Semi-empirical prediction of dam height and stability of dams formed by rock slope failures in Norway

Based on an inventory of 69 dams formed by rock slope failures in southwestern Norway and published inventories from other parts of the world, we developed semiempirical relationships linking the maximum dam height (HD.max in metres) to dam volume (VD in 106 m3) and other relevant parameters such as valley width (WV in metres) or dam area (AD in square kilometres). Power laws are obtained for HD.max = f (VD) and HD.max = f (VD, WV), while a linear relationship links HD.max to the ratio VD/AD. For dams in southwestern Norway, the linear relationship HD.max = 1.75×VD/AD has the least uncertainties and provides the best results when comparing predicted dam heights with a validation dataset composed of existing dams in northern Norway and numerically modelled dams for possible rock slope failures. To assess the stability of future dams, we use the predicted dam heights in the dimensionless blockage index (DBI) and relating this index to the probability of dam failure derived from our dataset and other published databases on landslide dams. This study underlines the potential of semi-empirical relationships for assessing dam height and stability that needs to be included in preliminary hazard and risk assessment for unstable rock slopes, because damming of a river is an important secondary effect of landslides due to upstream flooding and possible outburst floods in the case of dam failure.

The National landslide database of Norway (NVE, 2020) includes at least 181 historical landslides that caused damming of 28 rivers. Most of them were earth and debris slides (153) and only 22 events were rockslides or rock avalanches. Many of 29 those events created only minor damming of rivers without significant consequences. Yet, there were several major events 30 with significant consequences in terms of loss of life or long-lasting landscape changes: the worst natural disaster in 31 Norway's history occurred on 21 September 1345 when the Gaula River was dammed by a massive debris slide that created 32 a 14 km long lake. After only 2-3 days the dam breached leading to a huge outburst flood in the Gaula Valley burying 48 33 farms and killing at least 500 persons (Furseth, 2006). In 1823, a rock avalanche dammed the Frondøla River and formed 34 the Lintuvatnet Lake (NVE, 2020). The lake is still existing today, even though the dam partially breached leading to an However, these models require numerous input parameters and extensive calibration to obtain reliable results, which 50 precludes their cost-efficient use for characterization of many sites, as is required in regional studies. 51 Here we establish semi-empirical relationships for the rapid assessment of the maximum dam height, comparable to those 52 developed for landslide run-out (e.g. Scheidegger, 1973;Corominas, 1996)

Inventory and characteristics of landslide dams 61
Systematic mapping of RSF dams in southwestern Norway (approximately 120 000 km² in surface) was carried out by 62 Jakobsen (2015) using the online orthophoto map service "Norge i bilder" (Norwegian Mapping Authority, 2020b) and its 63 associated web map service (WMS) in a geographical information system (GIS) (Figure 1b). This aerial photo analysis 64 focused on present-day lakes as an indicator for possible dams, with the aim of identifying lakes that were impounded by 65 RSF. The analysis investigated therefore the immediate downstream surroundings of lakes, looking for deposits, debris and 66 scars of RSF, but also debris from a possible downstream flooding due to a dam breach. It must be noted that dams without 67 remaining lake are therefore not included in present inventory. 68 The detected dams were mapped and registered in a geospatial database, and their geomorphologic characteristics determined 69 based on orthophotos and the national 10-m digital elevation model (DEM) (Norwegian Mapping Authority, 2020a). These 70 dam characteristics include: 71 the type of landslide that formed the dam, chiefly rock avalanches (massive RSF with several hundred thousand to 72 millions of cubicmeters in volume and high mobility) and rockslides/rockfalls (RSF with several thousands to hundred 73 thousands of cubicmeters in volume, but without high mobility) or other landslide types; 74 -the morphologic dam classifications in plan view and in across-valley and along-valley profiles according to Hermanns 75 because the aim of these empirical relationships is to determine the maximum dam height of future RSF. 136 The frequency of rock avalanches in Norway was highest shortly after the last deglaciation, i.e. between 14 000 and 10 000 137 years BP depending on the location (e.g. Böhme et al., 2015; Hermanns et al., 2017). We therefore assume that also most of 138 the RSF dams in southwestern Norway were formed shortly after the retreat of the Scandinavian ice sheet. However, three 139 dams are most likely influenced by glaciers, notably by depositing on decaying glaciers or on dead-ice bodies in the valley. 140 For 10 other dams such a glacial influence is possible. We excluded these 13 dams from further analyses because their 141 dimensions may have been altered by glaciers and are thus not representative for the present-day situation. -5 -deposits in a symmetrical valley in 24 cases (Type i), and as asymmetrical with thickest deposits in the distal part in 19 cases 154 (Type ii) (Figure 3b). The classification of the along-valley profiles reveals 21 dams with low thickness and gentle slopes 155 (Type 1) due to the absence of constraints in the valley morphology (Hermanns et al., 2011b), and 29 dams with high 156 thickness and steep slope (Type 2) in a confined valley setting (Figure 3c). 157 Table 1 (Table 1). 166

Semi-empirical relationships 167
We created semi-empirical relationships for the 54 RSF dams in southwestern Norway that were not influenced by glaciers. Regarding the influence of the morphologic dam classification on the dam height (Table 2), dams classified as asymmetrical 187 with thickest deposits in the distal part (Type ii in across-valley profile) are higher than dams with symmetrical deposits in 188 a symmetrical valley (Type i), but smaller than those partially blocking a valley (Type iv). In along-valley profiles, Type 2 189 dams with high thickness and steep slope are higher than Type 1 dams with low thickness and gentle slopes. Too few data 190 are available for the other dam types in along-or across-valley profiles and in plan view . 191 In narrow valleys the RSF deposits are more confined leading to thicker deposits and thus to a higher dam compared to wide 192 valleys where the deposits are unconfined and spread out over a larger surface. We calculated therefore ratio V D (in 10⁶ m³) 193 -6 -over valley width W V (in m) and fitted following power-law with the exponent given by dimensional analysis (Figure 6

Prediction of maximum dam height 234
We use the semi-empirical relationships (Eq. (2), (3) and (4)) to predict the maximum dam height generated by a future rock 235 slope failure damming a valley. We thereby use following assumptions and methods: 236 -The dam volume V D is equal to the slide volume V S times a bulking factor of 1.25 (25% volume increase due to fracturing 237 of the rock mass and porosity of the deposits) (Hungr and Evans, 2004). This implies that the entire volume reaches the 238 valley and forms the dam. This is obviously the worst-case scenario as shown by Ermini and Casagli (2003) with an 239 average ratio V D /V S of 40% for rainfall-triggered landslides and 57% for earthquake-triggered landslides. In Norway, 240 however, numerical run-out modelling for the six unstable rock slopes used for the validation of the semi-empirical 241 relationships (see Table 3 4) is assessed iteratively based on the run-out area, which can be assessed using simple 245 modelling tools, such as the Fahrböschung or angle of reach (Scheidegger, 1973;Corominas, 1996)  (1) to assess the probability of failure p f . 258

Validation of semi-empirical relationships 259
To test the semi-empirical relationships for RSF dams in southwestern Norway, we analyzed four RSF dams in northern 260 Norway as validation dataset. Those dams are presently stable or infilled (Fig. 2b). In addition, the relationships were 261 validated by comparing predicted dam heights with results from detailed numerical run-out modelling for six unstable rock 262 slopes (see Böhme et al., 2016 for an example; NGU, 2020). 263 Table 3   unstable dams for given DBI-classes (Figure 8a, b). A possible reason is the relatively lower dam heights in the Andes 339 compared to other datasets (see discussion above), which leads to lower DBI-values. Other causes for this difference could 340 be the grain size of deposits, climatic conditions and the age of the Andean dams, which are up to 60 ka old (Hermanns et 341 al., 2004, 2011a, Costa and González Díaz, 2007 It would be interesting to perform this stability assessment for different geological, geomorphological and climatic 343 environments, in order to obtain lower and upper DBI-limits for different conditions. This requires however more complete 344 inventories, as at least 100 or 150 landslide dams are required to obtain a sufficient number of bins (10 to 15 bins) containing 345 each a sufficient number of dams (≥10). 346

Prediction of dam height using semi-empirical relations or numerical modelling 347
Two of the proposed semi-empirical relationships rely only on the dam volume V D (Eq. (2)), or on the ratio V D over valley 348 width W V (Eq. (3)). These equations are thus a quick tool to assess the dam height, yet comparison with numerical modelling 349 shows that these relationships overestimate the maximum dam height (Table 3). The third proposed semi-empirical 350 relationship using the ratio V D over dam area A D (Eq. (4)) provides a better match with numerical modelling results, requires 351 however a simple run-out analysis to assess the run-out area and estimate A D (Figure 9) Figure 9). However, these simulations require many input parameters and extensive calibration in order to obtain reliable 363 results. These requirements impede their cost-efficient use in regional studies, where a large number of potential landslide 364 dams need to be assessed. 365 The proposed semi-empirical relationships are a conservative method because they assess the maximum dam height and thus 366 not the lowest elevation where dam overtopping may occur. Numerical simulations on the other hand provide the dam height 367 and elevation where overtopping would occur. This difference partly explains the discrepancy between numerically modelled 368 and empirically predicted dam heights (Table 3). Another possible reason for this discrepancy is the difference between 369 observed and modelled run-out areas. The effective run-out area of a landslide can be significantly smaller than numerically 370 simulated ones: the latter generally cover the entire area potentially affected by a landslide, while the real run-out area of a 371 landslide event may only cover parts of the total area. As the landslide volume in reality may spread over a smaller area than 372 simulated, the average and maximum dam heights obtained by numerical simulations or by Eq. (4) may be too small. Yet, 373 the possible overestimation of A D is counterbalanced by conservative estimate of V D being the entire landslide volume V S 374 times a bulking factor of 1.25. More back-analyses of landslide-generated dams are required to ascertain these possible 375 differences between modelled and real run-out areas. In turn, this could lead to an improved workflow for assessing the dam 376 height and reducing uncertainties. 377 These considerations highlight the necessity to assess uncertainties on dam height and stability by using various approaches, 378 including different semi-empirical relationships, but also numerical simulations for critical areas. To assess uncertainties, 379 we calculate for example the DBI and p f using H D.max for the potential RSF dams of the validation dataset (see Table 3). 380 Compared to the results from numerical simulations, the DBI increases in average by 0.64 and 0.56 for Eq. (2) and (3), 381 respectively. This leads in turn to an average increase of p f of +16% and +14%, respectively. This comparison highlights 382 that despite large uncertainties, the influence on dam stability and thus on the consequences assessment is relatively 383 moderate. 384

Conclusions & perspectives 385
The semi-empirical relations presented here provide a rapid approach for predicting the maximum dam height of dams that 386 might result from the future failure of an unstable rock slope. All relations require only limited input parameters, chiefly the 387 slide volume, the valley width and the dam area based on simple run-out assessments. These semi-empirical relationships 388 are established from an inventory of 54 RSF dams in southwestern Norway with dam volumes ranging from 12 000 m³ to 389 135 × 10⁶ m³. Only dams generated by catastrophic rockslides or rock avalanches and without any glacial influence were 390 included in the analyses. Consequently, the semi-empirical relations presented here may be less or not applicable for other 391 landslide types (e.g. debris-flows, shallow landslides) and other volume classes. The upper bounds of the 90% prediction 392 intervals of these semi-empirical relationships range from 1.48 to 1.81, meaning that approximately 5% of the actual 393 maximum dam heights exceed the predicted value by 48% to 81% or more. 394 Validation of the semi-empirical relationships was performed using four RSF dams in northern Norway, but also results 395 from detailed numerical run-out simulations for six unstable rock slopes. The maximum dam heights predicted by the semi-396 -11 -empirical relations are generally in good agreement with the measured/modelled dam heights from the validation dataset. 397 Best validation results are obtained for the relationship linking maximum dam height to landslide volume and dam area with 398 only a modest overestimation of the maximum dam heights (average relative error of 18%). This semi-empirical relationship 399 provides thus an appropriate tool for the first-order assessment of dams generated by rock slope failures at a local to regional 400 scale. Using limited input parameters, this relationship allows the prediction of the maximum dam height and thus the 401 upstream inundation area, but also to quickly forecast the dam stability using the dimensionless blockage index. 402 Possible improvements of these semi-empirical relationships are the inclusion of additional datasets, notably existing 403 landslide dams from other regions in Norway. Similar datasets could be collected for other mountainous regions in the 404 World, possibly leading to semi-empirical relationships with different parameters than those presented here for dams from 405 rock slope failures in southwestern Norway. Another possible major improvement consists in the addition of those dams that 406 do not possess a lake or residual lake at present. This requires however very time-intensive screening over large regions to 407 detect the landslide deposits that might have blocked a river in the past. Furthermore, the presented semi-empirical 408 relationships are only valid for rockslides and rock avalanches. Similar semi-empirical relationships can be imagined for 409 other landslide types, but more complete datasets on those landslide dams are required first. We strongly suggest using the 410 predictive tools developed here to assess landslide dam formation and stability, which should be an integral part of risk 411 assessment for future landslide events. 412

Competing interests 419
The authors declared that they do not have any competing interests. 420