressure in joints and 2) earthquake shaking with PGA of 0.16 g will cause a significant increase in rockfall activity only if water levels in joints are greater than 50% of the joint height.Postprint (published version

Size distribution for potentially unstable rock masses and in situ rock blocks using LIDAR-generated digital elevation models

In this paper, two analytical procedures which are independent from the existence of empirical data are presented for the calculation of (1) the size distribution of potentially unstable rock masses that expresses the potential rockfall size distribution, including big volumes corresponding to potential rare events with low susceptibility of failure and (2) the in situ block distribution on the slope face. Two approaches are, respectively, used. The first one involves the detection of kinematically unstable surfaces on a digital elevation model (DEM) and on orthophotos and the calculation of the volumes resting on them. For the second one the in situ block volumes formed by the intersection of the existing discontinuity sets are calculated using a high-resolution DEM. The procedures are presented through an application example at the country of Andorra and in particular at the chute of Forat Negre. The results from the first procedure indicate that it is kinematically possible to have mobilized volumes of some thousands of cubic meters; however, these are considered rare events with low susceptibility of failure. The size distribution of potentially unstable rock masses for big volume events was well fitted by a power law with an exponent of -0.5. The in situ block distribution on the slope face from the second procedure, assuming three types of intersection between the joints of the existing discontinuity sets and two extreme cases of discontinuity persistence, was also found to follow a power law, but with an exponent of -1.3. The comparison with the observed in the field block volume distribution on the slope face indicates that in reality discontinuities have a very high persistence and that considering only their visible trace length overestimates volumes, which is conservative.Peer ReviewedPostprint (author’s final draft

A methodology to obtain the block size distribution of fragmental rockfall deposits

Peer ReviewedPostprint (author's final draft

Aproximación probabilística al número y tamaño de bloques en desprendimientos con fragmentación

La mayoría de las masas rocosas que se desprenden se fragmentan en los primeros impactos contra el terreno. El riesgo asociado a los desprendimientos debe calcularse, por lo tanto, teniendo en cuenta este proceso de fragmentación. Para incorporar la fragmentación en la simulación de trayectorias es necesario prever el número y el volumen de los fragmentos rocosos resultantes en cada desprendimiento. Se presenta aquí un método para estimar estas dos variables. Se parte de la hipótesis que la distribución del volumen de los bloques acumulados en el canchal puede ser utilizada para generar aleatoriamente conjuntos de bloques, cada uno de ellos simulando el conjunto de bloques resultantes de un desprendimiento. Esta aproximación se ha aplicado a un canchal del Solà d’Andorra, Andorra la Vella (Principado de Andorra). La validez del método se ha contrastado comparado los resultados obtenidos con lo observado en desprendimientos recientes, inventariados por un plan de vigilancia de la zona.Postprint (published version

Recommendations for the quantitative analysis of landslide risk

This paper presents recommended methodologies for the quantitative analysis of landslide hazard, vulnerability and risk at different spatial scales (site-specific, local, regional and national), as well as for the verification and validation of the results. The methodologies described focus on the evaluation of the probabilities of occurrence of different landslide types with certain characteristics. Methods used to determine the spatial distribution of landslide intensity, the characterisation of the elements at risk, the assessment of the potential degree of damage and the quantification of the vulnerability of the elements at risk, and those used to perform the quantitative risk analysis are also described. The paper is intended for use by scientists and practising engineers, geologists and other landslide experts.Peer ReviewedPostprint (published version

Rockfalls: analysis of the block fragmentation through field experiments

Fragmentation is a common feature of rockfall that exerts a strong control on the trajectories of the generated blocks, the impact energies, and the runout. In this paper, we present a set of four real-scale rockfall tests aimed at studying the fragmentation of the rocky blocks, from the global design of the field procedure to the data analysis and the main results. A total of 124 limestone, dacite, or granite blocks ranging between 0.2 and 5 m3 were dropped from different heights (8.5 to 23.6 m) onto four slopes with different shapes (single or double bench) and slope angles (42º to 71º). The characteristics of the blocks, in particular the size, surface texture and joint condition, were measured before the drops. The trajectories of the blocks and both the initial and the impact velocities were tracked and recorded by means of three high-speed video cameras. A total of 200 block-to-ground impacts have been studied. On average, 40% of the blocks broke upon impact on the slope or on the ground, making it necessary to measure the fragments. The initial and final sizes of the blocks/fragments were measured by hand with tape, though photogrammetric techniques (UAV and terrestrial) were also used for comparison purposes. The information gathered during the field tests provides a deep insight into the fragmentation processes. On the one hand, the high-resolution slow-motion videos help to describe when and how the block breakage takes place and the spatial distribution of the pieces. On the other hand, it is possible to compute the block trajectories, the velocities, and the energy losses using videogrammetry. The results include, for instance, a block average fragmentation of 54% and 14% for the limestone and granitoids, respectively; the systematic inventory of the size fragments, which may be used for fitting the power law distributions; and after each breakage, the total angle of aperture occupied by the fragments has been measured, with values in the range 25º–145º. To figure out the different behavior of the blocks in terms of breakage/no breakage, each block-to-ground impact has been characterized with a set of parameters describing the energy level, the robustness of the substrate, and the configuration of the block contact at the impact point, among others. All these terms are combined in a function F, which is used to adjust the field data. The adjustment has been carried out, first, for the whole 200 events and later for a subset of them. The procedure and the results are described in the paper. Although the discrimination capability of F is moderately satisfactory, it is very sensitive to the test site and setup. It must be highlighted that these field tests are a unique source of data to adjust the parameters of the numerical simulation models in use for rockfall studies and risk mitigation, especially when fragmentation during the propagation is considered.The authors acknowledge the support of the Spanish Ministry of Economy and Competitiveness for the research projects RockRisk (BIA2013-42582-P), RockModels (BIA2016-75668-P, AEI, ERDF/FEDER, UE) and GeoRisk (PID2019-103974RB-I00/AEI/10.13039/501100011033). GeoRisk is funded by the Agencia Estatal de Investigación (AEI) on the framework of the Plan for Scientific-Technical Research and Innovation. The support of the Spanish Ministry of Education (grants to the second and third authors, codes FPU13/04252 and BES-2014-069795, respectively) and the BBVA Foundation (thirteenth author’s contract) is also appreciated. The collaboration of Canteras Hermanos Foj and Canteras Ponderosa S.A., Marc Janeras, and S. Moreno is greatly acknowledged. Finally, we thank two anonymous reviewers and the Editor who helped to improve the structure and content of the final version. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Peer ReviewedPostprint (published version