Modelling of debris-flow deposition: terrain slope, mobility coefficient, and back-calculated basal friction coefficient

. Numerous simulation models have become available for the hazard delineation of gravitational mass movements such as debris flows in the potential depositional area. This study compares the basal friction coefficients of two modelling approaches as they have been used in other studies to simulate mass movement deposition. We show that both back-calculated basal friction coefficients are related to the fan or terrain slope in the deposition region and to a mass flow mobility coefficient. The latter is characterized by the geometric scaling between the deposition area and the deposition volume as well as by process type. Intended as a preliminary study, we aim to initiate systematic investigations of different model applications to determine the runout pattern of mass movement processes, that could provide guidance for appropriate parameter selection.


Introduction
Various studies proposed simulation models for the propagation and deposition of fast gravitational mass movements such as snow avalanches, debris flows, rock avalanches and landslides, and similar modelling approaches were proposed to some extent. In a very simplified approach, the simulation considers the motion of a mass point, which leads to mathematically simple solution methods [1].
Regarding numerical simulation models for debris flows, the mixture of water and sediment is often considered as a quasi-homogeneous fluid and its flow behaviour is described with a rheological model. A frequently used approach is the Voellmy Fluid (VF) model, which was applied already earlier for snow avalanches and includes a turbulent flow parameter and the basal friction coefficient SVF [e.g. 2,3].
A simple model to describe only the runout behaviour on the fan was proposed by Hungr et al. [4] and Takahashi [5]. This model was first developed for avalanches and then applied to debris flows. It assumes a constant inflow or mass flux (MF) of debris material at the fan apex, accounts for constant friction losses during the depositional phase described by the basal friction coefficient SMF. It can be solved analytically, for a given flow height and velocity at the fan apex and the known fan slope Sfan [1].
The basal friction coefficients (SVF, SMF) govern the flow behaviour during the depositional phase. A disadvantage of both the VF and the MF model is, that these coefficients cannot be determined directly from sample of the water-debris material. Thus, for hazard assessment studies of debris flows and related mass * Corresponding author: dieter.rickenmann@wsl.ch flows, these basal friction coefficients have to be backcalculated based on the known depositional pattern on the fan of past (reference) events [1,3,6].
For a very simple simulation approach, the Voellmy mass point model, the debris flow is expected to stop where the channel or fan gradient approaches the basal friction coefficient [1]; this is confirmed by some backcalculated debris-flow events in Switzerland [7]. A similar observation also applies to the mass flux (MF) model; the back-calculated basal friction coefficient SMF for some debris flow events from Switzerland and Japan was found to be slightly larger than the slope of the depositional terrain [1].
The dimensionless mobility coefficient kB was first proposed by Iverson et al. [8] to characterize the depositional behaviour of lahars and other mass flows, and its application to debris flows was further studied by Berti & Simoni [9] and Scheidl & Rickenmann [10]. From geometric scaling of mass-flow deposits, it can be postulated that the following power law exists between planimetric deposition area A and flow volume V such as: where kB is a mobility coefficient.
The objective of this preliminary study is to examine, whether the back-calculated basal friction coefficient SVF with a numerical simulation model based on the Voellmy fluid rheology shows also some correlation with the slope of the depositional terrain. Furthermore, we also checked whether the backcalculated basal friction coefficient SMF and SVF show a correlation with the mobility coefficient kB.

Datasets and Analysis
We used five datasets that include observations on mass flow characteristics, such as flow (event) volume, planimetric deposition area, fan slope (Sfan) and terrain slope in the depositional zone (Sdep), respectively, and back-calculated basal friction coefficients (SMF, SVF). The datasets concern debris-flow or rock-avalanche events in Austria, Italy, and Switzerland (Table 1, Table  2).  [15,16,17] The mass flux model of Hungr et al. [4] and Takahashi [5] was applied to dataset A to back-calculate the basal friction coefficient SMF. To do so, empirical equations of Rickenmann [18] were used, first one to estimate the maximum discharge as a function of flow volume, and second another one to estimate flow velocity and flow height for the given channel geometry at the fan apex. In the case of applying a numerical simulation model based on the Voellmy fluid rheology (datasets B, C, D, E), an input flow hydrograph at the fan apex was estimated, similarly as in the case of the mass flux model.

Results
First, we checked whether the geometric scaling of mass-flow deposits according to eq. (1) holds approximately for our datasets.  Next, we verified whether the back-calculated basal friction coefficients show a correlation with the terrain slope at the deposition zone. As can be seen, there is a rather strong linear relation between SMF and Sfan in the case of applying the mass flux model to dataset A (Fig.  2), with an R 2 value of 0.88. In the case of applying a numerical simulation model based on the Voellmy fluid rheology to the datasets B, C, D, E, also a linear relation exists between between SVF and Sdep, but with a somewhat larger scatter of the values for the debris-flow data around the linear trend line (Fig. 3); the reasonable R 2 value of 0.73 appears to be influenced by an extreme data point of the rock avalanches with a large value for both SVF and Sdep.  Given that there is an approximately linear relation between the basal friction coefficient and the terrain slope of the depositional zone, we may expect that some of the scatter around this mean trend may be explained by a mobility index of the mass flow material. Therefore, we first examined a possible relation between the ratio SMF/Sfan and the mobility coefficient kB, for the case of applying the mass flux model was applied to the debris flow events of dataset A. As is evident from Figure 4, there is a reasonable correlation between the two variables, defined by a power law regression with an R 2 of 0.62. Fig. 4. Relation between the ratio SMF/Sfan and the mobility coefficient kB. Here, the mass flux model was applied to the debris flow events of dataset A. Second, we examined a possible relation between the ratio SVF/Sdep and the mobility coefficient kB, for the case of using the Voellmy fluid rheology with a numerical simulation model applied to the mass flow events of datasets B, C, D, E. As is illustrated in Figure  5, there is also some correlation between the two variables for this case, however with a larger scatter around a linear trend line with an R 2 of 0.48.

Discussion
It is interesting to note that the back-calculated basal friction coefficient shows a correlation with the terrain slope in the deposition zone, and a linear regression can be defined. Using the mass flux model, the relation is SMF = 1.28 Sfan (Fig. 2), and applying the Voellmy rheology with a numerical simulation model, the relation is SVF = 0.86 Sdep (Fig. 3). In this context, the mass point model was applied to 21 debris flow events to back-calculate the related basal friction coefficient SMP (Genolet 2002), and a linear regression for these data gives: SMP = 1.02 Sdep. These results may be expected based on theoretical considerations. While the application of the one-dimensional mass flux model required basal friction coefficients that were larger than the fan slope, a two-dimensional numerical simulation model with the Voellmy rheology resulted in basal friction coefficients that were generally smaller than the terrain slope in the deposition zone.
Our analysis also showed that some of the remaining variability of the back-calculated basal friction coefficient SMF and SVF (after accounting for its dependence on the terrain slope) can possibly be explained with variations in the mobility coefficient kB (Fig. 4, 5). Our very small dataset of rock avalanches appears to be in line with the trend defined by the debrisflow events (Fig. 5). If the total runout distance of rock avalanches and debris flows are compared, it is found that for a given event volume and total descent height, the debris flows are more mobile than the rock avalanches [18]. However, if only the depositional phase of the mass flow propagation is considered, the mobility of debris flows and rock avalanches appears to vary in similar ranges, as indicated by studies of Griswold & Iverson [19] and Scheidl & Rickenmann https://doi.org/10.1051/e3sconf/202341507013 , 07013 (2023) E3S Web of Conferences 415 DFHM8 [10]. This mobility coefficient kB can be considered to be approximately similar to a result of a large-scale (prototype) rheometer test: The value kB is a characteristic value for the mean flow behaviour during the depositional phase of the debris flow (or rock avalanche) event from which it was determined. Therefore, it seems reasonable to use this value for the assessment of the depositional behaviour. Moreover, Scheidl & Rickenmann [10] postulated that steeper debris flow fans may be associated with smaller kB values than flatter and larger fans.
As a caveat it may be mentioned that the determination of the mobility coefficient kB may be most reliable if the observed deposition (area) was due to only one single debris flow surge. If multiple surges debris-flow surges with different material characteristics (water content, grain size distribution) contributed to the final deposition pattern, including a partial overlap of deposits from different surges, the estimation of the mobility coefficient kB may be more difficult without detailed knowledge of the flow characteristics.

Conclusions
Often, an important element in the hazard assessment of debris flows is to predict potential runout areas on the fan for a given event volume. Typically, twodimensional numerical simulation models are used, and a Voellmy fluid rheology may be applied [3,6,12]. For this modelling approach, and also for other fluid rheologies applied to debris flows, the parameters characterizing the rheological or frictional behaviour must be estimated based on experience from past events, i.e. on back-calculating appropriate model parameters to reproduce for example the depositional pattern (perimeter) of a reference event. Despite this fact, there are still only very few studies that attempted to provide a systematic comparison of the results when applying different modelling approaches, and to offer some guidelines for selecting the model parameters. We consider our preliminary study as a small step or stimulus for further similar efforts.