E3S Web of Conferences 415, 02012 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502012

Evaluation of a method to calculate debris-flow volume basedon observations of flow depth
Brian W. McArdell1*, Jacob Hirschberg1, Jordan Aaron1, and Alexandre Badoux1
1

Swiss Federal Institute for Forest, Snow and Landscape Research WSL, CH-8903 Birmensdorf, Switzerland

Abstract. The volume of a debris flow is a critical parameter in hazard analysis, yet accurate estimates ofvolume are often unavailable due to mixing with larger rivers at the downstream end of alluvial fans. We
describe a method to calculate the volume of debris flows using flow depth data collected at a check dam,using a Manning friction relation to describe the velocity of the debris flow as a function of flow depth, andthe geometry of the channel cross section. The method is evaluated using a published data set from the
USGS debris-flow flume where event volume and stage information have been accurately measured, andresults in volume estimates either somewhat smaller than or up to 50% larger than observed volumes. We
further demonstrate the method to single-surge and a multiple-surge debris flows observed at Illgraben.

1 Introduction
Accurate estimates of debris-flow volume are necessary
for many applications, including hazard analysis,
interpreting recent events, evaluating the performanceof mitigation structures, and a variety of applications ingeomorphology (e.g. [1-6]). In many Alpine torrent
channels, debris-flow deposits become mixed with flood
flows when they enter larger river channels, thereby
transporting sediment away from the depositional area.
Further, uncertainties in the elevation of the groundsurface before the event may introduce large errors into
volume estimates. Consequently, it is difficult toaccurately measure deposit volume. Many authors
provide volume estimates without stating which
methods were used to calculate the volume. Presumably,
reported volume values are based on e.g. estimates ofdeposit thickness and aerial extent, or expert judgement.
Herein we describe and test a method to calculate
event volume using flow depth data, a friction relation
to describe the velocity of the debris flow as a functionof flow depth, and the geometry of the channel crosssection where depth is measured. We sum the dischargeover the event duration to calculate the volume. To test
the method, we use published data from the USGS
debris-flow flume [7] where volume and depth havebeen accurately measured.

2 Methods
The method we use was briefly described in [1] tocompare the amount of sediment exported from
Illgraben catchment in comparison with measurementsfrom sediment traps on hillslopes. No independent
observations of debris-flow volume are available there
due to mixing of debris flows sediment with flood flows

in the Rhone River. The method has been in use since2007 [2] to estimate volume and discharge forinterpreting well-documented debris flows [1-6, 8-11].
The method has not yet been systematically tested at any
field sites, mainly due to a lack of accurate independent
volume estimates e.g. from measurements of sediment
trapped in retention basins.
Observations of debris-flow velocity along an
individual surge are uncommon [12-14] but it is clear on
publicly-available videos of debris-flow surges that thevelocity is largest at the front of the flow, and both depth
and velocity decrease with distance upstream of the flow
front. Many friction relations are available to describe
how the depth-averaged flow velocity varies as afunction of flow depth (e.g. [15]), however the Manning
friction approach used herein has the advantage that it
has been in widespread use for estimating flow velocity
in sediment-laden floods as well as debris flows [16-18].
The Manning relation describes how the depth-averaged
velocity V varies with the hydraulic radius R of the flow
for steady-uniform flow in channel with slope S:
V=kRmS1/2(1)where: k = a friction coefficient assumed to be constantover the duration of the event, and m is the exponent on
the hydraulic radius term, usually m=2/3. The friction
coefficient in the case of debris flows is interpreted toinclude both friction arising at the interface of the debris
flow and the channel bed, and the internal friction within
the flow. The hydraulic radius R is the ratio of the areaof the channel A and the wetted perimeter of the channel
P (Fig. 1). For wide channels, or for when the sidewallsof the channel are much smoother than the channel bed(as in the USGS experiments [7] discussed below), it isreasonable to approximate R~h, where h is the flow

* Corresponding author: brian.mcardell@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, 02012 (2023)
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https://doi.org/10.1051/e3sconf/202341502012

depth. Both A and P vary as a function of the flow depth,
which at Illgraben is a trapezoidal cross-section (Fig. 1),
at a distance 1 m upstream of the brink of a concrete
check dam.

unique problems that may arise from using an individual
experiment. Here, we use their results for the roughchannel bed.
The python data analysis script we use to calculate
the main event parameters V, Q, and M for Illgraben was
modified for the geometry of the USGS channel. The
front velocity was determined from the front trajectory
plot (Fig. 9 in [7] and the friction coefficient k was backcalculated and used to compute the parameters. A
timestep of 0.01 s was used, and we focus on results
reported at 32 m downstream from the headbox (Fig. 2).

Fig. 1. Sketch of the measurement cross section at the check
dam under the road bridge at Illgraben. The cross-sectional
area A (blue shaded area) and wetted perimeter P arefunctions of the flow depth h which is determined using the
distance from laser or radar sensors installed on the bridge(e.g. [2]); these sensors are aimed at the center of the force
plate, which is 4 m wide, and 2 m long (in the flow
direction).

The volumetric discharge Q (m3/s) of the flow
passing over the check dam is the product of the velocity
and the cross-sectional area: Q=VA. The influence ofchannel width on discharge is taken into account using
A. The total volume M (m3) of the event is the sum of
the discharge over the event:(2)
where: C1 is a correction coefficient with a value close
to or smaller than one to account for the fact that thevelocity is not uniform at the cross-section, or more
specifically because the flow, after the passage of thedebris-flow front, tends to be slower at the edges than in
the middle of the channel. Herein we use C1=1 because
this effect is most visible towards the tail of events when
the corresponding discharges are relatively small.

Fig. 2. Results for the USGS debris-flow flume at x=32m forcalculated discharge (light blue lines), and cumulative event
volume M (red) for flow depths h (dark blue) as reported in[7]. The initial volume of the flow is the horizontal red
dashed line, and the cumulative volume is shown for
Manning exponent m=2/3 (solid red line) and m=1 (reddotted line). Results are shown for the SGM (a) and SG (b)sediment mixtures.

2.1 Application to the USGS debris-flow flume

2.2 Application to the Illgraben

The debris-flow experiments results described by
Iverson et al. [7] are useful for testing our procedure
because the initial volume has been accurately
measured, the time-series of flow depth information isavailable, and results are provided for two mixtures, asand-gravel mixture (SG) and a sand-gravel-mud
mixture (SGM) which is qualitatively similar to thesediment at Illgraben. The sizes of the flows are large,
minimizing the influence of possible scale effects.
Additionally, the results are reported for aggregatedrepeats of nearly identical simulations, minimizing any

Two large-volume events from Illgraben were selected,
one debris flow one main surge (24 June 2021) and an
event with semi-periodic roll waves (5 June 2022). For
the 24 June 2021 event, the maximum measured front
depth corresponds to the measured front velocity. For
the 5 June 2022 event, the maximum flow depth andvelocity are from the first surge, however the maximum
flow depth was observed several minutes later in a
subsequent surge. The front velocity was calculated as
the travel time between two check dams along the2

E3S Web of Conferences 415, 02012 (2023)
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channel, and the flow depth was taken from the laser
depth sensor. These values were used to determine
appropriate k values for the volume analysis describedabove.
The data (Fig. 3) are from the downstream end of
the channel above the confluence with the Rhone River(the location of the force plate described elsewhere [2,4], near the brink of the check dam (Fig. 1). At thislocation, data was collected at 2400 Hz and wasexported as median values in bins of 1s duration.

measured volume asymptotically approaching the initial
volume of 10 m3. The SG mixture (without mud) shows
poorer performance, over-predicting the volume by afactor of 1.4. While the results for the SGM experiments
support our volume-estimation method, several factors
may be able to account for the lack of good agreement
for the SG sediment mixture.
Changes in bulk density can influence the volume.
As the flow transitioned from rest in the head box to thechannel, small changes in bulk density were reported(Fig. 19 in [7]). This indicates that calculating the massof the flow, not volume, in comparison with the initial
mass would provide a more accurate comparison.
Deposition within the surge, followed by reentrainment at the tail of the event (especially in the SG
rough bed experiments, may result in an apparent flow
depth which is systematically too large, resulting in an
over-estimation of debris-flow volume. We intend totest this idea by performing our calculations on the SG
experiments which were performed on a smooth bed.
3.2 Results at Illgraben
The results at Illgraben for single-surge debris flows(Fig. 3a) and multi-surge events with large roll waves(Fig. 3b) indicate an increase in event volume up tonearly M=120,000 m3 and M=52,000 m3, respectively.
It it is not known if deposition occurs on the channel
bed during an event, which could result in an overestimate of the flow volume similar to what may beoccurring at the USGS flume for the SG mixture,described above. Erosion (and a correspondingly larger
discharge and volume) can be excluded due to thepresence of the check dam and force plate at themeasurement section.
Some evidence for deposition during the flow isprovided by geophone measurements, (see [2, 4]) whichshow that the impulses (from impacting particles) on the
bed remain fairly large near the flow front, and thereafter they decrease significantly. This may indicate thata layer of sediment has deposited on the force plate,although independent measurements of deposit
thickness are not available. A more likely explanation
for the decrease in impulses is that the grain size
decreases (fewer particles create vibrations) and/or thatthe particles are being transported in a liquified flow
after the front passes [2] and only rarely interact with thechannel bed. We anticipate that the new results from 3D
LiDAR scanning analysis [13, 14] will help constrain
the problem.
The flow passing over the measurement section (Fig.
1) is influenced by the free-overfall for Froude numbers
F=V/(gh)1/2 <1, where g is the gravitational acceleration.
For F<1, we expect the free overfall to cause a small
hydraulic drawdown, which causes the flow toaccelerate near the overfall, thereby minimizing
deposition at that cross section. However, the drawdown
under these conditions also causes the flow surface to be
somewhat lower than expected at the measuring
distance 1 m upstream of the overfall. The consequenceof this systematic error is that the depth, and therefore
estimates of velocity, discharge, and volume are also

Fig. 3. Results from Illgraben torrent for debris flows with
one main surge (a) and with many roll waves (b). The dark
blue lines are measured flow depths h, the light blue lines are
the calculated discharges Q, the solid red line is the runningsum of event volume M for m=2/3, and the dashed red line is
the M for m=1, as discussed in the text.

3 Results & Discussion
3.1 Results USGS debris-flow flume
The results from the USGS flume experiments (Fig. 2)
are useful for testing our volume-calculation method,
because both initial volume and accurate stage data are
available. In the experiments, both the front velocity andthickness of the flow (Figs. 9 and 10 in [7], respectively)
were approximately constant, indicating that the flow
was not significantly accelerating as it moved down thechannel, similar to flows observed at Illgraben. The
results for the SGM mixture, which contains somemuddy sediments, show promising results, with the3

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somewhat underestimated. The concepts of hydraulicdrawdown, or backwater effects, were elaborated forwater flows in open channels (e.g. [16]). While we
expect that results for surface drawdown for debris
flows will be similar to water flows, debris flows are
non-Newtonian fluids and some differences may arise.
It is likely that measurements being collected by 3D
LiDAR scanners [13, 14] will help us address thisproblem and allow us to correct for the drawdown in aprocedure analogous to using the slope of the hydraulicgrade, rather than the surface of the flow, as in openchannel flow [16].

analysis of the USGS data suggest that event volume
estimated using the Manning relation and stage data at
non-erodible cross sections may yield flow magnitude
estimates M accurate to within 50%.

4 Conclusions
We described and tested a method for estimating thevolume of a debris flow based on flow depth
measurements collected from a check dam, combinedwith the Manning friction relation commonly used inriver engineering. The results are sensitive to thecalibration procedure, and result volumes estimated towithin 50% based on comparisons with calculations
made at the USGS debris-flow flume where both flow
volume and flow depth measurements are available.
Increasing the exponent on the hydraulic radius term in
the Manning equation does not clearly increase theaccuracy of the method.

3.3 Influence of exponent m on the results
Several factors may influence the accuracy of ourpredictions at both the USGS flume and Illgraben. First,
observations of objects floating on the surface of theflow at Illgraben indicate that the surface velocity for agiven depth decreases more rapidly than predicted using
the Manning relation, assuming a constant verticalvelocity profile. If the exponent on the hydraulic radius
term m in Eq. 1 is increased, the velocity predicted forsmaller flow depths will decrease, and estimates ofdischarge and volume will also consequently decrease.
There is some support for using a larger exponent m
in gravel-bed rivers where the size of the particles islarge compared with the flow depth [18]. While theequivalent roughness in debris flows is large (max.
particle diameters can be as large as the depth of theflow), the use of a larger m has not yet, to the best of ourknowledge, been evaluated for debris flows. Therefore,
the results below for m=1 must be treated as speculative.
To illustrate the influence of a larger Manning
exponent m, we examined all four events here using an
exponent m=1 and determined the cumulative volume M(the dashed red lines on Figs. 2 and 3). For all cases, thevalue of friction coefficient k for the front of the flow
was re-calculated using (1) with m=1.
The results for m=1 indicate that the total volume of
the flow at the USGS flume is underestimated for the
SGM case (Fig. 2a) and somewhat better predicted for
the SG case (Fig 2b). At the Illgraben, the calculated
volume is approximately 20% smaller for the singlesurge event (Fig. 3a) and roughly similar for the eventwith roll waves. While these results are somewhat
speculative, they illustrate some of the problems andassumptions inherent in calculating debris flow volume
from only stage measurements: A fundamental
assumption is that the flow friction is adequately
described using the Manning equation, and furthermore,that the friction coefficient is constant along the entire
duration of the debris flow. It is clear that particle grain
size certainly varies in a debris flow, but the flowing
bulk density remains somewhat constant over many
reported events (e.g. [2, 4, 12]), so the sedimentconcentration is approximately constant. So, it may bereasonable to assume that the friction coefficient k may
also be approximately constant over an event, however
this is contradicted by Hungr [19] who suggested alongitudinal variation of flow friction related tosediment concentration. In spite of these shortcomings,

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