E3S Web of Conferences 415, 07013 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341507013

Modelling of debris-flow deposition: terrain slope, mobility
coefficient, and back-calculated basal friction coefficient
Dieter Rickenmann1*, and Christian Scheidl2
1Swiss Federal Research Institute WSL, 8903 Birmensdorf, Switzerland

2University of Natural Resources and Life Sciences, 1190 Vienna, Austria

Abstract. 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.

1 Introduction
Various studies proposed simulation models for the
propagation and deposition of fast gravitational mass
movements such as snow avalanches, debris flows, rockavalanches and landslides, and similar modelling
approaches were proposed to some extent. In a very
simplified approach, the simulation considers themotion of a mass point, which leads to mathematicallysimple 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 andthe 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 foravalanches and then applied to debris flows. It assumesa 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, fora given flow height and velocity at the fan apex and theknown fan slope Sfan [1].
The basal friction coefficients (SVF, SMF) govern theflow behaviour during the depositional phase. A
disadvantage of both the VF and the MF model is, thatthese coefficients cannot be determined directly fromsample of the water-debris material. Thus, for hazard
assessment studies of debris flows and related massflows, 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 stopwhere 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 Japanwas found to be slightly larger than the slope of thedepositional terrain [1].
The dimensionless mobility coefficient kB was first
proposed by Iverson et al. [8] to characterize thedepositional 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 suchas:
A = kB V2/3

(1)

where kB is a mobility coefficient.
The objective of this preliminary study is toexamine, whether the back-calculated basal friction
coefficient SVF with a numerical simulation model basedon the Voellmy fluid rheology shows also somecorrelation with the slope of the depositional terrain.
Furthermore, we also checked whether the backcalculated basal friction coefficient SMF and SVF show acorrelation with the mobility coefficient kB.

* Corresponding author: dieter.rickenmann@wsl.ch© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 07013 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341507013

2 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, andback-calculated basal friction coefficients (SMF, SVF).
The datasets concern debris-flow or rock-avalanche
events in Austria, Italy, and Switzerland (Table 1, Table2).

Deposition area A (m2)

1E+06

Table 1. Data set used here. Model labelling: VF = Voellmy
Fluid, MF = Mass Flux. Dataset B is a subset of dataset A.
KeyDataset

CountryN

ModelA

Debris flow AT-IT

Austria, Italy

38MF

B

Debris flow IT-1

8VFC

Debris flow IT-2

7VF

DDebris flow CHItaly (SouthTyrol)
Italy (BolzanoProvince)
Switzerland

7VFE

Rock avalanche CH

Switzerland

4VF

ReferencesA

Hochschwarzer (2009)

[11]

B

Hochschwarzer (2009)[11]

CSimoni et al. (2012)

[12]

D

Raymond-Pralong et. (2018), Scheuner (2007)

E

Hungr & McDougall (2009), Steinemann(2012), Tognacca et al. (2019)

1E+05

1E+04

1E+03

1E+02
1E+02

1E+04

1E+06

Volume V (m3)

Fig. 1. Relation between planimetric deposition area A andflow volume V for our 5 datasets. For the scaling relation in
the graph, a mobility coefficient of kB = 20 was arbitrarilyselected.

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 arather strong linear relation between SMF and Sfan in thecase of applying the mass flux model to dataset A (Fig.
2), with an R2 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 asomewhat larger scatter of the values for the debris-flow
data around the linear trend line (Fig. 3); the reasonableR2 value of 0.73 appears to be influenced by an extreme
data point of the rock avalanches with a large value forboth SVF and Sdep.

Table 2. References for the different datasets.
Key

Debris flow IT-1, VF-model
Debris flow IT-2, VF-model
Debris flow CH, VF-modelRock avalanche CH, VF-model
Debris flow AT-IT, MF-modelA = 20 V^(2/3)

No. in list

[13, 14][15, 16, 17]

Debris flow AT-IT, MF-model0.5

The mass flux model of Hungr et al. [4] andTakahashi [5] was applied to dataset A to back-calculate
the basal friction coefficient SMF. To do so, empiricalequations of Rickenmann [18] were used, first one toestimate the maximum discharge as a function of flow
volume, and second another one to estimate flow
velocity and flow height for the given channel geometryat the fan apex. In the case of applying a numericalsimulation model based on the Voellmy fluid rheology(datasets B, C, D, E), an input flow hydrograph at thefan apex was estimated, similarly as in the case of themass flux model.

SMF0.4y = 1.28x
R² = 0.88

0.3
0.2
0.1

3 Results
First, we checked whether the geometric scaling ofmass-flow deposits according to eq. (1) holds
approximately for our datasets. Figure 1 illustrates
generally align around the scaling relation, although
there is some variability of the kB values for the debrisflow datasets. The mean kB values are 22.9 for dataset
A, 32.4 for datasets B, C, and D, and 18.0 for dataset D.

0.1

0.20.3Sfan

0.4

0.5

Fig. 2. Relation between the back-calculated basal friction
coefficient SMF and the fan slope Sfan. Here, the mass flux
model was applied to the debris flow events of dataset A.

2E3S Web of Conferences 415, 07013 (2023)
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https://doi.org/10.1051/e3sconf/202341507013

events of datasets B, C, D, E. As is illustrated in Figure
5, there is also some correlation between the twovariables for this case, however with a larger scatteraround a linear trend line with an R2 of 0.48.

Debris flow IT-1, VF-model
Debris flow IT-2, VF-model
Debris flow CH, VF-modelRock avalanche CH, VF-modelLinear (all)

0.8

SVF

2.00.4

1.5

0.00

0.2

0.4

Sdep

0.6

SVF / Sdep

y = 0.86x
R² = 0.73

0.2

0.0

y = 1.89x-0.13
R² = 0.62

1.2

02040

Mobility coefficient kB

204060

Mobility coefficient kB

80

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 canbe defined. Using the mass flux model, the relation is
SMF = 1.28 Sfan (Fig. 2), and applying the Voellmyrheology with a numerical simulation model, therelation is SVF = 0.86 Sdep (Fig. 3). In this context, themass 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 thesedata gives: SMP = 1.02 Sdep. These results may beexpected based on theoretical considerations. While theapplication 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 theterrain 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 beexplained with variations in the mobility coefficient kB(Fig. 4, 5). Our very small dataset of rock avalanchesappears to be in line with the trend defined by the debrisflow events (Fig. 5). If the total runout distance of rockavalanches and debris flows are compared, it is foundthat for a given event volume and total descent height,
the debris flows are more mobile than the rockavalanches [18]. However, if only the depositional
phase of the mass flow propagation is considered, themobility of debris flows and rock avalanches appears tovary in similar ranges, as indicated by studies ofGriswold & Iverson [19] and Scheidl & Rickenmann

1.6

1.004 Discussion

Debris flow AT-IT, MF-model

1.4

y = -0.017x + 1.40
R² = 0.48

Fig. 5. Relation between the ratio SVF/Sdep and the mobility
coefficient kB. Here, the Voellmy fluid rheology was used ina numerical simulation model applied to the mass flow eventsof datasets B, C, D, E.

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 someof the scatter around this mean trend may be explainedby 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 thecase of applying the mass flux model was applied to thedebris flow events of dataset A. As is evident from
Figure 4, there is a reasonable correlation between thetwo variables, defined by a power law regression withan R2 of 0.62.
1.8

1.0
0.5

0.8

Fig. 3. Relation between the back-calculated basal friction
coefficient SVF and the terrain slope in the deposition zone
Sdep. Here, the Voellmy fluid rheology was used in anumerical simulation model applied to the mass flow eventsof datasets B, C, D, E.

SMF / Sfan

Debris flow IT-1, VF-model
Debris flow IT-2, VF-model
Debris flow CH, VF-modelRock avalanche CH, VF-modelLinear (all)

2.5

0.6

60

Fig. 4. Relation between the ratio SMF/Sfan and the mobility
coefficient kB. Here, the mass flux model was applied to thedebris flow events of dataset A.

Second, we examined a possible relation between
the ratio SVF/Sdep and the mobility coefficient kB, for thecase of using the Voellmy fluid rheology with anumerical simulation model applied to the mass flow

3E3S Web of Conferences 415, 07013 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341507013[10]. This mobility coefficient kB can be considered tobe approximately similar to a result of a large-scale(prototype) rheometer test: The value kB is acharacteristic value for the mean flow behaviour during
the depositional phase of the debris flow (or rockavalanche) event from which it was determined.
Therefore, it seems reasonable to use this value for theassessment of the depositional behaviour. Moreover,Scheidl & Rickenmann [10] postulated that steeper
debris flow fans may be associated with smaller kBvalues than flatter and larger fans.
As a caveat it may be mentioned that thedetermination of the mobility coefficient kB may bemost reliable if the observed deposition (area) was due
to only one single debris flow surge. If multiple surges
debris-flow
surgeswithdifferent
material
characteristics (water content, grain size distribution)
contributed to the final deposition pattern, including apartial overlap of deposits from different surges, theestimation of the mobility coefficient kB may be moredifficult without detailed knowledge of the flow
characteristics.

4.

O. Hungr, G.C. Morgan, R. Kellerhals, Canadian
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Graf, R. Kaitna, Natural Hazards and Earth SystemScience, 15, 7, 1483–1492 (2015).
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Voellmy-Perla, Postgraduate thesis, Ecole
Polytechnique Fédérale de Lausanne, Suisse(2002).
8. R.M. Iverson, S.P. Schilling, J.W. Vallance,
Geological Society of America Bulletin, 110, 8,
972-984 (1998).
9. M. Berti, A. Simoni, A., Geomorphology, 90, 144161 (2007).
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Murgänge am Beispiel Südtiroler Ereignisse,Master thesis, University of Applied Life Sciences
and Natural Ressources, Vienna (2009).
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Back-analysis of field events in Italian Alps, inProceedings of the Annual International
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Schneider, Wasser Energie Luft, 110, 1, 39-48(2018).
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durch GIS-basiertes Fliessmodell, Diplomarbeit,
University of Zürich and Swiss Federal ResearchInstitute WSL, Birmensdorf, Switzerland (2007).
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5276 (2008).

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

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