E3S Web of Conferences 415, 01015 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501015

A dilatant two-phase debris flow model with erosion validated
by full-scale field data from the Illgraben test station
Guillaume Meyrat1*, Perry Bartelt1, and Brian McArdell2
1WSL Institute for Climate Change, Extremes and Natural Hazards in Alpine Regions CERC,

Davos, Switzerland
2Swiss Federal Institute for Forest, Snow and Landscape Research WSL,Birmensdorf, Switzerland
Abstract. We propose a dilatant, two-layer debris ﬂow model validated by full scale density/saturationmeasurements obtained from the Swiss Illgraben test site. Like many existing models we suppose the debris
ﬂow consists of a matrix of solid particles (rocks, boulders) that is surrounded by muddy ﬂuid. However,we split the muddy ﬂuid into two fractions. One part, the inter-granular ﬂuid, is bonded to the solid matrix
and ﬁlls the void space between the solid particles. The combination of solid material and inter-granular
ﬂuid forms the ﬁrst layer of the debris ﬂow. The second part of the muddy ﬂuid is not bonded to the solid
matrix and can move independently from the ﬁrst layer. This free ﬂuid forms the second layer of the debris
ﬂow. During ﬂow the rocky particulate material is sheared which induces dilatant motions that change thesolid/fluid concentration of the first layer and then his density. As suggested by real data of Illgraben, therheology used is not constant and uniform but a function of the flow composition/ density. The model isthen compared to real debris flow data of Illgraben and tested on a real event in Ritigraben for which erosion
data are available.

1 Introduction
The assessment of debris ﬂow hazard relies on both
numerical simulation models and empirical methods.
Most numerical approaches solve shallow-water typeequations [1-5] and therefore can be eﬀectively applied topredict ﬂow heights and debris ﬂow runout distances.
Nonetheless, the application of numerical models inhazard engineering practice remains limited. This is dueto two salient problems. Firstly, it is diﬃcult to accurately
quantify the initial starting and entrainment masses for aspeciﬁc torrent. And secondly, historical case studies arestill necessary to calibrate the rheological parameters thatgovern debris ﬂow motion at a speciﬁc site, and therefore
possible inundation area. Without this information, themotion of a debris ﬂow is diﬃcult to model because it
depends strongly on the relative amounts of solid and ﬂuid
masses. In this contribution we present a two-phase debris
flow model, that splits the fluid phase into bonded andnon-bonded (free) parts. The dilatancy of the solid phase
controls the distribution of bonded fluid in the debris flow.
Field data from Illgraben and Ritigraben are used tocalibrate and validate the model. These two torrents arelocated in Canton Wallis, Switzerland.

2 Model definitions and equations
We model debris flows with a two-layer, depth-averaged
approach [6], see Fig.1. The debris flow contains twomaterial components: a solid component (subscript s)

consisting of coarse granular sediment, associated with adensity ρ𝑠 , and a fluid component (subscript m) consistingof fine sediment likely to behave as suspended sediment(e.g. sand, silt, clay), hereafter referred to as the muddy
fluid content, whose density is denoted by ρ𝑚 .

Fig. 1. Sketch of a debris flow. We divide the flowing
material in two layers. The first one is a mixture consistingof all the solid material and a part of the fluid calledinterstitial fluid. The second one is composed by the restof the fluid, called free fluid.

The solid and fluid components are divided into twolayers. The first phase/layer (subscript 1) contains the
granular solid material and a part of the fluid (interstitialfluid). The fluid is contained in the interstitial space
between particles and is assumed to be bonded to the solid
particles (subscript b). The second phase/layer (subscript2), is formed by the fluid which can flow independently
from the first layer. We term it the free fluid (subscript f).
As we are dealing with a two-phase model with three* Corresponding author: guillaume.meyrat@slf.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, 01015 (2023)
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components (solid, interstitial fluid and free fluid), massconservation equations are of the number of three and canbe written, per unit of area and density:𝜕ℎ1+ 𝛻⃗ (ℎ1 𝒗𝟏 ) =

𝜕𝑡
𝜕ℎ𝑏
𝜕𝑡

ρ𝑚

ρ

𝑄+ 𝑒𝐸

(first phase)(1)(second phase)

(2)

ρρ −ρ
+ 𝛻⃗ (ℎ𝑏 𝒗𝟏 ) = 𝑚 𝑄 + 𝑒 𝑏 𝐸 . (bonded fluid)

(3)

𝜕𝑡
𝜕ℎ2ρ𝑠

ρ𝑠+ 𝛻⃗ (ℎ2 𝒗𝟐 ) = −𝑄ρ𝑠ρ𝑏 −ρ𝑠Fig. 2. Sketch of two different configurations. The left one isa dilated one, happening during flowing. The right one is theso-called co-volume configuration, which happens at rest.
Even if the two configurations contain the same amount offluid and solid, the first layer density varies from the left tothe right one, due to the dilatant action of the solid matrix.

The symbol 𝛻⃗ is the divergence operator in Cartesian
coordinates. The right-hand side of the equations contains
the term Q, [6], which is the mass exchange rate between
the inter-granular and free fluid because of dilatant actionsin the solid matrix (see following). It also contains theerosion rate, denoted by E and ρ𝑒 is the density of themass entrained. In order to compute the erosion rate E, westarted with the model introduced in [7] and we adapted it
for a two-phase model. In this model, the erosion rate E is
not uniform, but considered to be a function of the flow
composition. Note that E does not appear in the second
layer equation, because we assume that the second layer
cannot entrain mass.
Two momentum conservation equations are associatedwith each phase, which can be written, per unit of area anddensity:𝜕(ℎ1 𝒗𝟏 )ℎ1+ 𝛻⃗ ⋅ (ℎ1 𝒗𝟏 ⊗ 𝒗𝟏 + 𝑔 𝐼) +𝜕𝑡2
ρ𝑺
ρ𝑔ℎ1 𝛻⃗ (𝑏 + 𝑚 ℎ2 ) = − 𝟏 + 𝑚 𝑃(first phase)(4)𝜕(ℎ2 𝒗𝟐 )ℎ2+ 𝛻⃗ ⋅ (ℎ2 𝒗𝟐 ⊗ 𝒗𝟐 + 𝑔 𝐼) +𝜕𝑡2
𝑺𝑔ℎ2 𝛻⃗ (𝑏 + ℎ1 ) = − 𝟐 − 𝑃 . (second phase)

(5)

ρ𝑠

ρ𝑠

ρ𝑠

ρ𝑠

flow is at rest, it is in the co-volume configuration, whichcorresponds to the one where the solid matrix is the mostcollapsed, right sketch on Fig. 2. However, underinteractions with the rough bed of the channel, the solid
matrix can expand its volume during flowing, leading todifferent flowing configurations, associated with different
densities, left sketch on Fig. 2. These different
configurations are obtained by exchanging fluid from one
phase to the other, which is reflected in the eq. 1-3 by Q.
Note that, even if the solid mass is conserved, the firstlayer density will vary because of the inter-granular fluid
concentration, which is given by,
ρ1 =𝑺𝒊 = (𝜇𝑖 𝑔𝑧 ℎ𝑖 +𝜉𝑖2𝑣 )𝒆𝒗 -ℎ𝑠 +ℎ𝑓= ρ𝑠 (1 − 𝑓) + ρ𝑓 𝑓 .

(7)

The symbol f represents the volumetric fluid
concentration in the first layer. Therefore, in our model,
the dilatancy controls the entire space and time evolutionof the debris flow density. For more details about themathematical structure of the dilatancy, see [6].

The vectors 𝒗𝟏 and 𝒗𝟐 represent the velocities of the firstand second layers, respectively; b denotes the bottom
topography. Note that the term 𝑔ℎ𝑖 𝛻⃗ 𝑏 represent thegravitation acting on the i-th layer. The symbol ⊗ denotes
the tensor product and I is the two-dimensional unity
matrix. The left side is the total variation of themomentum with respect to time, including the effect ofgravitation and the influence of each phase on the other[8,9]. The right-hand side represents the change inmomentum due to external forces (excluding gravitation).
𝑺𝒊 is the shearing forces acting on the i-th layer and 𝑷𝒊 are
the momentum exchanges due to mass exchanges Q
between the two phases. We use a Voellmy-Salm type
friction for the debris flow,
𝜌𝑖 𝑔

ρ𝑠 ℎ𝑠 +ρ𝑓 ℎ𝑓

Fig.3. The ratio of the shear and the normal stress, plotted
as a function of the volumetric fluid concentration. The
curve indicates clearly a decrease of the shear stressintensity when increasing fluid content in the debris flow.
Data are measured for real debris flow event in Illgraben,[10,11].

(6)

The symbol 𝑔𝑧 denotes the slope perpendiculargravitational acceleration, 𝒆𝒗 is a unit vector parallel to theflowing direction and 𝜇𝑖 and 𝜉𝑖 the coulomb and turbulent
coefficients friction.

Real debris flow data of Ill graben, a debris flow test site
located in Wallis (CH), [10,11], shows that the intensityof the shear forces, represented by the ratio of the shear
and the normal stress, decreases with increasing thevolumetric fluid fraction of a debris flow, Fig 3. Similar

The fluid mass exchange Q is a result of the dilatantactions of the solid particles in the first layer. When the2E3S Web of Conferences 415, 01015 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501015

considerations have already been pointed by Iverson in hispaper the debris flow rheology myth [12]. Therefore, inour model, the shear force acting on the first layer 𝑺𝟏 is afunction of the debris flow composition. Mathematically,
it is given by the evolution of the shearing coefficient µ1and 𝜉1 :𝜇(𝑓) = 𝜇𝑠 (1 − 𝑓) + 𝜇𝑓 𝑓 = 𝜇𝑠 (1 − 𝑓)
𝜉(𝑓) = 𝜉(1 − 𝑓) + 𝜉𝑓 𝑓

density profile but also the right density structure of adebris flow.

(8)
(9)

The friction parameters 𝜇𝑠 , 𝜉𝑠 represent the coefficient of
the solid material while 𝜇𝑓 , 𝜉𝑓 are the one valid for thefluid. We generally assume 𝜇𝑓 = 0. As the second layer
is always composed only by fluid, its shearing coefficientare constant and uniform: 𝜇2 = 𝜇𝑓 = 0 and 𝜉2 = 𝜉𝑓 .

3 Results
Fig. 5. Spatial density evolution from the front (top left corner)
to the tail (bottom right corner) of a debris flow. As expected,
the density varies from a high value in the front of the debris
flow, composed by large boulders to a lower value in themuddy tail. The red circle in the lower left coner map indicates
the Ritigraben torrent location.

Using the data (Shear and normal stress and flow height)
measured at the Illgraben test site, [10,11], we can extract
the debris flow density and therefore compare themeasured density with the density computed by our modelusing the dilatancy theory [6]. The comparison is shownin Fig. 4 where the density of both experimental data
(spared dots) and numerical output (circles) are plotted asa function of the norm al stress. This result indicates thatthe proposed model is able to simulate the longitudinaldensity profile of a real debris flow. The blue dots fallingunder the dataset represent the watery surges flowing
before the front of the flow, which is not capture by themeasurements.

According to Iverson [12], and to Fig. 3, the rheology ofa debris flow is not constant and uniform but evolves withthe flow composition, eq.8 and eq. 9. The spatial
evolution of the Coulomb friction µ is shown on Fig. 6.
The front, associated with high value of the density, isassociated with a value of the coulomb friction 𝜇 ≃ 0.15
while it goes to zero as we go to the tail. Fig. 5 and 6 comefrom the same simulation.
The event of Ritigraben was special from a debris flow
mitigation point view because two drones flights, one
before and one after the event, have been performed.
Therefore, it gives accurate and rare data about erosion.
The comparison with the model is shown on Fig. 7. The
red curve is obtained with drone flight while the blue oneis the model outputs. One can see that the general erosion
pattern, as well as the entire eroded volume, can becaptured by the model.

Fig. 4. Denisty profile of four real event in Illgraben, [10,11].
The spared dotes are the measured value of the density and thecircles represent the numerical outputs.

Fig. 5 shows a simulation of a real event in the torrent ofRitigraben located in Wallis (CH). The density evolves
from high value in the front (top left corner) to the tail(bottom right corner). This result agrees with the fact thatthe front of the debris flow is composed by large boulders
and becomes more and more wet as we go toward the tail.
Therefore, the model is not only able to predict the correct

Fig 6. As suggested by real debris flow data, the rheology is afunction of the flow composition. This fact is shown on thisfigure by the decrease of the coulomb friction coefficient μ,given by eq. 8. This figure comes from the same simulation
as the one of Fig. 5.

3

E3S Web of Conferences 415, 01015 (2023)
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5.

T. Takahashi, Annual review of fluid mechanics
13(1), 57 (1981)
6. Meyrat G, McArdell B, Ivanova K, Müller C,Bartelt P. A dilatant, two-layer debris flow model
validated by flow density measurements at the Swiss
illgraben test site. Landslides. (2022)
7. F. Frank, B.W. McArdell, C. Huggel, A. Vieli,
Natural Hazards and Earth System Sciences
15(11), 2569 (2015)
8. E. Savary, Y. Zech, Journal of Hydraulic Research
45(3), 316 (2007)
9. K.T. Mandli, Finite volume methods for themultilayer shallow water equations withapplications to storm surges. Ph.D. thesis (2011)
10. F. Schlunegger, A. Badoux, B.W. McArdell, C.
Wendeler, D. Schnydrig, D. Rieke-Zapp,
P. Molnar, Quaternary Science Reviews 28(11-12),1097 (2009)
11. F. Frank, B.W. McArdell, C. Huggel, A. Vieli,
Natural Hazards and Earth System Sciences
15(11), 2569 (2015)
12. R.M. Iverson, Debris-flow hazards mitigation:mechanics, prediction, and assessment 1,303 (2003)

Fig. 7. Erosion pattern towards the channel. The red curve isobtained by drone flights while the blue curve is thenumerical model outputs.

4 Conclusion
We applied a dilatant flow model to reproduce not only
the correct shape of the flow but also the right structure,i.e. a flow with a dense front which become more andmore wet as we move towards the tail. The model
contains two types of fluid: bond and non-bonded (free).
The fluid is not constrained to the solid matrix, and,
therefore can flow well beyond the solid matrix. Thus,
the model allows for de-watering simulations. We
adapted the well-known Voellmy-Salm rheology to thedebris flow problem, using real debris flow data as aguide. Finally, the erosion can be accurately simulatedusing a simple erosion model adapted for two-phaseflow.
We conclude by noting that it is unlikely that the debris
flow problem will be solved by purely theoreticalconsiderations. More likely is the continual monitoringof debris flow sites coupled with the documentation ofdebris flow events and their persistent back-calculationand modelling. Central in this undertaking will be theapplication of a numerical model that effectively
simulates the motion of debris flows of different
muddy/granular (i.e. mass) compositions. Work on theexperimental front is on-going.

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