E3S Web of Conferences 415, 02015 (2023)
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https://doi.org/10.1051/e3sconf/202341502015

Coupling Depth-Averaged and 3D numerical models to study
debris flow: Saint-Vincent event
Andrea Pasqua1,2*, Alessandro Leonardi2,1, and Marina Pirulli1
1 Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, 10129, Italy
2 The University of Sheffield, Western Bank, Sheffield, S10 2TN, United Kingdom

Abstract. Debris flows are extremely rapid and unpredictable phenomena whose rheology is poorly
understood. Moreover, human settlements are often located in areas prone to debris flows. The combinationof these features makes debris flows hazardous phenomena. Barriers are usually installed in debris flow
paths to mitigate risk. However, their design is still based on empirical methods. In order to base the designof barriers on a more reliable approach, the understanding of debris flows must be improved. Continuum
numerical models have proved to be a helpful tool for studying debris flows. In particular, numerical models
can predict the speed and the flow depth in debris flows paths, and roughly estimate the forces and thepressure acting on a mitigation structure. Currently, two main groups of continuum numerical models areavailable to study debris flows (i) depth-averaged (DA) models and (ii) three-dimensional (3D) models.
Although DA models can study a real-scale event, they may over-simplify the flow-structure interaction.
On the other hand, 3D models can be very reliable for studying flow-structure interaction but studying awhole phenomenon (from triggering to deposition) would require enormous computational resources. This
work aims to show how the coupling of a DA and a 3D model allows an effective and performing analysis
of a debris flow dynamics. The study is focused on the 2014 Saint-Vincent event (Aosta Valley, Italy).

1 Introduction
Debris flows consist of poorly sorted material saturated
with water and flowing in channelized paths. They arecharacterized by high speed and the absence ofpremonitory signs. Moreover, human settlements areoften located in areas prone to debris flows. The
combination of unpredictability, high speed, andpresence of human settlements make debris flows
hazardous phenomena.
In order to mitigate the risk, often barriers are
installed along debris flows paths (Fig. 1) [1,2].
However, nowadays design of barriers is based on
empirical or simplified methods, mainly because of apoor understanding of debris flows dynamics [3] andtheir interaction with solid obstacles [4]. In want of abetter knowledge of the dynamics of debris flow, andthus, to better design the barriers, it is necessary to gain
knowledge of this type of event behaviour. Continuum
numerical models have proved to be efficient andreliable to study debris flows. In this frame, two groupsof models can be efficiently employed: (i) depthaveraged (DA) models and (ii) three-dimensional (3D)
models. DA models are based on depth-averaging the̅ ) along the flow depth (ℎ) [5]. This leads
flow speed (𝒖to very quick analyses even on very large topographies.
However, the depth-averaging procedure leads to theloss of information along the direction perpendicular tothe flow. Hence, the flow-structure interaction is studied
as a flow of depth-averaged velocity impinging on aFig. 1 Example of filter barrier, Saint-Vincent, Aosta Valley(Italy)

mitigation structure. Since flow-structure interaction isan intrinsically three-dimensional phenomenon, thisapproximation may be too simplistic. Thus, DA models
should be employed with care when studying flowstructure interaction. On the other hand, 3D models canstudy flow-structure interaction very accurately, withfull resolution of pressure and velocity variation along
the direction perpendicular to the flows. Nevertheless,studying a whole flowing process (made of billions ofnumerical points to be solved) is extremely timespending and complex.
Since the flow-structure interaction depends on how
flows evolve upstream of the barrier, this work aims tomerge DA and 3D models to study a flow-structure* Corresponding author: andrea.pasqua@polito.it© 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/).

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interaction without renouncing of understanding of how
the flow develops upstream the barrier. A few studies
have been dedicated to coupling two different numericalmodels. Su et al. (2022) coupled a discrete element
method to a depth-averaged method at the site scale [6].
Domnik et al. (2013) coupled a bi-dimensional model toa depth-averaged method at the laboratory scale [7].
Pasqua et al. (2022) coupled a DA model to a 3D model[8]. Unfortunately, these studies do not apply thecoupling at the full scale of the event yet.
This work discusses an application where a DA anda 3D model are coupled to study an event at full scale.
The event studied is the one that occurred in SaintVincent, Aosta Valley (Italy), in 2014. To achieve that,the approach proposed in [8] is modified. Primarily, theVoellmy rheology is employed. The reason is that
Voellmy rheology has been widely proven reliable insimulating debris flows at full scale, and it has twoparameters to calibrate, which makes back analysis easy
[9,10].
This paper is structured as follows. In Section 2, thebasis of DA, 3D models, and coupling are discussed. Insection 3, the coupling and its application to the real casestudy of Saint-Vincent (Aosta Valley) are discussed.
Finally, some conclusions are drawn.

equation and leads to thermodynamic equilibrium[12,13].
2.1 Coupling depth-averaged and 3D models
To couple DA and 3D models rationally, it is essential
to define which part of the domain each model solves.
The DA model solves the domain area, where nostructures are assumed to be present. As soon as the flow
approaches a structure, the DA results (depth-averaged
velocity and flow depth) are converted into an input forthe 3D model: a 3D velocity profile.
Coupling a DA and a 3D model is difficult toachieve. The main issue is obtaining a 3D velocity
profile from a depth-averaged profile (the velocity along
the perpendicular to the flow direction). The 3D velocity
profile is unknown in advance and is a rheology
function. The approach proposed by [8], where theauthors use the µ(I) rheology, is modified in this work.
The µ(I) rheology is difficult to calibrate at the site scale
since it requires 5 parameters. Moreover, the µ(I)rheology has been validated for dry granular flows at thelaboratory scale. To bypass these issues, in this paper,
the employed rheology is the Voellmy. This is because
the Voellmy rheology has been widely validated tostudy debris flow events at a site scale. The 3D steadystate Voellmy velocity profile must be imposed at thecoupling section to achieve the coupling. However, toour best knowledge, the 3D Voellmy velocity profile has
not been proposed yet. This work discusses obtaining
such a profile from the depth-averaged velocity and flow
depth.
The Voellmy expression in the DA framework is thefollowing:

2 Numerical models
As mentioned in Section1, continuum numerical models
can be DA or 3D. Only the fundamental details of themodels are here discussed. Please, refer to the following
references for a complete discussion of the difference
between DA and 3D models.
DA models approximate debris flows, which consistof a heterogeneous mixture of solid and fluid material,
as an equivalent fluid [5,11]. DA models solve thedepth-averaged version of the Navier-Strokes equations(i.e., mass and momentum conservation). DA models
can predict the flow path, speed, and depth evolution intime. However, the depth-averaging process is a criticalpoint. Studying flow-structure interaction with DA
implies that every flow property (i.e., velocity, pressure)is depth-averaged along the flow depth. Hence, thedesign of a mitigation structure is grounded on averaged
values. However, flow-structure interaction is
intrinsically three-dimensional, and depth-averaging
may lead to underestimating the magnitude of the event.
3D models, on the other hand, can study shear
deformation within the flow and the flow structureinteraction very precisely. In this study, the employed
3D model is Lattice-Boltzmann method (3D LBM), a
relatively new fluid solver compared to the moreclassical fluid solver (finite differences, finite volumes,finite elements). LBM does not discretise the NavierStrokes equations but relies on a probability density
function 𝑓(𝒙, 𝑡, 𝒄) representing the probability offinding fluid particles with speed 𝒄 at location 𝒙 andtime 𝑡. LBM discretises the space using a regular grid inwhich 19 velocities 𝒄𝑖 are allowed (D3Q19). The
macroscopic variables density and velocity are obtainedby summation of ci. The equation governing theevolution of 𝑓𝑖 is a discretised form of the Boltzmann

𝜏 = 𝜇𝑃 + 𝜌𝒈

̅2𝒖
𝜉,

(1)

̅ , and 𝜉 are the shear resistance,
where 𝜏, 𝜇, 𝑃, 𝜌, 𝒈, 𝒖
the friction coefficient, the pressure, the bulk density,
the gravity acceleration, the depth-averaged flow
velocity, and the turbulent coefficient that considers thedissipation due to the velocity.
Unfortunately, Eq. (1) cannot be employed as it is in
a 3D framework because the 𝜉 parameter has beendeveloped and validated only in DA contexts. To
overcome this issue, see the following expression ofVoellmy rheology in a 3D framework is proposed:𝜏 = 𝜇𝑃 +𝜌𝑔𝑙2 𝛾̇ 2𝜉,(2)where 𝑙, and 𝛾̇ are a length scale related and the internal
shear rate, respectively. Assuming steady-state from Eq.
(2), one can obtain both the depth-averaged Voellmy
velocity expression:
32 ℎ2√(tan 𝜃 − 𝜇)𝑔 cos 𝜃𝑢̅ =5𝑑
and the 3D Voellmy velocity profile:

(4)

32 ℎ2
√(tan 𝜃 − 𝜇)𝑔 cos 𝜃 ∙𝑢(𝑧) =3𝑑𝑧 3⁄2∙ [1 − (1 − ) ].
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By substituting Eq. (4) in Eq. (5) and rearranging theterms, the 3D velocity profile can be rewritten as afunction of 𝑢̅:
5𝑧 3⁄2𝑢(𝑧, 𝑢̅) = 𝑢̅ [1 − (1 − ) ].
(6)3ℎAs stated above, the Voellmy rheology fits the studyof debris flow event. However, to rationally couple theDA and the 3D model, the rheological parameters must
be the same in both frameworks. We propose to calibratefirst the DA model and then obtain the rheologicalparameters in the 3D model. Since 𝜇 has the same
physical meaning in both frameworks, its value does notFig. 2 Drainage area of Saint-Vincent, Aosta Valley (Italy).
The DA model resolves the domain outside the purple oval. In
contrast, the 3D model solves the volume inside the purple
oval. The 3D volume representation, for graphical reasons, is
not represented to scale.

The velocity profile in Eq. (6) can be used in a 3D model
if the flow is in or close to a stationary condition.

Fig. 3 Rack dam and its numerical schematisation. All thedimensions are in meters

3 Saint-Vincent event and results

Table 1 Depth-Averaged and 3D Voellmy rheologicalparameters

In this section, the DA-3D coupling is applied at theevent that occurred in Saint-Vincent, Aosta Valley,Italy, in 2014. A complete description of the event isreported in Leonardi et al. (2021) [14]. Since this area isprone to debris flows, local authorities installed aconcrete rack dam. In the rack dam, 3 m long steel
profiles IPE 270 are imbedded in a 1 m concrete base.
The steel profiles are 0.5 m spaced. Fig. 3 sketches thebarrier characteristics mentioned.
The drainage basin extends for 5.22 km 2 (Fig. 2).
The DA model carries out the analyses where nostructure exists (jet-coloured area in Fig. 2). The 3Dmodel run analyses where the rack dam is present(volume within the purple oval in Fig. 2). The 3D model
converts the results of the DA model (depth-averaged
velocity and flow depth) into a 3D velocity profile (Eq.
6). The 3D model studies a rectangular natural channel
18.0 m long and 9.0 m wide (see Fig. 4). Because thebarrier is 3.5 m high, the numerical domain is 4.0 m highto contain the whole barrier.
The DA model here employed is RASH3D [15].
RASH3D uses an unstructured grid whose dimension is
5.0 m, guaranteeing a good approximation of thetopography and short computing time. On the otherhand, the 3D model uses a squared grid of 8.5 ∙ 10−2 m,
and the time discretisation is 1 ∙ 10−4 s. These twoparameters guarantee the stability of LMB method anda good approximation of the natural channel and thebarrier.

Depth- Averaged Voellmy
𝜇 = tan 11.3°
𝜉 = 500 𝑚/𝑠 2

3D Voellmy
𝜇 = tan 11.3°
𝑙 = 0.285 𝑚change. The value of 𝑙 in the 3D framework is related tothe particle diameter and to 𝜉. In this study the 𝑙 value
was calibrated with a back-analysis process, and its best
fit-value is 0.285 m. Table 1 shows the rheologicalparameters for both frameworks.
In Fig. 4, the reader can see the results downwards:(i) the 3D domain when the flow front reaches thecoupling section, (ii) the flow impinging on the rackdam, and (iii) the flow crossing the rack dam. At theinitial stage, the flow enters the 3D domain; the DA
analysis is over, and the depth-averaged velocity andflow depth are converted through Eq. 6 in a 3D velocity
profile. In the second frame, the flow impinges against
the barrier. This frame may be the most important.
Simulating impact as accurately as possible to better
design barriers is crucial. The DA-3D coupling here
shows all its potential. In their original paper, Leonardi
and Pirulli (2020) [14] studied the flow-structureinteraction simulating a dam break. The model
employed was based on the discrete element method(DEM). Although the study [14] is solid, some variables
must be assumed, i.e., particle numbers and theirvelocity. These assumptions must be made with care andverifying them with data field is difficult. On the otherhand, coupling DA-3D can bypass this issue. Since the3E3S Web of Conferences 415, 02015 (2023)
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studied flow first fills the rack dam and then crosses thebarrier. The impact with the barrier is likely similar towhat was observed on the field, the run-up mechanism
appears realistic, and no unphysical behaviour appears
in the simulation. Eventually, the flow crosses the rack
dam as expected. Stopping a debris flow event with only
one mitigation structure would be hard to achieve due tothe magnitude of the velocity and pressure. Common
risk

without renouncing to understanding how the flow
evolves upstream.
The results reported in the present work areencouraging. As mentioned in Section 3, DA-3D
coupling may bypass the difficulties linked to moreclassical approach methods like dam break simulatedwith DEM. Nonetheless, further developments should
be carried out. Multi-barrier numerical models should becompared to field.

References
1.

Leonardi, A., Wittel, F.K., Mendoza, M., Vetter,
R., Herrmann, H.J., 2016. Comput.-Aided Civ.
Infrastruct. Eng. 31 (5), 323–333 (2016)
2. Leonardi, A., Pirulli, M., 2020. Comput. Geotech.
118, 103311 (2020)
3. Iverson, R.M. International Conference on DebrisFlow Hazards Mitigation. Vol. 1. pp. 303–314(2003)
4. Iverson, R.M., Rev. Geophys. 35 (3), 245–296(2007)
5. Savage, S.B., Hutter, K., J. Fluid Mech. 199, 177–215 (1989)
6. Su X., Xia X., Liang Q., Hou J., Comput. Geotech.
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7. Domnik B., Pudasaini S.P., Katzenbach R., MillerS.A., Journal of Non-Newtonian Fluid Mechanics,
201 56-58 (2013).
8. Pasqua A., Leonardi A., Pirulli M., Comput.
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9. Pirulli, M., Sorbino G., Natural Hazards and Earth
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11. Hungr O Canadian Geotechnical Journal 32 (1995)
12. Mohammed A A New York: Springer ISBN 9780-85729-454-8 (2011)13. Succi S Oxford University Press ISBN
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Fig. 4 Evolution of the flow against the rack dam (initial state,impact, and overflow)

mitigation practice prescribes installing more mitigation
structures. Each mitigation structure slows down theflow and retains boulders. This multi-structure strategy
reduces the dimension of each mitigation structure.

4 Conclusions
In this work, a DA-3D coupling to study debris flows atfull scale was proposed.
The present work studied the event that occurred in2014 in Saint Vincent (Italy). The study showed that it
is feasible to couple a DA model with a 3D model.
Moreover, it was possible to study the event at the fullscale. Especially where the absence of mitigation
structure is assumed, the domain is solved with a DAmodel. By contrast, the 3D model converts the depthaveraged velocity and flow depth where a mitigation
structure is present into a 3D velocity profile. The flowstructure interaction is thus studied via a 3D model4