Landslide-generated impulse waves

Nubia Andrea Jara Barrera

Advisor: prof. Miguel Cabrera

A thesis presented for the degree ofMSc in Civil Engineering

Departamento de Ingenierıa Civil y AmbientalUniversidad de los Andes

Bogota, ColombiaJune, 2021

Table of contents

1 Summary 2

2 Introduction 3

3 Literature Review 53.1 Fundamentals of a mass flow interacting with a water body . . . . . . . . . . . . . . . . . 5

3.1.1 Oscillatory and translatory waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.2 Shallow, intermediate and deep-water waves . . . . . . . . . . . . . . . . . . . . . . 63.1.3 Linear and non-linear waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Mass flows wave generation: physics and current theories . . . . . . . . . . . . . . . . . . 93.2.1 Subaerial flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.2 Submerged flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Mass flows wave generation: Laboratory experiments . . . . . . . . . . . . . . . . . . . . . 123.3.1 Subaerial flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Submerged flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Mass flows wave generation: Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 163.4.1 Subaerial flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.2 Submerged flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 r.avaflow: Depth averaged methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Methodology 204.1 Subaerial flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Submerged flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Results and discussion 285.1 Subaerial flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1.1 Landslide characteristics at impact . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.2 Wave characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.3 Deposit shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1.4 Maximum amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.5 Virtual mass evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Submerged flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Conclusions and future perspectives 466.1 Subaerial flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Submerged flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7 References 47

8 Nomenclature 49

A Codes 51A.1 Python function to create experimental setups DEM as raster (ASC) maps . . . . . . . . 51A.2 Code to create solid and fluid phases with r.lakefill . . . . . . . . . . . . . . . . . . . . . . 53A.3 Code to run r.avaflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.4 Python function to process r.avaflow output raster maps. . . . . . . . . . . . . . . . . . . 56

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1 Summary

Landslide generated waves, often refered as tsunamis in coastal engenieering, are a latent hazard to peopleand infrastructure near water bodies in mountainous areas. Therefore, models that allow for predictionsof the wave behavior have been developed in order to perform risk assessments. A depth averaged modelis employed to model the motion of multiphase mass flows, including the possibility of the landslide tointeract with a reservoir. In this work, the laboratory experiments made by Miller et al. (2017) andBullard (2018) of subaerial flows impacting a water reservoir are recreated with r.avaflow 2.3, explor-ing the similarities between the experimental and the numerical results. Landslide, wave, and depositcharacteristics are compared. The numerical results are in good agreement with the experimental resultsparticularly in the near-field and for shallow still water depths. The virtual mass arise as an importantcomponent to be considered in the mass flow model as it can take substantial larger values than the dragcomponent.

Additionally, a series of laboratory experiments are performed to provide the first insights in the re-lationship between landward waves and column level of submergence. The maximum landward wavetrough a−m shows an inverse relation with the columns submergence, and the position where a−m is gener-ated shows a constant relation with the submergence ratio.

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2 Introduction

Landslides are natural phenomena that occurs in mountainous areas such as the Andes cordillera, theAlps, or the Himalayas (Prada-Sarmiento et al., 2019, Desrues et al., 2019, Marc et al., 2019). Thiskind of mass movements may generate a wave if it reaches (or occurs inside) a water body, resulting incasualties and damages to infrastructure. Examples of tsunamis generated by landslides interacting witha water body are the Chehalis lake, the Anak Krakatau and the Vajont dam events (Robbe-Saule, 2019,Barla & Paronuzzi, 2013). The Chehalis lake event occurred in 2007, where rainfalls made the terrainunstable and 3× 106 m3 of material impacted the lake producing a 38 m wave (Robbe-Saule, 2019). Thewave traveled 7.5 km and the wave run-up reached 8 m (Robbe-Saule, 2019). In 2018, 0.28 km3 of theAnak Krakatau volcano flank detached and impacted the sea generating a 43 m wave leaving more than1000 casualties (Robbe-Saule, 2019). Finally, the Vajont landslide occurred in 1963 due to a slope failuretriggered by the dam filling process (Barla & Paronuzzi, 2013). The mass flow consisted of debris andsoil with a volume of 2.7 − 3 × 109 m3 that impacted the dam reservoir and generated a 260 m wave(Barla & Paronuzzi, 2013). The wave overtopped the arch dam and the flooding destroyed the town ofLongarone, leaving more than 2000 casualties (Barla & Paronuzzi, 2013). Because landslide-generatedwaves can cause large damage, their dynamics are of great interest. To take effective defensive measures,aspects such as how the landslide velocity and thickness are related to the wave amplitude, celerity, andlength, how far the landslide will travel across the water, and how the wave changes in space and timeneed to be addressed.

In order to develop predictive models on how a landslide-generated wave spreads across a water body,depending on the rheology of the mass flow and the characteristics of the terrain, physical models, nu-merical simulations and theoretical assessments have been made. The main objectives of these studieshave been: (1) to solve the model equations that describe the interaction between the phases present inthe landslide, and the water body (Savage & Hutter, 1989, Pudasaini & Mergili, 2019, Pudasaini, 2019,Pudasaini, 2020), (2) to validate the model with well-controlled laboratory tests (Mulligan & Take, 2017,Bullard, 2019, Cabrera et al., 2020), and (3) to implement the model in numerical simulations of realevents (Mergili et al, 2018, Grilli et al., 2017).

The present study aims to investigate the behavior of two types of landslide-generated waves: the wavegenerated by a subaerial landslide, by reproducing the laboratory experiments made by Miller et al.(2017) and Bullard (2018), and the collapse of a wide submerged column generating landward waves.The objectives of the present study are, thus:

1. To assess the results obtained with r.avaflow 2.3 when simulating waves generated by dry andsaturated landslides impacting a water body,

2. To evaluate the effect of the landslide initial solids to fluid ratio in the amplitude of the generatedwave with r.avaflow 2.3

3. To explore the waves generated by the collapse of a submerged column in a planar setup at 1 g.

r.avaflow 2.3 is an open source numerical tool which implements a depth-averaged method to modelthe motion of a mass flow that may impact a water body, and the corresponding generated tsunami(Pudasaini & Mergili, 2019).

The text is organized as follows: In Chapter 3 the wave types are described, and current theories oflandslide-generated waves are presented, along with previous laboratory experiments and numerical sim-ulations results. In Chapter 4 the numerical and experimental setups are detailed, and the variables ofthe study are summarized. In Chapter 5 the results obtained in this study numerical simulations, for a

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granular mass flowing down an inclined channel impacting a water reservoir, and laboratory experiments,for a granular mass flowing inside a water reservoir, are presented. The numerical landslide, wave, anddeposit characteristics are analyzed, whereas the experimental landward wave trough amplitude and itsrelation to the position of the deposit crown is presented. Finally, conclusions and future perspectivesare presented in Chapter 6.

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3 Literature Review

3.1 Fundamentals of a mass flow interacting with a water body

When describing the behavior of an impulse wave generated by a landslide, the parameters of interest areprimarily the fluid surface elevation η and its celerity c, as a function of the initial height of the reservoirh, the landslide velocity vs, the landslide thickness s, and the terrain slope α. Figure 1 shows a schemeof a granular mass interacting with a water body and the quantities associated with the problem.

Figure 1. Granular material impacting a water body where s and vs are the landslide thickness and velocity, αis the terrain slope, η is the function that describes the fluid surface elevation, am is the maximum amplitude ofthe generated wave, and h is the initial height of the reservoir or still water depth. After Mulligan et al. (2017).

There are three types of flows that can generate an impulse wave: subaerial, partially submerged, andfully submerged slides (Fig. 2). Also, the wave type generated depends on the flow type: seaward wavesare generated for the subaerial, partially submerged, and submerged landslides, but landward waves aregenerated only for the submerged landslide type. The present study will be focused in the impulse wavesgenerated by subaerial and submerged landslides. Waves can be classified by considering four criteria:(i) the waves are oscillatory or translatory, (ii) the waves are shallow, intermediate or deep-water, (iii)the waves are periodic or non-periodic, (iv) the waves are linear o non-linear. These criteria are usefulwhen comparing the waves generated in the nature and in the laboratory, and to choose the relevanttheoretical equations in the formulation of predictive models.

Part. submergedSubaerial Submerged

seawardseaward seaward

landward

Figure 2. Initial position of slides before impulse wave generation and wave types generated for each case.

3.1.1 Oscillatory and translatory waves

Oscillatory and translatory waves differ in that the first type does not transport mass fluid whereas thesecond one does, due to the type of motion of the water particles under the crest (Le Mehaute, 1976).Figure 3 shows that in the oscillatory waves, the water moves horizontally with a velocity c, and theparticles inside the wave move elliptically, whereas in the translatory waves the water particles movehorizontally with the velocity of the fluid (Le Mehaute, 1976).

5

𝑐

Translatory

𝑐

Oscillatory

Figure 3. Water particle motion in oscillatory and translatory waves.

3.1.2 Shallow, intermediate and deep-water waves

The distinction between shallow, intermediate, and deep-water waves is made with the ratio betweenthe wavelength L, and the reservoir depth h. When this parameter is greater than 20 (L/h > 20), thewaves are classified as shallow, when the parameter is between 2 and 20 (2 < L/h < 20), the waves areintermediate, and when the parameter is less than 2 (L/h < 2), the waves are classified as deep waterwaves.

3.1.3 Linear and non-linear waves

The theory of linear waves states that the ratio between the wave height H and the reservoir depth h t(H/h) must be less than 0.03 and the slope H/L must be less than 0.0006 (Le Mehaute, 1976). Whenthis criteria is accomplished, the equations describing the movement of the wave are the Airy theory

equations. According to this theory (Airy, 1845), the wave celerity is c =√

gL2π for deep-water waves,

c =

√gL2π tanh

(2πhL

)for intermediate water waves, and c =

√gh for shallow water waves (Fig. 4).

Figure 4. Dimensionless wave celerity against relative wave length for linear waves. Taken from Heller (2007).

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However, waves generated by a landslide impacting a water body are, in general, highly non-linear.The non-linearity of a wave can be established with the Ursell number

(U = HL2/h3

)which indicates

the ratio between the amplitudes of the second and first-order terms in the Boussinesq’s solitary wavefree surface elevation. If U < 10, the theories used to describe the wave behavior are the Stokes andMiche/Dubreil-Jacotin theory, if 2 <= L/h <= 20, and the Gerstner theory if L/h > 20. WhenU ≈ 1 and L/h > 10, the Cnoidal and solitary wave theories are applicable (both Scott-Russell andBoussinesq). Finally, if U >> 1 and L/h >> 20, then the long wave, tidal bore and flood wave andmonoclinical theories are applicable (Fig. 5).

Figure 5. Linear and non-linear wave theories for water wave classification. Hyd.: hydrostatic pressuredistribution, irro.: irrotational flow, nhyd.: non-hydrostatic pressure distribution, osc.: oscillatory wave, ro.:

rotational flow, trans.: translatory wave. Taken from Heller (2007).

According to several laboratory experiments (Fritz, Hager & Minor, 2004, Zweifel, Hager & Minor,2006, Heller & Hager, 2010, Bougouin et al, 2020), the non-linear waves, generated by a landslide impact-ing a water body, are reported to be Stokes, cnoidal, solitary or bore-like waves (Fig. 6) . The first twotypes correspond to periodic waves and the remaining are non-periodic waves. The Stokes waves developin intermediate an deep water depths and are steeper than the sinusoidal linear waves. In these type ofwaves, the particles do not move in a closed orbit, thus there exists a small fluid mass transport. Thecnoidal waves develop in shallow to intermediate water depths, and the particles also have an ellipticallynon-closed orbit.

The solitary waves are a single crest without troughs, and because it is a non-periodic wave, the particlesmove horizontally allowing a great mass transport. When the amplitude of a solitary wave is greaterthan 0.78h, the wave breaks and a bore is generated because air enters the crest of the wave causing itto curve. A bore is non-symmetrical and has a steep front and a smooth tail.

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Stokes Cnoidal

Solitary Bore

Figure 6. Stokes, cnoidal and solitary wave shapes, and bore shape. L and T are the length and period of thewave, a and H are the amplitude and the height of the wave, c is the celerity of the crest and h is the still water

depth. Taken from Heller (2007).

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3.2 Mass flows wave generation: physics and current theories

3.2.1 Subaerial flows

Mulligan & Take (2017) propose an expression for the maximum amplitude of an impulse wave, generatedby a dry subaerial granular slide impacting a water reservoir, by replacing the landslide momentum fluxin the hydrostatic and hydrodynamic momentum balance equations.

The change in momentum in time due to a granular slide impacting a water body is

∂msvscosα

∂t=ρsVsvscosα

∆te(1)

where ms is the slide mass, ρs is the slide bulk density, Vs is the slide volume, (vscosα) is the horizontalcomponent of flow velocity and ∆te is the effective time between the impact and the detachment of theflow. By dividing the slide volume by its width and length, the momentum flux can be expressed as

Ms =ρssvscosα

∆te(2)

The hydrostatic momentum balance can be found by expressing the pressure gradient between the land-slide impact zone and the undisturbed zone as

∂P

∂x=

1

Le

[1

2ρfg (h+ am)2 − 1

2ρfgh

2]

(3)

where ρf is the water density and Le is the effective flow length. The momentum flux of the fluid is thus

Mfs =ρfg

(ham + 1

2a2m

)Le

(4)

And because there is a conservation of momentum between the landslide and the water reservoir (Mfs =Ms) hence

ρssvscosαL

∆teρfg= ham +

1

2a2m (5)

Thus, the maximum wave amplitude is

am =

√h2 +

2ρssvscosαL

ρg∆te− h (6)

When the difference between the granular flow velocity and the celerity of the wave inhibit the verticaldisplacement of the water, the hydrodynamic momentum flux can be expressed as

Mfd =ρfamc

∆te(7)

where c =√gh. Equating the landslide momentum flux and the hydrodynamic momentum flux Mfd =

Ms, yields

ρfamc

∆te=ρssvscosα

∆te(8)

thus,

am =ρssvscosα

ρ√gh

(9)

9

Additionally to the momentum balance equations, the wave breaking criterion states that if am/h > 0.78the wave will break in the near field. Similarly, the continuity limit establishes that am

h < svscosα∆tehL + 1

due to the conservation of fluid mass. The last classification criteria is the densimetric Froude number(Frd = vs

c

(shρs−ρfρf

)1/2)

that determines a critical depth hcr, when Frd = 1, where a change in the

tendency of the am vs h curve is observed (Fig. 7). With this critical reservoir depth, a maximumwave amplitude is expected. The densimetric Froude number is more useful to classify flows than theconventional Froude number, because it takes into account the density difference between the slidingmaterial and the fluid in the reservoir, and the slide thickness relative to the still water depth.

Figure 7. Theoretical momentum flux equations for the maximum wave amplitude in the near-field and theexperimental results from Miller et al. (2017). Taken from Mulligan & Take (2017).

3.2.2 Submerged flows

Following the hydrostatic momentum balance proposed by Mulligan et al. (2017), Cabrera et al. (2020)developed an expression to find the maximum seaward wave amplitude generated by the collapse of agranular column. The authors start by defining the momentum flux in the collapse of a granular columnFg as

Fg = ρsgA (10)

where g is the gravitational acceleration, ρs is the bulk density of the granular column and A is thetriangular area above the collapsed column dependent on the cutback length Lc (Fig. 8) expressed as

A =1

2

(LcH0

2

)(11)

thus

Mgc =FgH0

=ρsgLc

4(12)

Because there exist a momentum exchange between the granular column and the fluid, equating Mfs

(Equation 4) and Mgc

1

2a2m +Hwam −

ρsρf

L2c

8= 0 (13)

10

Equation 13 has a positive root of

am =

√H2w +

ρsρf

L2c

4−Hw (14)

Cabrera et al. (2020) approximate the cutback length Lc as

Lc ≈ H0 − C (15)

where C is the compound center of mass of the granular column expressed as

C =Csρs + φCw (ρs − ρf )

ρs(16)

Figure 8. Sketch of the landslide generated waves mechanism for a submerged granular column. Taken fromCabrera et al. (2020).

Cabrera et al. (2020) proposed that Equation 13 could be used as a first-order approximation fortsunami hazard assessment, and demonstrated a good agreement (± 30%) between similar experimentalsetups presented by Robbe-Saule (2019) and Huang et al. (2020) with Equation 13.

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3.3 Mass flows wave generation: Laboratory experiments

3.3.1 Subaerial flows

Physical modelling has been widely implemented to study landslide-generated impulse waves: from pistonand block slide models (Noda, 1970, Monaghan & Kos, 2000, Walder et al., 2003, Harbitz et al., 2003,Ataie-Ashtiani & Nik-Khar, 2008) to collapses of granular columns and granular wedges, both in 2D and3D configurations (Mohammed & Fritz, 2012, Viroulet, Sauret & Kimmoun, 2014, McFall & Fritz, 2016,Jara & Cabrera, 2020). The results of laboratory experiments have been used in predicting wave types,and wave characteristics, especially maximum wave amplitude.

Fritz, Hager & Minor (2003) used cylindrical particles of 4 mm in diameter made of SD 104-C, Polycom-pound AG, 87% barium-sulfate y 13% polypropylene, grain density ρg = 2640 kg/m3, effective frictionangle φ′ = 43◦, and dynamic bed friction δ = 24◦ as a release volume to generate waves at VAW. Theslide density was ρs = 1620 kg/m3 and corresponded to a calculated porosity inside the release box ofn = 0.39. Also, the slope of the inclined channel used by the authors was 45◦and the still water depth hwas varied as 0.3, 0.45 and 0.675 m.

The physical model at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW) at ETH, con-sisted of a prismatic channel of 3 m long and 0.5 m wide, connected to a horizontal channel of 11 mlong. The instrumentation of the model consisted in laser distance sensors, particle image velocimetryand capacitance wave gages.

Fritz, Hager & Minor (2004) classified the generated waves as a function of the Froude number Fr =vs/√gh, and the dimensionless slide thickness S = s/h, where s is the slide thickness. A nonlin-

ear oscillatory wave pattern was observed when Fr < (4 − 7.5S), a nonlinear transition wave regionwas observed when (4 − 7.5S <= Fr < (6.6 − 8S)), a solitary-like wave region was observed when(6.6− 8S <= Fr < (8.2− 8S), and a dissipative transient bore was generated when Fr >= (8.2− 8S).

Similarly, Miller et al. (2017) released a dry granular material, composed of ceramic spherical beadsof 3 mm in diameter, with a grain density of 2400 kg/m3 and an internal friction angle of 33.7◦, downthe Queen’s University flume, varying the water still depth from 0.05 to 0.50 m. The landslide flume atQueen’s University consisted of a 6.7 m long channel, with a slope of 30◦, connected to a 33 m long hor-izontal channel and a run-up zone of 2.4 m long with a slope of 27◦. The width of the prismatic channelwas 2.09 m, and the inclined section was covered in an aluminum plate coated with a coarse frictionalpaint, whereas the horizontal section had a smooth concrete bottom. The flume was instrumented with 9capacitance wave probes to measure the water surface elevation, an acoustic doppler current profiler anda camera to measure the velocity, and two additional cameras to measure the landslide flow thicknessand to record the behavior of the near-field wave.

Miller et al. (2017) found that the experiments with shallow water depths (h = 0.05 − 0.10m) gen-erated turbulent bores that evolved into a long train of waves and break during their generation. On thecontrary, the experiments with deep water depths (h = 0.25 − 0.50m) generated long and stable wavesthat did not break, but increased in amplitude when travelling across the near-field. The intermediatewater depths (h = 0.17− 0.20m) generated steep waves that broke at the end of the near-field zone. Theauthors observed that the theoretical time series of solitary wave solution expressed as

η = amsech2

[(√3

2

)√amh3

(x− ct)]

(17)

12

where c =√g (h+ am), closely represented the waves generated with the intermediate and deep water

depths, but did not represent the bores formed with shallow water reservoir depths.

Additionally, Bullard et al. (2019) generated impulse waves by releasing water down the Queen’s Univer-sity landslide flume, to evaluate the influence of highly mobile landslides. Bullard et al. (2019) varied thestill water depth from 0.15 to 0.65 m in 0.05 m increments, and found that, in the maximum amplitudecalculated with the hydrostatic momentum balance equation by Mulligan et al. (2017), the effectivelength of a high mobility slide can be expressed as Le = 1

2vscosα∆te, thus the maximum amplitude canbe calculated with

aq =

√h2 +

ρss (vscosα)2

ρg− h (18)

The equation was also in good agreement with the experimental observations of Miller et al. (2017).

Regarding fluidized flows, Bougouin et al. (2020) released a volume of glass beads, with diameter of65 µm and grain density of 2550 kg/m3, down an inclined channel of 1 m long, 0.8 m high and 0.2 mwide, with α = 15◦, connected to a horizontal channel of 7 m long. The model was instrumented withtwo cameras located where the still water crosses the inclined plane, and at 2.4 m from this point inthe direction of the horizontal channel. The inclined channel allowed the entrance of air to simulate thefluidized flow with a volume fraction of 0.56.

Bougouin et al. (2020) observed that, after the release, the flow accelerated until it reached a con-stant speed down the inclined channel. Then, the slide impacted the water and generated a jet, and awave with a large amplitude (the ratio between the wave amplitude and the slide thickness is up to 8),which broke quickly after the impact, followed by waves with smaller amplitudes. A turbidity currentwas also developed above the slide, after it penetrated the water, and the turbidity current propagatedthrough the whole horizontal channel.

Bougouin et al. (2020) observed, first, that for release volumes larger than 4.7 dm3 the wave ampli-tude reached a constant value, second, that the maximum wave amplitude was reached in less than 0.5s after the impact, and it evolved in a crest with constant celerity, and, third, that the wave amplitudepredictive models should include the vertical component of the slide velocity.

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Table 1. Properties of materials used to perform parametric studies of subaerial landslide-generated waves: particle shape, particle material, and particlediameter (d), and properties of landslides for all experiments: Froude number (Fr), slide thickness (s), slide velocity (us), porosity (n) and slide bulk density

(rhos). h: reservoir height, am maximum amplitude of the generated wave.

Reference Physical model Type of slide Part. shape Part. material d (mm)

Fritz, Hager & Minor (2004) VAW Granular 2D Cylinder PP-BaSO52 4Zweifel, Hager & Minor (2006) VAW Granular 2D Cylinder PP-BaSO52 4Heller & Hager (2010) VAW Granular 2D Cylinder PP-BaSO52 2, 4, 5, 8Miller et al. (2017) Queen’s University Granular 2D Sphere Ceramic 3Bullard et al. (2019) Queen’s University Fluid WaterBougouin, Paris & Roche (2020) Laboratoire Magmas et

Volcans, UCAGranular 2D Sphere Glass 0.065

Reference Fr (-) s (m) us(m/s) n (-) ρs(kg/m3)

Fritz, Hager & Minor (2004) 1.08-4.66 0.050-0.199 2.76-8.20 0.39 1620

Zweifel, Hager & Minor (2006) 1.25-4.89 0.075-0.189 2.60-8.77955, 1590,2111, 2640

Heller & Hager (2010) 0.86-6.83 0.053-0.249 2.06-8.34 590, 1720Miller et al. (2017) 8.25 0.035 4.5 0.38 1296Bullard et al. (2019) 2.07-5.37 0.02-0.06 4.19-8.11 1000Bougouin, Paris & Roche (2020) 0.03-0.074 0.44 1400

Reference h (m) am/h (-)

Fritz, Hager & Minor (2004) 0.3, 0.45, 0.675 0.25

(vs√gh

)1.4 (sh

)Zweifel, Hager & Minor (2006) 0.15, 0.30, 0.45, 0.60,

0.675

13

(vs√gh

) (sh

)1/2 ( msρwlh2

)1/3

Heller & Hager (2010) 0.15, 0.60 49

(vs√gh

)4/5 (sh

)2/5 ( msρwlh2

)1/5(cos[ 6

7α ])2/5

Miller et al. (2017),Mulligan & Take (2017)

0.05, 0.08, 0.10, 0.17,0.20, 0.25, 0.38, 0.50

√1 + 2ρsvscosαL

ρg∆teh2− 1, ρssvscosα

ρ√gh3

Bullard et al. (2019) 0.15-0.65

√h2+

2ρss(vscosα)2

ρf g

h − 1

Bougouin, Paris & Roche (2020) 0.131-0.385 0.15ln[(

vsgh

) (sh

) ( msρf bh2

)sinα

]+ 0.88

14

3.3.2 Submerged flows

Impulse waves generated by underwater landslides have been studied with laboratory experiments by twomeans: the release of a rigid or deforming mass down an inclined channel filled with water (Liu et al.,2005, Ataie-Ashtiani & Najafi-Jilani, 2008, Grilli et al., 2017, Romano et al, 2020), and by the collapseof a submerged granular column (Pinzon & Cabrera, 2019, Cabrera et al., 2020). Grilli et al. (2017)released a wedge of glass beads with a mass of 1.5-25 kg down an inclined channel with a slope of 30◦.The channel width was 0.25 m, and the still water depth was set between 0.32, and 0.37 m. Grilli et al.(2017) reported that two different waves are generated when the glass beads are released: one moving inthe same direction of the released material (seaward wave), and the other traveling onshore (landwardwave). Ataie-Ashtiani & Najafi-Jilani (2008) performed 120 laboratory tests in a 2.5 m wide channel witha variable inclination of 15-60◦. The authors used solid blocks with squared, triangular, and paraboliccross sections, and a granular material with a particle mean diameter of 7-9 mm, and a density between1800 and 2000 kg/m3 as a released mass. The authors found that the generated impulse waves werestrongly influenced by the bed slope angle, the still water depth position above the released mass, andthe released mass thickness, and are weakly affected by the landslide length and shape. Ataie-Ashtiani& Najafi-Jilani (2008) also found that the energy conversion from landslide into the generated waveswas influenced mainly by the bed slope and the initial submergence of the slide, showing that, when theinitial submergence of the slide decreased, the converted energy increased.

On the other hand, Cabrera et al. (2020) performed laboratory experiments with a planar setup, where acolumn of ceramic beads collapse generated impulse waves. The column used by the authors had a fixedaspect ratio of 1.5 and a variable relative depth of submergence Hw/H0, where Hw is the still water depth,and H0 is the column height. The authors found that the amplitude and propagation direction of thegenerated wave was mainly controlled by Hw/H0, where the seaward wave maximum amplitude increasedwhen Hw/H0 decreased, and the generation of landward waves was observed when 0.87 < Hw/H0 < 1.5.

15

3.4 Mass flows wave generation: Numerical simulations

3.4.1 Subaerial flows

The Navier-Stokes equations have been used to develop numerical models that simulate subaerial-landslide-induced tsunamis (Ataie-Ashtiani & Shobeyri, 2008, Abadie et al., 2010). These equationshave been depth averaged and the shallow water equations have been obtained to accurately describethe flow of a thin layer of material down an inclined plane impacting a water reservoir (Ma et al., 2015,Franz et al., 2020, Bregoli et al., 2020).

Ma et al. (2015) presented a model where a saturated landslide flowing down an inclined plane wassimulated with a granular flow model, coupled with a Coulomb frictional model, to compute the inter-granular stresses. The tsunami wave was simulated with a three-dimensional non-hydrostatic WAVEmodel described by Ma et al. (2012). Ma et al. (2015) observed that the model overestimated the waveamplitudes. In the near field, the overestimation of the amplitude of the leading wave was between 23.7and 42.7%, whereas in the far field this value reduced to ≈ 17%. The authors explained that this is dueto an overestimation of the slide front thickness of about 5 cm.

Franz et al. (2020) simulated landslide-tsunami waves using, also, the shallow-water equations witha Coulomb rheology model, but assuming the momentum flux between the landslide and the water asa perfectly elastic collision. The authors noted that this is not valid from a physical perspective butmay be useful in evaluating the momentum transfer between two cells, each with an assigned mass andvelocity. Thereby, the momentum transfer from the water to the slide and the momentum transfer fromthe slide to the water, during the rigid collision, was computed as

MTsdt = SF2(

1hρw

1sρs

) (uwb − usb) (19)

MTfdt = −Sf2(

1hρf

1sρs

) (usb − uwb) (20)

Where uwb is the water velocity in the x direction before collision, usb is the landslide velocity in the xdirection before collision, and SF = 0.145 (smax/h)1.465 is a fitting parameter to reproduce the experi-ments presented by Miller et al. (2017), where smax is the maximum landslide thickness.

Regarding the wave amplitude, Franz et al. (2020) noticed that this parameter was larger in the numer-ical model than in the laboratory experiments, because in the numerical simulations the wave did notbreak nor generated a wave train, thus energy was not lost on those processes.

Bregoli et al. (2020) presented a 1D model that solved the motion problem of a landslide impactinga water body by evaluating the energy transfer as

dEsdx

=dEfdx

dEddx

(21)

Where Es is the total landslide energy at impact, Ef is the energy loss due to friction between thelandslide and the bottom, and Ed is the drag energy transferred to the water body. The drag energy wascalculated by Bregoli et al. (2020) as Ed = Ew–Et where Ew is the total wave energy, and Et are thedissipative energies (turbulence, water surface tension breaking and compressibility). With a set of 20laboratory experiments made by Bregoli et al. (2017), Bregoli et al. (2020) calibrated the basal frictionangle, and the drag coefficient, and found that 52% of the total landslide energy was dissipated by basalfriction, 42% was dissipated by turbulence, water surface tension breaking, and compressibility, and only6% was converted into wave energy.

16

3.4.2 Submerged flows

Grilli et al. (2017) simulated tsunamis generated by submarine mass failures with NHWAVE, a 3Dnon-hydrostatic model. Then, FUNWAVE-TVD, a 2D nonlinear dispersive long wave Boussinesq model,was used to propagate the wave. For a deforming landslide, the granular medium was represented as adense fluid, or as a saturated Savage-Hutter mass type. The water was simulated as a layer above thegranular medium, and it was modeled by 3D Euler equations, while the bottom layer was modeled witha depth-integrated, dense Newtonian fluid, or granular flow.

Grilli et al. (2017) recreated laboratory experiments of submerged landslide generated waves by slidesmade of glass beads flowing down an inclined channel. The inclined channel had an inclination of 35◦, itwas connected to a horizontal channel, and both had a width, and length of 0.25, and 6.27 m. The glassbeads had a diameter of 4-10 mm and the still water depth was set to h = 0.32-0.37 m. The authorsapplied the granular flow, and the dense fluid models to simulate the glass beads. For the granularmodel, the Manning bottom friction coefficient was defined trough the internal, and basal friction angles(φ = 41◦ and δ = 24◦), and for the dense fluid model, the Manning bottom friction coefficient wasdefined as the dynamic viscosity µs. The void fraction 1− αs was set to 36.6%, and the internal frictionangle was slightly modified from 34 to 41◦. The grid resolution was set to ∆x = ∆y = 0.01m. Grilliet al. (2017) pointed out that the depth-integrated method did not reproduce the bulbous shape of thelandslide when it is flowing down the inclined channel, but they argue that this did not strongly affectthe wave generation.

The dense fluid results showed that the two leading waves were well simulated, whereas the trailingwaves showed increasing discrepancies. They also observed that the maximum root mean square (RMS)difference slightly increased with the horizontal distance from 7.2 to 8.4%. On the other hand, granularflow model results showed a better agreement between experiments in the far field, being the maximumRMS 10.5% in the numerical wave gage WG1, located at the start of the horizontal channel.

Grilli et al. (2017) also performed a sensitivity analysis of tsunami generation to slide submergence.The authors varied the still water depth between 0.31 and 0.35 m, for a wedge with a 0.29 m height.Grilli et al. (2017) observed that the leading wave amplitude increased nonlinearly as the slide submer-gence decreased. The nonlinearlity was explained by the authors by free surface effects, generated by thelarger wave amplitude-to-depth ratio, resulting in an additional increase in wave amplitude.

17

3.5 r.avaflow: Depth averaged methods

When describing the behavior of a landslide, the mass and momentum balance equations are the startingpoint in the model derivation. Pudasaini (2012), presented a model where the parameters describing thefluid phase are its density ρf , viscosity ηf , and isotropic stress distribution, while for the solid phase areits density ρs, internal friction angle φ, basal friction angle δ, and anisotropic stress distribution. Thus,the mass balance equations for the solid and fluid phases are, with subscript s and f , respectively,

∂αs∂t

+∇ • (αsus) (22)

∂αf∂t

+∇ • (αfuf ) (23)

where αi is the volume fraction, and ui is the velocity. Additionally, the momentum balance equationsare expressed as

∂

∂t(αsρsus) +∇ • (αsρsus × us) = αsρsf −∇ • αsTs + p∇αs + Ms (24)

∂

∂t(αfρfuf ) +∇ • (αfρfuf × uf ) = αfρf f–αf∇p+∇ • αfτf + Mf (25)

where f is the body force density, Ts is the Cauchy stress tensor, τf is the viscous stress tensor, p∇αsis the buoyant force, p is the fluid pressure, and M is the interfacial force density. This last term canbe expressed as the sum of the force associated with the viscous drag and the force due to the virtualmass, the forces that appear when solid and fluid phases interact. Additionally, because there exist aninterfacial momentum transfer, Ms = −Mf ,

Ms = MD + MVM = CDG (uf − us) |uf − us|+ CV GMd

dt(uf − us) (26)

where CDG is the generalized drag coefficient and CVMG is the generalized virtual mass coefficient. Thesetwo coefficients are a function of the volumetric fraction, and Pudasaini & Mergili (2019) expressed themas

CDG =αsαf (ρs − ρf )g

[UT {PF + (1− P)G}+ SP ]j(uf − us)|uf − us|j−1 (27)

where F = γ (αf/αs)3Rep/180, G = αM(Rep)−1, and P are parameters between 0 and 1 that combines the

solid-like and fluid-like drag contributions to flow resistance, 0 when solid particles are moving througha fluid and 1 when fluid flows through a dense packing of grains. UT =

√gd/γ and Rep = ρfdUT /ηf

are the particle Reynolds number, j is 1 or 2 when a laminar flow or a turbulent flow is considered,respectively, g is the gravitational acceleration, d is the particle diameter and γ = ρf/ρs. And

CVMG = αsρfCd

dt(uf − us) (28)

where the virtual mass coeficient C, is expressed as

C =N0vm(l + αns )− 1

αs/αf + γ(29)

To obtain a reduced two-dimensional debris flow model, Pudasaini (2012) used depth-averaging in thez direction, assuming that one of the other two dimensions of the flow are larger than its depth. Thedepth-averaging method consists in the integration of the mass, and momentum balance equations, be-tween the base and the flow surface, in order to remove their dependence on z dimension.

18

These equations have been implemented in r.avaflow, an open-source computational tool (https://www.landslidemodels.org/r.avaflow/), along with the finite difference method, to simulate the routing ofa mass flow from a release area down to a deposition area (Mergili et al., 2017).

r.avaflow allows the construction of a flow as a mixture of three phases: a coarse solid material, modelledas a Mohr-Coulomb continuum, a fine-solid phase, modelled as a Coulomb-viscoplastic material, anda fluid phase modelled as a viscoplastic material. Hence, the fluid and fine-solid phases are shear-ratedependent materials, whereas the coarse-solid phase is a shear-rate independent material. Because thefluid phase is composed of water and very fine particles (colloids, clay and silt), the material is viscousat high velocities and plastic at low velocities (e.g. during deposition).

19

4 Methodology

4.1 Subaerial flows

In this work, two sets of numerical simulations with r.avaflow 2.3 were performed: the first set where a drygranular flow impacts a still water reservoir, and the second where a saturated granular flow impacts thesame still water reservoir. The key input parameters to run the simulations in r.avaflow are the cellsize,the number of phases, the elevation raster map, the height raster map, density, viscosity, internal andbase friction angles of each phase, and the time step length control (Courant-Friedrichs-Lewy condition).All parameters used to run the simulations are summarized in Table 3 and in Table 2.

The experimental results performed by Miller et al. (2017), and the experimental results performedby Bullard (2018) were compared with the numerical results for the dry granular flow, and for the satu-rated granular flow, respectively. Thus, the Queen’s University landslide flume was reconstructed (Fig.9) as a raster map of 2 cm squared cells, which was used as the elevation model. The code to generate theraster maps is presented in Appendix A1. The landslide flume cross section was modified from a squareto a trapezoid, with a walls inclination of 85◦, to avoid numerical issues (Fig. 10). When a simulationwith vertical walls was tested, a progressive mass loss was observed, where half of the solids reached thereservoir (Fig. 11). Likewise, the phase 1 (solids) height, and the phase 3 (fluid) height were generatedwith r.lakefill as 2 cm raster maps (Appendix A2).

The height of the released material is 0.47 m for both dry and saturated cases, yet the phase 1 (solids)height for the saturated case varies between 0.23 and 0.34 m, and the phase 3 (fluid) height varies between0.23 and 0.13 m, thus fluid volume fractions between 0.5 and 0.73 are achieved. Also, the phase 1 (solids)maximum height is 0.3 m to take into account the porosity of 0.38 reported by Mulligan et al. (2017).

In view of the above, three sets of simulations were performed. In the first set, the released mate-rial corresponds only to the phase 1 raster map, whereas in the other two sets the released materialcorresponds to the raster maps of phases 1 and 3. In all three sets the variable is the reservoir depth (h),modelled as a phase 3 raster map. In Table 4 the reservoir depths (h), and solids to fluid ratio (P:W)for each set of simulations is displayed. The P:W ratio is calculated based on solids to fluid height. Forexample, a P:W of 0.64:0.36 means that in a pixel with a total release height of 0.47 m, 0.3 m correspondsto phase 1 (solids) and 0.17 m corresponds to phase 3 (fluid).

r.avaflow outputs are flow height and flow velocity in x and y, for each time step and for each phase. Toprocess the flow height outputs, the raster maps are loaded with the open function in Python3, and athreshold of 0.003 m (the diameter of the glass beads used in Miller et al. (2017) experiments) is applied,thus the solids and fluid heights below this value are defined as NaN.

After the fluid phase raster is loaded, two arrays are defined: one for the total fluid height or veloc-ity (rows f) and one where the fluid height or velocity at time t = 0 is subtracted from the total fluidheight or velocity (rows f am) at time t. The median and the standard deviation along the y axis (Fig. 9)is calculated for both fluid and solid phases arrays. The median was chosen over the mean value to havea better measure of the central tendency, because the last one is particularly susceptible to the influenceof outliers and skewed data.

The virtual mass raster for each time step is created by determining first the virtual mass coefficientCV GM , and then the convective derivative, second term in the left-hand side of Equation 26. CV GM iscomputed using the output height rasters from r.avaflow 2.3 as αs and αf , and using the following pa-

rameters: N0vm = 10, l = 0.12, γ =

ρfρs

= 1000kg/m3

2400kg/m3 = 0.42, n = 1. The convective derivative is calculated

20

as follows

d

dt(uf − us) =

dufdt− dus

dt=

(∂uf∂t

+ uf • ∇uf

)−(∂us∂t

+ us • ∇us

)(30)

If uf = (ufx, ufy) y us = (usx, usy)

d

dt(uf − us) =

∂uf∂t

+ ufx∂uf∂x

+ ufy∂uf∂y− ∂us

∂t− usx

∂us∂x− usy

∂us∂y

(31)

The output velocity rasters are used to calculate the time derivatives and the spatial derivatives bysubtracting ut+1 and ut, and by using the Numpy gradient function with axis=1 for the derivative in x,and with axis=0 for the derivative in y.

The drag force raster is created for each time step by calculating first the generalized drag coefficientCDG, using the output height rasters from r.avaflow 2.3 to calculate αs and αf , and the following param-etes: UT = 0.1 m/s, γ = 0.42, P = αns , with n = 1, F = γ(αf/αs)

3Rep/180, with Rep = 1, G = αM−1f ,

with M = 3, SP = ( Pαs + 1−Pαf

)K, with K = 1 m/s. The output velocity rasters from r.avaflow 2.3 are

used to calculate (uf − us)|uf − us|j−1, with j = 1.

Table 2. Input parameters to run mass flows simulations with r.avaflow 2.3. Inputs of type ”raster” areindicated in italics and the parameters modified to run the simulations of the present study are indicated in bold.

Default values were taken from r.avaflow 2.3 site (Mergili & Pudasaini, 2014-2021).

Symbol Input parameter Units Default value Used value

ρs,-,ρf density kg/m3 2700,1800,1000 2400,1800,1000φ′, δ,-,-,-,- friction degree 35,20,15,5,0,0 34,24,0,0,0,0

-,-,-,-,ηf ,-

viscositym2/s,Pa,m2/s,

Pa,m2/s,Pa-9999,-9999,-3.0,

-9999,-3.0,0.0-9999,-9999,-3.0,

-9999,-3.0,0.0ambient - 0.02,0.004,-7.0,0.0 0.02,0.004,-7.0,0.0transformation - 0.0,0.0,0.0 0.0,0.0,0.0

special-,-,-,-,m/s,-,-,m/s,

-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-

10,0.12,1,1,1,3,1,0.1,1,1,1,1,1,0,0,

0,1,1,1,10,0,0,1,1,1

10,0.12,1,1,1,3,1,0.1,1,1,1,1,1,0,0,

0,1,1,1,10,0,0,1,1,1dynfric degree,J 2.0,1000000 2.0,1000000controls - 0,0,0,0,0 0,0,0,0,0thresholds m,J,Pa,m 0.1,10000,10000,0.001 0.1,10000,10000,0.00001sampling - - -time s 10,300 0.05,15slomo - 1 1cfl -,s 0.04,0.001 0.4,0.001profile - - -ctrlpoints - - -reftime - - -phexagg - 1 1orthophoto - - -

21

Table 3. Input parameters to run mass flows simulations with r.avaflow 2.3. Mandatory inputs are highlightedin red and inputs of type ”raster” are indicated in italics. For the release heights (hrelease1, hrelease2, hrelease3and hrelease) at least one of the four input rasters is mandatory. Default values were taken from r.avaflow 2.3 site

(Mergili & Pudasaini, 2014-2021).

Symbol Input parameter Units Default value Used value

cellsize m - 0.02limiter - 1 1phases - m maoicoords m - -elevation m - Queen’s University landslide flume DEMhrelease1 m - Phase 1 (solids) heighthrelease2 m - -hrelease3 m - Reservoir + phase 3 (fluid) heighthrelease m - -rhrelease - - -vhrelease - - -trelease s - -trelstop s - -vinx m/s - -viny m/s - -

usx vinx1 m/s - -usy viny1 m/s - -

vinx2 m/s - -viny2 m/s - -

ufx vinx3 m/s - -ufy viny3 m/s - -

hentrmax1 m - -hentrmax2 m - -hentrmax3 m - -hentrmax m - -rhetrmax1 - - -vhentrmax m - -

φ′ phi1 ◦ - -phi2 ◦ - -

δ delta1 ◦ - -delta2 ◦ - -ny1 m2/s - -ny2 m2/s - -

ηf ny3 m2/s - -ambdrag - - -flufri ◦ - -centr - - -ctrans12 - - -ctrans13 - - -ctrans23 - - -impactarea - - -hdeposit m - -hydrograph - - -hydrocoords m - -

22

Figure 9. Scheme of the Queen’s University landslide flume. Taken from Miller et al. (2017).

Figure 10. Scheme of the Queen’s University landslide flume, released material and reservoir reconstructed asraster maps. hT : total height of the released material.

23

Figure 11. Normalized volume change of the solid phase for each time step of the simulation. The vertical blackline shows the time at which the solids impact the reservoir. WS: walls slope.

Table 4. Variable ranges for each set of simulations.

ID Flow type Variable Range

sim1 Dry h 0.05, 0.08, 0.10, 0.15, 0.17, 0.20, 0.25, 0.30, 0.35, 0.38, 0.40,0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.95 m

sim2 Saturated h 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65 m

sim3 Saturatedh 0.15, 0.30, 0.45, 0.60 m

P:W 0.50:0.50, 0.75:0.25

24

4.2 Submerged flows

To evaluate the generation of landward waves generated by submerged landslides, two procedures werefollowed: one using r.avaflow 2.3, and the other using a planar model for granular column collapsesdescribed in detail by Pinzon & Cabrera (2019) and outlined in Figure 12. The numerical simulationsperformed with r.avaflow recreated the experiments made by Grilli et al. (2017) at the Ecole Centralede Marseille (IRPHE), where 2 kg of glass beads were released from a triangular reservoir located on aninclined channel of 35◦connected to a horizontal channel. The dimensions of the IRPHE experimentalsetup are shown in Figure 13 where the water depth in Grilli et al. (2017) is set to 0.332 m. As for thesubaerial simulations, the inclination of the channel walls was modified from 90◦to 85◦in order to avoidmass losses. However, a maximum volume loss of about 35% is still present, thus an inclination of thechannel walls of 80◦was tried, and the same maximum volume loss was observed (Fig. 14). Hence, theresults of the numerical simulations with r.avaflow, for the submerged landslide, are discarded from thepresent study.

Figure 12. Planar setup used for the collapse of granular columns. Taken from Pinzon & Cabrera (2019).

Regarding the laboratory experiments, a parametric study varying the column level of submergenceHw/H0 was performed. In the column level of submergence, Hw corresponds to the water height measuredfrom the column base, and H0 corresponds to the column height. The granular column (Fig. 15) hada fixed aspect ratio (H0/L0) of 0.16 and the column level of submergence Hw/H0 was varied from 0.8to 1.4, range for which landward waves were observed in Cabrera et al. (2020). The ceramic beadshad a diameter between 1.8 and 2 mm, a density of ρcb = 3600 kg/m3, and were manufactured bySigmund-Linder GmbH. The acrylics containing the ceramic beads were back-illuminated with a 3000lm LED panel and the collapses were recorded with a Mikrotron MotionBLITZ Cube 4 camera at aframe rate of 500 fps. The images were corrected to avoid a perspective error, thus orthogonal imageswere produced. Then, the orthogonal images were calibrated and spatio-temporal images were created.Spatio-temporal images were made by extracting a column of interest for each orthogonal image, andthen stacking the extracted pixels (Cabrera & Wu, 2017). Because the main analysis was to evaluate thecorrelation between the column level of submergence and the horizontal coordinate where the maximumlandward wave trough (a−m) is found, five spatio-temporal images were generated for each collapse: thefirst three in arbitrary points where the landward wave is observed, and the last two near the point wherethe largest trough is found. The maximum trough (a−m) is thus the largest of the five troughs.

25

Figure 13. Ecole Centrale de Marseille (IRPHE) landslide flume for submerged experiments. Taken from Grilliet al. (2017).

Figure 14. Normalized volume for each time step of the submerged simulations based on the experiments madeby Grilli et al. (2017). CFL: Courant–Friedrichs–Lewy condition, MHF: threshold for minimum flow height, WS:

walls slope.

26

x (mm)

y (m

m)

45

90

290145

45

90

y (m

m)

Seaward

Landward

xf,crown

x(am-)

am-

Sliding gate

HwH0

Figure 15. Initial and final state snapshots of the experiment for Hw/H0 = 1.4, with dimensions and coordinatesystem. a−m corresponds to the landward wave largest trough.

27

5 Results and discussion

5.1 Subaerial flows

5.1.1 Landslide characteristics at impact

Figure 16 and Figure 17 show, as a solid line, the median of the solids thickness (s) and the median ofthe velocity magnitude (〈vs〉), for each time step, measured at the intersection of the still water surfaceand the inclined channel. I calculated the median of s and 〈vs〉 along the cross section of the channel.Figure 16 and Figure 17 also display the standard deviation of s and vs as a shadow with transparency.First, it can be noted that the median of s and 〈vs〉 is the same for all reservoir depths (h), indicatingthat the landslide behaves similarly prior to impact, and guaranteeing the repeatibility of simulations.Second, for both dry and saturated cases, the thickness rapidly increases after impact, from less than 1mm up to 5 mm in approximately 1 s for the dry case, and from approximately 1 mm up to 5 mm in lessthan 0.3 s for the saturated case. Then, for both dry and saturated cases, s decreases to the thicknessbefore the impact in approximately 2 s. Finally, 〈vs〉 remains constant throughout the impact for thedry case, and shows a slight decrease from 3 m/s to 1 m/s for the saturated case.

The numerical results follow the tendency of the laboratory observations, thus the momentum trans-fer between the solids in the flow and the water in the reservoir is well represented in the numericalsimulations.

0.00

0.02

0.04

0.06

0.08

0.10

s (m

)

1 0 1 2 3x (m)

0.0

0.5

1.0

heig

ht (m

)

A

h = 0.95 mh = 0.50 mh = 0.25 mh = 0.05 m

0 1 2 3 4 5t (s)

0

2

4

6

8

v s (m

/s)

h = 0.95 mh = 0.50 mh = 0.25 mh = 0.05 m

h = 0.25 mMiller et al. (2017). h = 0.25m

0 1 2 3 4 5t (s)

h = 0.25 mMiller et al. (2017). h = 0.25m

Figure 16. Thickness (s) and velocity (vs) of the solids measured at point A. Dry case.

28

0.00

0.02

0.04

0.06

0.08

0.10

s (m

)

1 0 1 2 3x (m)

0.0

0.5

1.0

heig

ht (m

)

A

h = 0.95 mh = 0.60 mh = 0.40 mh = 0.20 m

0 1 2 3 4 5t (s)

0

2

4

6

8

v s (m

/s)

h = 0.95 mh = 0.60 mh = 0.40 mh = 0.20 m

h = 0.20 mBullard (2018), sat.

0 1 2 3 4 5t (s)

h = 0.20 mBullard (2018), sat.

Figure 17. Thickness (s) and velocity (vs) of the solids measured at point A. Saturated case, P:W 0.73:0.27.

5.1.2 Wave characteristics

Figure 18 shows time snapshots for a variable solids to fluid volume ratio (P:W). t* indicates the timeafter impact, thus Figure 18 shows, first, that the waves generated by the saturated flow propagates fasterthan the wave generated by the dry flow, being the wave generated by the landslide with 73% of solidsthe faster one; second, that the wave generated by the landslide with 50% of solids get progressivelycloser to the faster wave, meaning that the wave generated by the landslide with more fluid will reach thefar field earlier. Finally, that the relation between the solids to fluid ratio with the wave characteristicsin the near-field is non-trivial: the slower wave is the one generated by the dry flow, the faster wave isthe one generated by the flow with 73% of solids, the wave generated by the dry flow, and the flow with64% of solids reach the same celerity after t*=2.5 s.

It can be also observed that the landslide experiencing the strongest phase separation at the slide front isthe one with 50% solids. This means that at impact, and at the front (2 m uphill impact point), one thirdof the channel is water-dominated, one third is solids-dominated, and the central part does not have anyof the phases. On the other hand, the flow with 64% of solids experiences the strongest phase separationat the slide tail, and the front it is dominated by the fluid phase, even though the reservoir is first reachedby the solid phase. Lastly, the flow with 73% of particles experiences the weakest overall phase separa-tion, and the greatest influence of fluid at the slide front. Also, Figure 18 shows that the wave generatedby the dry landslide breaks before the waves generated by the saturated landslides, and that the tail andfront slopes of the generated bores are nearly identical for all cases. Refer to https://bit.ly/37ny564

for video simulations.

29

0.0

0.5

1.0 t*=0.00sP:W 0.50:0.50P:W 0.64:0.36P:W 0.73:0.27Dry

t*=0.50s

0.0

0.5

1.0 t*=1.00s t*=1.50s

0.0

0.5

1.0 t*=2.00s t*=2.50s

2 0 2 4 60.0

0.5

1.0 t*=3.00s

2 0 2 4 6

t*=3.40s

x (m)

heig

ht (m

)

Figure 18. Snapshots of landslide impact over time after impact (t*) for a still water depth h=0.20 m and avariable solids to fluid ratio (P:W). Solid line: water surface elevation, dashed line: solids height.

Regarding the wave propagation, Figure 19 shows that for shallow water depths (0.05-0.10 m), thewave breaks in the experiments closely after the maximum amplitude is reached. This behavior is ob-served in the numerical results for the dry case (Fig. 20), when h is greater than 0.08 m, and for allsuccessive depths. Figure 20 and Figure 21 show the median of the wave traces calculated along the y axis.

For the dry case, and for relatively deep water depths (0.25 - 0.50 m), the numerical am is reachedin the inclined channel (x < 0m), and as the wave moves further into the horizontal channel the wave-length increases. For the saturated case (Fig. 21), am is reached in the horizontal channel for still waterdepths h between 0.15 and 0.65 m, and in the inclined channel for h = 0.95 m. Regarding the wavelengthL, Figure 22(a) shows that, for the dry case, L increases between 0.15 and 2 times faster in the numericalsimulations than in the experiments because bores are generated in the simulations instead of solitarywaves. For all still water depths the wave propagates along the horizontal channel as a bore that has asteeper tail for shallow water depths than for intermediate and relatively deep water depths. Similarly,for the saturated case, Figure 22(b) shows that the wavelength increases with time at the same rate thanfor the dry case.

30

Figure 19. Evolution of the water surface over time with distance from the beginning of the horizontal channelat x = 0m at 15 Hz. Taken from Miller et al. (2017).

The difference between the wave type generated during the impact (bores in the numerical simulationsand solitary waves in the experiments) is related to the mass propagation in the method. Because it is notat the fluid molecule level, the water cannot move in circles and the generated wave must be translatoryand irrotational (i.e. solitary waves cannot be generated). Also, due to the constant vertical velocityprofile assumed in the depth averaged method the wave shape is the same throughout the simulation.

It is worth noting that the depth averaged method implemented in r.avaflow is not capable of repro-ducing the air entrapment that causes the wave to break. Thus, the generation mechanism of the boresin the experiments and the generation mechanism of the bore-like waves in the simulations is not thesame. Even though, for intermediate and relative deep water depths the wave shape is dissimilar, fortsunami hazard assessment the wave amplitude and celerity are the main parameters to calculate retainingstructures heights, forces over infrastructure, and evacuation times.

31

0.0

0.1

0.2 h = 0.05m h = 0.08m

0.0

0.1

0.2 h = 0.10m h = 0.17m

0.0

0.1

0.2 h = 0.20m h = 0.25m

1 0 1 2 3 40.0

0.1

0.2 h = 0.38m

1 0 1 2 3 4

h = 0.50m

x (m)

(m)

Figure 20. Evolution of the water surface over time with distance from the beginning of the horizontal channelat x = 0m at 20 Hz, from the time of impact to 1.6 s after the impact. Dry case.

0.00.10.20.3 h = 0.15m h = 0.20m

0.00.10.20.3 h = 0.25m h = 0.30m

0.00.10.20.3 h = 0.40m h = 0.55m

2 0 2 4 6 80.00.10.20.3 h = 0.65m

2 0 2 4 6 8

h = 0.95m

x (m)

(m)

Figure 21. Evolution of the water surface over time with distance from the beginning of the horizontal channelat x = 0 m at 10 Hz, from the time of impact to 3 s after the impact. Saturated case.

32

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4t (s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

L (m

)h (m)

0.050.250.50

r.avaflowMiller et al. (2017)

(a) Solid lines: numerical simulations (dry case), dashedlines: experiments performed by Miller et al. (2017).

(b) Solid lines: numerical simulations (saturated case).

Figure 22. Evolution of wavelength L in time.

Figure 23 shows that the wave propagation is slower in the numerical simulations than in the labora-tory experiments, although, the difference in time decreases as h increases, and it is lower at the near-fieldprobe of x = 2 m. Also, the time series show that the wave shape obtained in the laboratory experimentsis well represented by the numerical simulations, especially in the shallow water depths (h=0.05 andh=0.10) where the wave becomes less symmetrical with distance. Hence, the difference in the relativepeak amplitude η/h is between -0.023, for h = 0.05 m at x = 2 m (the numerical wave peak is 2% greaterthan the experimental wave peak), and 0.45 for h = 0.10 m at x = 25 m (the numerical wave peak is65% lower than the experimental wave peak).

For the saturated flow, η/h shape is well represented for x between 0 and 17 m where the wave asymmetryincreases with x in both experimental and numerical results, and the numerical peak relative amplitudeis between than 25 and 35% smaller than the experimental η/h (Fig. 24). However, after x = 17 m, theexperimental wave becomes more symmetric with x and the peak relative amplitude for the simulationsis 50% smaller than the experimental η/h. One of the reasons for the faster decrease in η/h in thenumerical simulations is the flume walls inclination. Because the walls are not completely vertical, thetotal water volume in the channel is up to 4% greater in the numerical setup than in the experimentalsetup, meaning more mass available to move an thus the need of a greater momentum imparted by thelandslide to generate the same wave amplitude. Finally, despite the faster propagation of the wave in thelaboratory than in the simulations, the difference between the time at which the peak in the normalizedsurface elevation (η/h) is reached (t∗) differs only between 4 and 16% (Fig. 25), thus a factor of safetyof 1.2 is proposed to be used when calculating the water surface arrival time.

33

0.50.00.51.0

/h (-

)x = 2m h = 0.05 m

Miller et al. (2017), dry, h = 0.05m

0.50.00.51.0

/h (-

)

x = 15m h = 0.05 mMiller et al. (2017), dry, h = 0.05m

0 5 10 15 20 25 30time (s)

0.50.00.51.0

/h (-

)

x = 25m h = 0.05 mMiller et al. (2017), dry, h = 0.05m

(a) h=0.05 m

0.50.00.51.0

/h (-

)

x = 2m h = 0.10 mMiller et al. (2017), dry, h = 0.10m

0.50.00.51.0

/h (-

)

x = 15m h = 0.10 mMiller et al. (2017), dry, h = 0.10m

0 5 10 15 20 25 30time (s)

0.50.00.51.0

/h (-

)

x = 25m h = 0.10 mMiller et al. (2017), dry, h = 0.10m

(b) h=0.10 m

0.50.00.51.0

/h (-

)

x = 2m h = 0.20 mMiller et al. (2017), dry, h = 0.20m

0.50.00.51.0

/h (-

)

x = 15m h = 0.20 mMiller et al. (2017), dry, h = 0.20m

0 5 10 15 20 25 30time (s)

0.50.00.51.0

/h (-

)

x = 25m h = 0.20 mMiller et al. (2017), dry, h = 0.20m

(c) h=0.20 m

0.50.00.51.0

/h (-

)

x = 2m h = 0.50 mMiller et al. (2017), dry, h = 0.50m

0.50.00.51.0

/h (-

)

x = 15m h = 0.50 mMiller et al. (2017), dry, h = 0.50m

0 5 10 15 20 25 30time (s)

0.50.00.51.0

/h (-

)

x = 25m h = 0.50 mMiller et al. (2017), dry, h = 0.50m

(d) h=0.50 m

Figure 23. Time series of relative water surface elevation observed at numerical and experimental wave probesat x = 2.3 m, x = 15 m, and x = 25 m. Dry case. Time values are modified such that t = 0 s corresponds to the

time at impact.

1.0 0.5 0.0 0.5 1.0t* (s)

0.0

0.5

1.0

1.5

/h (-

)

x = 2.0m

1.0 0.5 0.0 0.5 1.0t* (s)

x = 9.0m

1.0 0.5 0.0 0.5 1.0t* (s)

x = 22.0m h = 0.20 m, P:W 0.73:0.27Bullard (2018), h = 0.20 m

Figure 24. Time series of relative water surface elevation. t∗ corresponds to the adjusted time such that themaximum amplitude is located at t∗ = 0.

34

Figure 25. Normalized maximum amplitude time t∗ for numerical simulations (r.ava) and experiments (exp).timp corresponds to the time of impact.

5.1.3 Deposit shape

From Figure 26 to Figure 28 the median of the slide solid phase, calculated along the y axis, is presentedas a solid line, and the standard deviation is presented as a shadow of the same color. For both dry andsaturated cases, the Figures show that the volume of the numerical solid phase stopping in the inclinedchannel increases with the still water depth, and that the deposits became thinner. This kind of shapediffers from the lobate shape of the deposits observed in the laboratory. However, the runout in thesimulations Lf,r.ava differs from the experimental runout Lf,exp in less than 20% (Fig. 30), thus a factorof safety of 1.2 when calculating this value is also suggested.

The difference in shape between numerical and laboratory experiments may be related, primarily, tothe value of the basal friction angle δ. Pudasaini & Hutter (2007) perform a series of numerical simula-tions where δ is varied from 23 to 30◦and observe that the run-out distance L and the deposit shape arevery sensitive to changes in δ, being L twice the distance when 23◦was used over 30◦. In a lesser extent,the deposits may be dissimilar due to a variation in the internal friction angle. The variation of the anglein the laboratory experiments occur due to the collisions between the solid particles, which generate anincrease in the mean distance between them (Pudasaini & Hutter, 2007). This allows the solid materialto slightly travel further into the horizontal channel. Also, the water penetrates into the slide voids,allowing for a lubrication of the particles and a decrease in the effective forces between bead-bead, andbead-channel surface. For instance, hydroplaning may occur in the laboratory experiments (Blasio et al.,2004).

35

0.5 0.0 0.5 1.0 1.5 2.0 2.5x (m)

0.0

0.1

0.2

0.3

0.4

0.5

0.6he

ight

(m)

h = 0.20 mh = 0.17 mh = 0.10 mh = 0.08 mh = 0.05 m

(a) Numerical deposits (b) Experimental deposits. Taken from Miller et al. (2017)

Figure 26. Shape of the partially submerged deposits (h=0.05-0.20m). Dry case.

0.5 0.0 0.5 1.0 1.5 2.0 2.5x (m)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

heig

ht (m

)

h = 0.50 mh = 0.38 mh = 0.25 m

(a) Numerical deposits (b) Experimental deposits. Taken from Miller et al. (2017)

Figure 27. Shape of the submerged deposits (h=0.25-0.50m). Dry case.

When varying the solids to fluid ratio P:W, the saturated landslides deposits display a more lobateshape and stop further into the horizontal channel than the deposit for the dry flow (Fig. 29). However,the median of all three saturated deposits fall in the same standard deviation area, thus it is not possibleto identify the solids to fluid ratio P:W based on the deposit shape or runout.

36

2 1 0 1 2 3 4x (m)

0.0

0.2

0.4

0.6

0.8

1.0he

ight

(m)

h = 0.95 mh = 0.60 mh = 0.40 mh = 0.20 m

2 1 0 1 2 3 4x (m)

Bullard (2018), sat, h = 0.20 mh = 0.20 m

Figure 28. Shape of numerical landslide deposits. Saturated case P:W 0.73:0.27.

1 0 1 2 3 4x (m)

0.0

0.2

0.4

0.6

0.8

1.0

heig

ht (m

)

P:W 0.50:0.50P:W 0.64:0.36P:W 0.73:0.27Dry

Figure 29. Shape of numerical landslide deposits for h = 0.20 m.

37

0.1 0.2 0.3 0.4 0.5h (m)

0.9

1.0

1.1

1.2

1.3

1.4

1.5

L f,r

.ava

/Lf,

exp (

-)

Dry landslideSat. landslide, P:W 0.73:0.27

Figure 30. Difference in deposit runout Lf for each reservoir depth.

5.1.4 Maximum amplitude

Figure 31 and Figure 32 show the maximum wave amplitude (am), with its corresponding standard devia-tion, calculated along y axis for the dry and saturated P:W 0.73:0.27 cases. For the dry case, am increaseswith h for shallow still water depths (h = 0.025-0.15 m), and then decreases for the intermediate stillwater depths (h = 0.20 - 0.38 m). Then, starting from h = 0.5 m, am reaches a constant value of ≈ 0.12m. For the saturated case, the maximum amplitude shows a constant behavior where the median of amfor all still water depths is at least 5 cm larger than am for the dry case. Also, the standard deviation ofam for the saturated flows is between one and three times greater than the standard deviation of am forthe dry case because, prior to impact, the water in the flow deforms the solid phase resulting in a hetero-geneous front (both in height and solid fraction) and thus in a more uneven water surface at the reservoir.

Compared with the experimental am, for h between hcr and 0.25 m, the median of the numerical max-imum amplitude is maximum 35% lower, and the standard deviation does not include the experimentalvalues, thus using a factor of 1.5 when calculating the maximum wave amplitude generated in interme-diate still water depths is proposed. The reason for the median of numerical am to be lower for thesestill water depths may be related to the fluid type r.avaflow is simulating (a mixture of water and fineparticles rather than only water) as explained in Section 5.1.2. Figure 33 shows that the lower maximumamplitude for all h is observed for a dry flow, and for the saturated cases the median of am increases withthe slide volume fraction. However, am for the saturated cases groups such that all standard deviationsoverlap. There is no statistical difference between the maximum amplitude observed in the simulationswhen the saturated landslide initial fluid to solid ratio varies.

38

0.0 0.2 0.4 0.6 0.8 1.0h (m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

a m (m

)

Hydrostatic MBHydrodynamic MBBreaking limitContinuity limithcr

Miller et al. (2017)Avaflow dry median

0 1 2 3 4 5 6aq/h (-)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

a m/h

(-)

Figure 31. Maximum amplitude for each reservoir depth. MB: momentum balance. Dry case.

0.0 0.2 0.4 0.6 0.8 1.0h (m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

a m (m

)

0 1 2 3 4 5 6aq/h (-)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

a m/h

(-)

Breaking limitContinuity limitBullard (2018)Avaflow sat. median, P:W 0.73:0.27

Figure 32. Maximum amplitude for each reservoir depth. MB: momentum balance. Saturated case.

0.0 0.2 0.4 0.6 0.8 1.0h (m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

a m (m

)

0 1 2 3 4 5 6aq/h (-)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

a m/h

(-)

Breaking limitContinuity limitP:W 0.50:0.50P:W 0.64:0.36P:W 0.73:0.27Dry

Figure 33. Median of the maximum amplitude for each reservoir depth obtained with r.avaflow 2.3.

39

Figure 34 shows, first, for both dry and saturated cases, that am along with its standard deviationdecreases with x. In average, for all h, am is 2.5 times lesser at x = 27 m than at x = 2 m for thesaturated case, displaying a nonlinear relationship where a greater decrease is observed for shallow stillwater depths. For the dry case, the standard deviation decreases, in average, 5.5 times comparing thevalues at x = 27 m, and the values at x = 2 m. The relationship is also nonlinear and a greater decreaseis observed for shallow still water depths. Second, Figure 34 shows that the curves of both dry andsaturated cases draw closer with x, however the standard deviation leaves space for a distinction betweenboth to be made. Third, different from the experimental results obtained by Bullard (2018), am does notincrease and then decrease with h, but it reaches a state where only increases instead. Also, excludingthe plot at x = 33 m, where reflected waves are influencing the results, Figure 34 shows that the breakinglimit in the numerical simulations is lower than the breaking limit calculated with the experimentalresults. With a regression using the value of am for 0.025 m, in the dry case, and the value of am for 0.15m, in the saturated case, at x = 27 m, the breaking limit is 0.33, near half the am/h reported by Bullard(2018).

0.00

0.05

0.10

0.15

0.20

0.25

0.30

a m (m

)

x = 2m

r.avaflow dry medianr.avaflow sat. median, P:W 0.73:0.27

x = 3m x = 6m

0.00

0.05

0.10

0.15

0.20

0.25

0.30

a m (m

)

x = 9m x = 11m x = 17m

0.0 0.2 0.4 0.6 0.8 1.0h (m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

a m (m

)

x = 22m

0.0 0.2 0.4 0.6 0.8 1.0h (m)

x = 27m

0.0 0.2 0.4 0.6 0.8 1.0h (m)

x = 33m

Figure 34. Maximum amplitude recorded at nine different digital probes along the horizontal channel. Thedotted line indicates the breaking limit found by Bullard (2018) where am/h = 0.6

40

5.1.5 Virtual mass evolution

Figure 35 shows the drag force magnitude (CD) and the virtual mass (MVM ). These two forces werecalculated at the channel section where am was observed, and then the forces were normalized by thesolids weight of the landslide slice in the same section. Figure 35 shows the drag force magnitude andthe virtual mass against the flow Froude number, calculated as Fr = vs√

gh, where vs is the median of

the solids front velocity. Figure 35 shows, first, that 39 of the 42 numerical flows are subcritical (93%),meaning that the wave celerity is greater than the slide velocity in most cases. Second, the values clusternear M/W = 0, meaning that the drag and virtual mass forces are negligible at this point comparedto the solids’ slide weight (MVM , CD <3%). Also, MVM is negative in 15 of the 42 simulations (36%),meaning that the acceleration of the solid phase is greater than the acceleration of the fluid phase and/orthat the fluid is decelerating.

Regarding negative virtual mass forces, Mciver & Evans (1984) observe this when free-surface effectsplay a major role in the fluid behavior. This effects are found to be important when the submergencedepth of the body is sufficiently small such that the kinetic energy of the fluid above the body is verysmall. For a cylinder, Mciver & Evans (1984) relate the kinetic and potential energy with the added massas T − V = 1

4AU2 where T and V ar the kinetic and potential energies, A is the added or virtual mass

and U is the velocity of the fluid. Negative virtual mass implies, considering the mass balance equation,that the relative velocity between solid and fluid phases is decreasing.

0.0 0.5 1.0 1.5 2.0Fr (-)

0.01

0.00

0.01

0.02

0.03

C D,M

VM/W

(-)

P:W0.50:0.500.64:0.360.73:0.27Dry

CD/WMVM/W

Figure 35. Normalized drag and virtual mass forces vs. Froude number for dry and saturated numericalsimulations. W: weight of the slice accounting for the row where the maximum wave amplitude am is observed.

41

Figure 36(a) shows the maximum virtual mass observed in each simulation, and the drag force atthe time the maximum virtual mass is observed. Similarly, Figure 36(b) shows the maximum drag forcecalculated in each simulation, and the virtual mass at the time maximum drag force is observed. Figure36(a) shows that the virtual mass is in average 120 times greater than the drag force, and even when themaximum drag force is observed, the virtual mass is up to 35 times greater than CD,max (Fig. 36(b)).This suggests that the virtual mass is a relevant force in the momentum balance equations, and it shouldnot be neglected from the model scheme.

0.0 0.5 1.0 1.5 2.0 2.5Fr (-)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

C D,M

VM,m

ax/W

(-)

P:W0.50:0.500.64:0.360.73:0.27Dry

CD/WMVM/W

(a) Virtual mass.

0 1 2 3 4Fr (-)

0.01

0.00

0.01

0.02

0.03

C D,m

ax,M

VM/W

(-)

P:W0.50:0.500.64:0.360.73:0.27Dry

CD/WMVM/W

(b) Drag force.

Figure 36. Maximum virtual mass (MVM,max) and drag force (CD,max).

42

5.2 Submerged flows

Figure 37 shows the time frames for the collapse with Hw/H0 = 1. The water level moves below the initialstill water level Hw at the same time that the column starts to collapse, and it reaches the same level atthe same time that all the particles cease their movement (Fig. 38), suggesting that all the moving massis tsunamigenic. As Hw/H0 increases the landward wave becomes marginal so that it can be observedin the spatio-temporal image only. Figure 39 shows that the maximum landward wave trough amplitudea−m decreases with Hw/H0, which is in agreement with the observations made by Cabrera et al. (2020).However the authors observe a maximum a−m for Hw/H0 = 1 of 10.6 mm, contrasting with the 4 mmobserved in the present study, due to the greater mobilized granular mass in the tall column collapse(H0/L0 = 2.5) used by Cabrera et al. (2020). It is worth noting that a−m for Hw/H0 < 1 seems to beinfluenced by capillary effects.

a) t = 0 s

b) t = 0.06 s

c) t = 0.12 s

d) t = 0.24 s

e) t = 0.48 s

f) t = 0.60 s

0 1 2 cm

Figure 37. Time frames for the collapse with Hw/H0 = 1. a) shows the release time, c) displays the maximumlandward wave trough amplitude (a−m) and f) shows the granular mass final position.

Figure 40(a) shows that the relative horizontal distance at which the maximum amplitude is observed(x(a−m)/xf,crown) decreases slightly with Hw/H0, but the normalized maximum trough amplitude val-ues do not vary from each other more than 12%. Furthermore, all x(a−m)/xf,crown values group near1, meaning that the maximum negative trough amplitude occurs near the final horizontal coordinate ofthe deposit’s crown, similar from tall granular columns, where x(a−m) approaches the coordinate origin(Cabrera et al., 2020), because particles at x = 0 are also moving. Figure 40(b) shows that x(a−m) is inmost cases less than the initial column length (in 9 out of 12 experiments), and there is no relationshipbetween the relative horizontal distance at which the maximum amplitude is observed (x(a−m)/L0) andthe column level of submergence Hw/H0.

The next steps in the research of landward waves is, first, to develop a momentum transference analysis inwhich a relationship between the mobilized granular mass and the maximum negative trough amplitudea−m could be established. Then, the experimental results need to be compared with the analytical resultsto evaluate the deviation and establish if the analytical expression could be used to an approximationfor tsunami hazard assessment. Also, additional experiments considering the same range of parametersin the present study should be performed to compare the results and evaluate the repeatability of the

43

experimental procedure. Finally, additional experiments varying the aspect ratio H0/L0 should be per-formed to evaluate if the maximum negative trough amplitude a−m does not change its position with thecolumn level of submergence, if it is observed in the deposit’s crown final coordinate, and how its positionchanges with the column initial length.

b) x = 256 mma) x = 248 mm

c) x = 264 mm d) x = 268 mm

e) x = 272 mm

Hw

yw(t)

Figure 38. Spatio-temporal images for Hw/H0 = 1 at different horizontal distances.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Hw/H0 (-)

0

2

4

6

8

10

12

14

a m (m

m)

Cabrera et al. (2020)

Figure 39. Landward wave maximum trough amplitude (a−m) as a function of the column level of submergence(Hw/H0).

44

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Hw/H0 (-)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4x(

a m)/x

f,cr

own (

-)

(a) xf,crown corresponds to the final horizontal coordinate ofthe crown deposit.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Hw/H0 (-)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x(a m

)/L0 (

-)(b) L0 corresponds to the initial column length.

Figure 40. Relative position at which the landward wave maximum trough amplitude is observed x(a−m) as afunction of the column level of submergence (Hw/H0).

45

6 Conclusions and future perspectives

6.1 Subaerial flows

In this study, the laboratory experiments performed by Miller et al. (2017) were recreated using r.avaflow.The comparison between experimental and numerical results shows that r.avaflow is an accurate soft-ware to perform an approximation for tsunami hazard assessment, as the maximum amplitudes (am),and the numerical water surface elevation (η) time series are similar to the experimental observationsmade by Miller et al. (2017) and Bullard (2018). The highly accurate numerical results for am and ηderive from the equivalent numerical and experimental landslide thickness and velocity at impact, andfrom the virtual mass force taken into account in the mass flow model approach. For hazard assessmentpurposes, a factor of safety of 1.5 for the intermediate still water depths (h = [0.18, 0.25] m) maximumamplitudes am, and a factor of safety of 1.2 for all still water depths time of arrival and solids runout isproposed. Also, to improve the correspondence between deposit shapes, a parametric study in which thebasal friction angle δ and the maximum value of passive earth pressure coefficient Kp is varied shouldbe performed. Because r.avaflow models a completely discrete process using continuum mechanics and afew input parameters, the simulations are low-time consuming compared with other numerical methods(e.g. discrete element methods) and real-size event simulations are approachable to perform a hazardassessment.

Simulations varying the solids to fluid ratio of the released mass were also performed. The resultsshows that it is possible to distinguish between maximum wave amplitudes generated by dry and satu-rated landslides, although it is not possible to differentiate between maximum wave amplitudes generatedby saturated landslides with different solids to fluid ratios. Additionally, virtual mass is negative at theposition and time where am is observed for all saturated simulations, meaning that the fluid inside thelandslide dampens the momentum transfer by producing a relative deceleration between solids in thelandslide and water in the reservoir. Nevertheless, the simulations generating a faster propagating waveare the simulations with a saturated slide, when compared with a dry slide.

6.2 Submerged flows

Laboratory experiments of a submerged wide granular column collapse using a planar setup were per-formed. The experiments show an inverse relation between the maximum wave trough a−m and the columnlevel of submergence Hw/H0, and a constant relation between the position at which the maximum land-ward wave trough is observed x(a−m), relative to the deposit crown and the initial column length, andthe column level of submergence. Similar from tall granular columns, the position at which the maxi-mum landward wave trough is observed corresponds to the horizontal coordinate at which the depositinclined profile starts (i.e. near the final horizontal coordinate of the crown deposit xf,crown). With theseexperiments results and with the results of additional experiments varying the column aspect ratio, thenext steps in the research of landward waves are: (1) develop a momentum transference analysis to findthe relationship between the mobilized mass and the maximum wave trough a−m and (2) evaluate if a−mposition does not change with xf,crown nor with L0.

46

7 References

Ataie-Ashtiani, B., & Najafi-Jilani, A. (2008). Laboratory investigations on impulsive waves caused byunderwater landslide. Coastal Engineering, 55(12), 989-1004. https://doi.org/10.1016/j.coastaleng.2008.03.003

Barla, G., Paronuzzi, P. The 1963 Vajont Landslide: 50th Anniversary. Rock Mech Rock Eng 46,1267–1270 (2013). https://doi.org/10.1007/s00603-013-0483-7

Bougouin, A., Paris, R., & Roche, O. (2020). Impact of fluidized granular flows into water: Implica-tions for tsunamis generated by pyroclastic flows. Journal of Geophysical Research: Solid Earth, 125 (5),e2019JB018954. https://doi.org/10.1029/2019JB018954

Bregoli, F., Bateman, A., & Medina, V. (2017). Tsunamis generated by fast granular landslides: 3Dexperiments and empirical predictors. Journal of Hydraulic Research, 55 (6), 743-758. https://doi.

org/10.1080/00221686.2017.1289259

Bregoli, F., Medina, V., & Bateman, A. (2020). The energy transfer from granular landslides towater bodies explained by a data-driven, physics-based numerical model. Landslides, 18 (4), 1337-1348.https://doi.org/10.1007/s10346-020-01568-3

Bullard, G. (2018). Wave Characteristics of Tsunamis Generated by Landslides of Varying Size andMobility (Doctoral dissertation).

Bullard, G. K., Mulligan, R. P., Carreira, A., & Take, W. A. (2019). Experimental analysis oftsunamis generated by the impact of landslides with high mobility. Coastal Engineering, 152, 103538.https://doi.org/10.1016/j.coastaleng.2019.103538

Cabrera, M. A., Pinzon, G., Take, W. A., & Mulligan, R. P. (2020). Wave generation across acontinuum of landslide conditions from the collapse of partially submerged to fully submerged granularcolumns. Journal of Geophysical Research: Oceans, 125, e2020JC016465. https://doi.org/10.1029/

2020JC016465

Cabrera, M. A., & Wu, W. (2017). Space–time digital image analysis for granular flows. InternationalJournal of Physical Modelling in Geotechnics, 17 (2), 135-143. https://doi.org/10.1680/jphmg.16.

00018

Desrues, M., Lacroix, P., & Brenguier, O. (2019). Satellite pre-failure detection and In situ monitoringof the landslide of the tunnel du Chambon, French Alps. Geosciences, 9 (7), 313. https://doi.org/10.3390/geosciences9070313

Franz, M., Jaboyedoff, M., Mulligan, R. P., Podladchikov, Y., & Take, W. A. (2021). An efficienttwo-layer landslide-tsunami numerical model: effects of momentum transfer validated with physical ex-periments of waves generated by granular landslides. Natural Hazards and Earth System Sciences, 21 (4),1229-1245. https://doi.org/10.5194/nhess-21-1229-2021

Fritz, H. M., Hager, W. H., & Minor, H. E. (2003). Landslide generated impulse waves. Experimentsin Fluids, 35(6), 505-519. https://doi.org/10.1007/s00348-003-0659-0

Fritz, H. M., Hager, W. H., & Minor, H. E. (2004). Near field characteristics of landslide generatedimpulse waves. Journal of waterway, port, coastal, and ocean engineering, 130 (6), 287-302. https:

//doi.org/10.1061/(ASCE)0733-950X(2004)130:6(287)

Grilli, S. T., Shelby, M., Kimmoun, O., Dupont, G., Nicolsky, D., Ma, G., & Shi, F. (2017). Modelingcoastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation,and case studies off the US East Coast. Natural hazards, 86 (1), 353-391. https://doi.org/10.1007/

s11069-016-2692-3

Heller, V. (2007). Landslide generated impulse waves: Prediction of near field characteristics (Doc-toral dissertation, Eth Zurich).

Heller, V., & Hager, W. H. (2010). Impulse product parameter in landslide generated impulse waves.Journal of Waterway, Port, Coastal, and Ocean Engineering, 136 (3), 145-155. https://doi.org/10.

1061/(ASCE)WW.1943-5460.0000037

Huang, B. L., Wang, J., Zhang, Q., Luo, C. L., & Chen, X. T. (2020). Energy conversion and

47

deposition behaviour in gravitational collapse of granular columns. Journal of Mountain Science, 17(1),216-229. https://doi.org/10.1007/s11629-019-5602-9

Ma, G., Shi, F., & Kirby, J. T. (2012). Shock-capturing non-hydrostatic model for fully dispersivesurface wave processes. Ocean Modelling, 43, 22-35. https://doi.org/10.1016/j.ocemod.2011.12.

002

Ma, G., Kirby, J. T., Hsu, T. J., & Shi, F. (2015). A two-layer granular landslide model for tsunamiwave generation: Theory and computation. Ocean Modelling, 93, 40-55. https://doi.org/10.1016/j.ocemod.2015.07.012

McIver, P., & Evans, D. V. (1984). The occurrence of negative added mass in free-surface problemsinvolving submerged oscillating bodies. Journal of engineering mathematics, 18 (1), 7-22. https://doi.org/10.1007/BF00042895

Mergili, M., Pudasaini, S.P., 2014-2021. r.avaflow - The mass flow simulation tool. r.avaflow 2.3Software. https://www.avaflow.org/software.php

Miller, G. S., Andy Take, W., Mulligan, R. P., & McDougall, S. (2017). Tsunamis generated by longand thin granular landslides in a large flume. Journal of Geophysical Research: Oceans, 122 (1), 653-668.https://doi.org/10.1002/2016JC012177

Mulligan, R. P., & Take, W. A. (2017). On the transfer of momentum from a granular landslide to awater wave. Coastal Engineering, 125, 16-22. https://doi.org/10.1016/j.coastaleng.2017.04.001

Pinzon Forero, G. A. (2019). Transitional granular flows: an experimental approach (Master’s thesis,Uniandes).

Pinzon, G., & Cabrera, M. (2019). Planar collapse of a submerged granular column. Physics ofFluids, 31 (8), 086603. https://doi.org/10.1063/1.5099494

Prada-Sarmiento, L. F., Cabrera, M. A., Camacho, R., Estrada, N., Ramos-Canon, A. M. (2019).The Mocoa Event on March 31 (2017): analysis of a series of mass movements in a tropical environ-ment of the Andean-Amazonian Piedmont. Landslides, 16 (12), 2459-2468. https://doi.org/10.1007/s10346-019-01263-y

Pudasaini, S. P. (2012). A general two-phase debris flow model. Journal of Geophysical Research:Earth Surface, 117 (F3). https://doi.org/10.1029/2011JF002186

Pudasaini, S. P., & Hutter, K. (2007). Avalanche dynamics: dynamics of rapid flows of dense granularavalanches. Springer Science & Business Media.

Pudasaini, S. P., & Mergili, M. (2019). A multi-phase mass flow model. Journal of GeophysicalResearch: Earth Surface, 124 (12), 2920-2942. https://doi.org/10.1029/2019JF005204

Robbe-Saule, M. (2019). Modelisation experimentale de generation de tsunami par effondrementgranulaire (Doctoral dissertation, Universite Paris-Saclay).

Zweifel, A., Hager, W. H., & Minor, H. E. (2006). Plane impulse waves in reservoirs. Journal ofwaterway, port, coastal, and ocean engineering, 132 (5), 358-368. https://doi.org/10.1061/(ASCE)

0733-950X(2006)132:5(358)

48

8 Nomenclature

α terrain slopeαs solid volume fractionαf fluid volume fractionam seaward wave maximum amplitudea−m landward wave maximum trough amplitudec wave celerityC compound center of mass of the granular columnC virtual mass coefficientCD magnitude of drag forceCDG generalized drag coefficientCs center of mass of the solidsCVMG generalized virtual mass coefficientCw center of mass of the fluidδ dynamic bed friction∆te effective time between the impact and the detachment of the slide and the waterη fluid surface elevationηf viscosity of fluidEd drag energy transferred from the landslide to the water bodyEf energy loss due to friction between the landslide and the surfaceEs total landslide energy at impactEt dissipative energies: sum of turbulence, water surface tension breaking and compressibilityEw total wave energyf body force densityF fluid-like contribution in generalized dragFg momentum flux in the collapse of a granular columnFr slide Froude numberFrd slide densimetric Froude numberg acceleration of gravityG solid-like contribution in generalized dragγ fluid to solid density ratioh still water depthhT Total flow heightH wave heightH0 granular column heightHw water heightK hydraulic conductivity.L wavelengthLc cutback length of granular columnLe slide effective lengthLf slide runoutms slide massM interfacial force densityMs momentum flux of landslideMfs hydrostatic momentum flux of waterMfd hydrodynamic momentum flux of waterMTf momentum transfer from the slide to the waterMTs momentum transfer from the water to the slideMVM magnitude of virtual mass force

49

n porosityφ′ effective internal friction anglep fluid pressureRep particle Reynolds numberρf water densityρg grain densityρs slide bulk densitys slide thicknessS dimensionless slide thicknessτf viscous stress tensorTs Cauchy stress tensoruf fluid velocityus solids velocityusb landslide velocity in the x directionUs terminal velocity of a particleuwb water velocity in the x directionU Ursell numbervs slide velocity parallel to the terrainVs slide volumex(a−m) horizontal coordinate where the maximum landward wave trough is observedxfcrown final horizontal coordinate of the crown deposit

50

A Codes

A.1 Python function to create experimental setups DEM as raster (ASC) maps

import numpy as np

def DEM_setup(setup=’Queens ’, cellsizex =0.02, cellsizey =0.02, walls_inc =85,

walls_height =10, fill=False ,

h_fill_wall = 10):

’’’ Generates the Queen ’s University landslide flume or the IRPHE landslide flume

as a raster (ASC) map

Parameters

----------

setup: str (’Queens ’ or ’IRPHE ’) indicating the setup to be generated

cellsize: float indicating the cellsize in m

walls_inc: float indicating the landslide flume walls inclination in degrees

walls_height: float indicating the landslide flume walls height in m

fill: boolean. Activate to generate the elevation model to use with r.lakefill

h_fill_wall: float indicating the height of the wall used to generate the

solids map with r.lakefill

Output

----------

Raster map (ASC) with the Queen ’s or IRPHE flume elevation ’’’

’’’Horizontal channel ’’’

if setup ==’Queens ’:

lenx_HC = 33 #m

leny_HC = 2.09 #m

elif setup ==’IRPHE ’:

lenx_HC = 12.87 #m

leny_HC = 0.25 #m

cols_HC = int(lenx_HC/cellsizex)

rows_HC = int(leny_HC/cellsizey)

HC = np.zeros ((rows_HC ,cols_HC))

’’’Inclined channel ’’’

if setup ==’Queens ’:

inc_IC = 30 #degrees

lenx_IC = 6.73*np.cos(np.radians(inc_IC)) + 2*(0.47/ np.tan(np.radians(inc_IC)))

#m

leny_IC = 2.09 #m

elif setup ==’IRPHE ’:

inc_IC = 35 #degrees

lenx_IC = 10.00 #m

leny_IC = 0.25 #m

cols_IC = int(lenx_IC/cellsizex)

rows_IC = int(leny_IC/cellsizey)

IC = np.zeros ((rows_IC ,cols_IC))

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xhstep_flume = -cellsizex/np.tan(np.radians (90- inc_IC))

for i in range(0,cols_IC):

xhstep_flume += cellsizex/np.tan(np.radians (90- inc_IC))

for j in range(0,rows_IC):

IC[j][i] = elev

’’’Stack horizontal and inclined channels ’’’

flume_nowalls = np.hstack ([np.fliplr(IC), HC , IC])

’’’Add walls to the channels ’’’

leny_wall = walls_height /(np.tan(np.radians(walls_incl))) #m

cols_wall = np.size(flume_nowalls ,1)

rows_wall = int(leny_wall/cellsizey)

wall = np.zeros ((rows_wall ,cols_wall))

yhstep_wall = np.tan(np.radians(walls_inc))*cellsizey

for i in range(0, rows_wall):

wall[i,:] = flume_nowalls [0 ,:]+i*yhstep_wall

flume_walls = np.vstack ([np.flipud(wall), flume_nowalls , wall])

’’’Add column to fill with r.lakefill ’’’

if fill:

if setup==’Queens ’:

cols_fill = [int(2*np.ceil (0.47/( np.tan(np.radians(inc_IC))*cellsizex)))]

elif setup ==’IRPHE ’:

cols_fill = [int (9.72/ cellsizex)]

fill_flume = flume_walls.copy()

for col in cols_fill:

fill_flume [:,col] += h_fill_wall

’’’Save file ’’’

filename = setup + ’_2017_DEM_ ’ + str(cellsizex) + ’m_trapez.asc’

if fill:

filename = filename [:-4] + ’_fill.asc’

outfile = open(filename ,’w’)

outfile.write("NCOLS " + str(np.shape(flume)[1]) + "\n")

outfile.write("NROWS " + str(np.shape(flume)[0]) + "\n")

outfile.write("XLLCENTER " + str (0) + "\n")

outfile.write("YLLCENTER " + str (0) + "\n")

outfile.write("CELLSIZE " + str(cellsizex) + "\n")

outfile.write("NODATA_VALUE -9999" + "\n")

outfile.write(’\n’.join(’ ’.join(’%0.3f’ %x for x in y) for y in flume))

outfile.close()

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A.2 Code to create solid and fluid phases with r.lakefill

r.in.gdal -o input="Queens_2017_DEM_0 .02 m_trapez_fill.asc" output="Queens_2017_DEM_0 .02

m_trapez_fill"

r.lakefill cellsize =0.02 elevation="Queens_2017_DEM_0 .02 m_trapez_fill" lakedepth="

Miller_2017_solids_0 .02 m_trapez" level =3.819 seedcoords =1.61 ,1.67

g.region raster="Queens_2017_DEM_0 .02 m_trapez_fill" -p

r.out.gdal input="Miller_2017_solids_0 .02 m_trapez" output="Miller_2017_solids_0 .02

m_trapez.asc"

r.lakefill cellsize =0.02 elevation="Queens_2017_DEM_0 .02 m_trapez_fill" lakedepth="

Miller_2017_reservoir_hm_0 .02 m_trapez" level=h seedcoords =31.14 ,1.72 #replace h with

the reservoir still water depth in m

Because the water in the reservoir may oscillate, r.avaflow was run with the water in the reservoir asa fluid phase until the oscillations were less than 1 cm above or below the initial still water depth h (6s). The last phase 3 raster map generated by the software was taken as the input reservoir map for thesimulations.

r.avaflow -t -a prefix="Miller_2017_reservoir_hm_0 .02 m_trapez" cellsize =0.02 phases=m

elevation="Queens_2017_DEM_0 .02 m_trapez_fill" hrelease3="Miller_2017_reservoir_hm_0

.02 m_trapez" time =0.05 ,6 #replace h with the reservoir still water depth in m

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A.3 Code to run r.avaflow

#Launching r.avaflow computational experiments for Miller et al. (2017) laboratory

experiments}

prefix="Miller_2017_hm" #replace h with the reservoir still water depth

cellsize="0.02"

phases="m"

r.in.gdal -o input="Queens_2017_DEM_0 .02 m_trapez.asc" output="Queens_2017_DEM_0 .02

m_trapez_fill"

r.in.gdal -o input="Miller_2017_solids_0 .02 m_trapez.asc" output="Miller_2017_hrelease1"

r.in.gdal -o input="Miller_2017_reservoir_hm_0 .02 m_trapez_hflow0120.asc" output="

Miller_2017_reservoir_hm_0 .02 m_trapez_hflow0120"

elevation="Queens_2017_DEM_0 .02 m_trapez"

hrelease1="Miller_2017_hrelease1"

alpha="0.64" #solid volume fraction

r.mapcalc --overwrite "hrel1 = $hrelease1*$alpha" #Phase 1 release height

hreserv="Miller_2017_reservoir_hm_0 .02 m_trapez_hflow0120"

density="2400 ,1800 ,1000"

friction="34,24,0,0,0,0"

cfl="0.4 ,0.001"

controls="0,0,0,0,0,0"

thresholds="0.1 ,10000 ,10000 ,0.00001"

g.region raster="$elevation"

r.avaflow -t -a prefix="$prefix" cellsize="$cellsize" phases="$phases" elevation="

$elevation" hrelease1=hrel1 hrelease3="$hrel3" density="$density" friction="

$friction" controls="$controls" cfl="$cfl" thresholds="$thresholds" time =0.05 ,40

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#Launching r.avaflow computational experiments for Bullard (2018) laboratory

experiments

prefix="Bullard_2018_hm" #replace h with the reservoir still water depth

cellsize="0.02"

phases="m"

r.in.gdal -o input="Queens_2017_DEM_0 .02 m_trapez.asc" output="Queens_2017_DEM_0 .02

m_trapez_fill"

r.in.gdal -o input="Miller_2017_solids_0 .02 m_trapez.asc" output="Miller_2017_hrelease1"

r.in.gdal -o input="Miller_2017_reservoir_hm_0 .02 m_trapez_hflow0120.asc" output="

Miller_2017_reservoir_hm_0 .02 m_trapez_hflow0120"

elevation="Queens_2017_DEM_0 .02 m_trapez"

hrelease1="Miller_2017_hrelease1"

hreserv="Miller_2017_reservoir_hm_0 .02 m_trapez_hflow0120"

alpha="0.73" #prop. de part.

r.mapcalc --overwrite "hrel1 = $hrelease1*$alpha" #Phase 1 release height

r.mapcalc --overwrite "hrel3 = $hrelease1 *(1- $alpha)+$hreserv" #Phase 3 release height

density="2400 ,1800 ,1000"

friction="34,24,0,0,0,0"

cfl="0.4 ,0.001"

controls="0,0,0,0,0,0"

thresholds="0.1 ,10000 ,10000 ,0.00001"

g.region raster="$elevation"

r.avaflow -t -a prefix="$prefix" cellsize="$cellsize" phases="$phases" elevation="

$elevation" hrelease1=hrel1 hrelease3="$hrel3" density="$density" friction="

$friction" controls="$controls" cfl="$cfl" thresholds="$thresholds" time =0.05 ,40

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A.4 Python function to process r.avaflow output raster maps.

def flume_profiles(name_of_elev_map , list_of_ascii_files_fluid ,

list_of_ascii_files_solids , asc_type=’h’, plot = True , timestep = 1,

params = "", figsize = (15 ,7.5), dpi = 200, directory = ’’, flowtype

= "dry", pdf = True ,

proftype = "max", dis2res = 7.46, volplot = True , allprof = False ,

allprof_for_each_time = False ,

wave_in_horizontal_channel_only = False , threshold_f = 0.003,

threshold_s = 0.003 , prop_p =0.64 , fps=1,

ws = 85):

’’’ Reads a set of ASC files and calculates the cross section statistics

Parameters

----------

name_of_elev_map: str indicating the elevation map path

list_of_ascii_files_fluid: list with fluid phase asc maps paths

list_of_ascii_files_solids: list with solid phase asc maps paths

asc_type: str (’h’, ’v’, ’VM’, ’drag ’) indicating the type of the raster maps

(flow height , flow velocity , virtual mass , drag force)

plot: boolean to create a mp4 video showing the cross section of how the solids

impact the reservoir

timestep: float simulation time step

params: str in the format

reservoirdepth_solidsdensity_solidsinternalfrictionangle_fluiddensity

figsize: tuple indicating the figures width and length in inches

dpi: float indicating the resolution of the figure in dots -per -inch

directory: str indicating the path to save the figures

flowtype: str (’dry ’, ’sat ’) dry for Miller et al. (2017) lab experiments , sat

for Bullard (2018) lab experiments

pdf: boolean save plots both as jpg and pdf

proftype: str (’max ’, ’mean ’, ’median ’, ’center ’) type of measure of central

tendency to analyze the data

dis2res: float inclined channel horizontal distance (7.46 m for the Queen ’s

University landslide flume and 2 m

for the IRPHE landslide flume)

volplot: boolean to generate the plot showing the solids volume at each time

step

allprof: boolean to generate a single plot showing the fluid surface traces

allprof_for_each_time: boolean to generate one plot for each fluid surface

trace

wave_in_horizontal_channel_only: boolean to ignore the fluid data before the

beginning of the horizontal channel

threshold_f: float fluid values less than the threshold are set as nan

threshold_s: float solid values less than the threshold are set as nan

prop_pa: float solid volume fraction

fps: int frames per second for mp4 video

ws: float walls inclination in degrees

Returns

----------

list with channel statistics calculated accross the yz section arranged as

max_f_am: float maximum fluid

height above still water depth , max_s: float maximum solid height , vol_s: list

volume of solids at each time step ,

vol_f: list volume of fluid at each time , max_f: float maximum fluid height ,

max_f_am_std: float max_f_am

standard deviation , max_f_std: float max_f standard deviation , data_f: list of

lists each list contains the fluid

height for each time step , data_f_am: list of lists each list contains the

56

fluid height above still water depth

for each time step , data_f_std: list of lists data_f standard deviation ,

data_f_am_std: list of lists data_f_am

standard deviation , data_s: list of lists each list contains the solids height

for each time step ,

data_s_std: list of lists data_s standard deviation , ind_max_f_am_time: int to

multiply by the time step and

obtain the time at which the maximum amplitude is observed , ind_max_f_am_x: int

to multiply by the cellsize

and obtain the position at which the maximum amplitude is observed ’’’

today = date.today()

fs_label = 15

fs_legend = 15

fs_ticks = 15

t_0 = int(list_of_ascii_files_fluid [0][ -8: -4])

t_f = int(list_of_ascii_files_fluid [-1][-8:-4])

time = [i for i in range(t_0 ,t_f)] + [t_f]

rows_elev , ncols , nrows , cellsize , nodata_value = loadASCII(winapi_path(

name_of_elev_map))

data_elev = list(rows[int(np.shape(rows)[0]/2) ,:])

ind = data_elev.index (0) #x coordinate where the horizontal channel begins

cross_section = list(rows_elev[:,int(np.shape(rows_elev)[1]/2) ])

ind_CS = cross_section.index (0)

data_f = []

data_f_std = []

data_f_am = []

data_f_am_std = []

data_s = []

data_s_std = []

vol_s = []

vol_f = []

for i in range (len(list_of_ascii_files_fluid)):

print("File num: " + str(i))

f_f = open(winapi_path(list_of_ascii_files_fluid[i]),"r")

f_s = open(winapi_path(list_of_ascii_files_solids[i]),"r")

out_f = f_f.readlines ()

out_s = f_s.readlines ()

if asc_type == ’v’:

filename_fluid_splitted = list_of_ascii_files_fluid[i]. split(’vflow’)

elif asc_type == ’h’:

filename_fluid_splitted = list_of_ascii_files_fluid[i]. split(’hflow’)

elif asc_type == ’VM’:

filename_fluid_splitted = list_of_ascii_files_fluid[i]. split(’

virtualmassflow ’)

elif asc_type == ’drag’:

filename_fluid_splitted = list_of_ascii_files_fluid[i]. split(’dragflow ’)

57

hreserv = float(filename_fluid_splitted [0][ -6: -2])

rows = range (6+ ind_CS ,len(out_f)-ind_CS)

rows_f = []

rows_s = []

vol_row_f = []

vol_row_s = []

for j in range(0,len(rows)):

row_f = list(out_f[rows[j]]. split(" "))

row_s = list(out_s[rows[j]]. split(" "))

row_f [-1] = row_f [-1]. replace(’\n’,’’)

row_s [-1] = row_s [-1]. replace(’\n’,’’)

if ’’ in row_f:

row_f.remove(’’)

if ’’ in row_s:

row_s.remove(’’)

row_f = [float(element) if float(element) != nodata_value and float(element

) > threshold_f else np.nan for element in row_f]

row_s = [float(element) if float(element) != nodata_value and float(element

) > threshold_s else np.nan for element in row_s]

row_f_1 = row_f.copy()

row_s_1 = row_s.copy()

if asc_type == ’h’:

contador = 0

for element_f ,element_s in zip(row_f ,row_s):

if element_f == element_s:

row_f[contador] = np.nan

row_s[contador] = row_s[contador ]/ float(prop_p)

contador +=1

if ’Miller ’ in filename_fluid_splitted [0]:

theta = 30

elif ’Bullard ’ in filename_fluid_splitted [0]:

theta = 30

elif ’Grilli ’ in filename_fluid_splitted [0]:

theta = 33

delx = hreserv*np.tan(np.radians (90-theta))

if wave_in_horizontal_channel_only:

row_f [0:ind] = [np.nan for i in range(ind)]

rows_f.append(row_f [:int(ncols)])

vol_row_f.append(np.nansum(row_f_1)*cellsize **2)

rows_s.append(row_s [:int(ncols)])

vol_row_s.append(np.nansum(row_s_1)*cellsize **2)

vol_f.append(np.sum(vol_row_f))

vol_s.append(np.sum(vol_row_s))

58

if i == 0:

if list_of_ascii_files_fluid [0][ -5] == str (0):

rows_0 = rows_f.copy()

else:

print(’The first file must be in time 0’)

rows_f_am = np.array(rows_f) - np.array(rows_0)

rows_f = np.array(rows_f)

if proftype == "max":

data_f.append(np.nanmax(rows_f , axis =0))

data_f_am.append(np.nanmax(rows_f , axis =0)-np.nanmax(rows_0 ,axis =0))

data_s.append(np.nanmax(rows_s , axis =0))

elif proftype == "mean":

data_f.append(np.nanmean(rows_f , axis =0))

data_f_am.append(np.nanmean(rows_f , axis =0)-np.nanmean(rows_0 ,axis =0))

data_s.append(np.nanmean(rows_s , axis =0))

elif proftype == "center":

half_row = int(((len(out_f) - 6)/2))

data_f.append(rows_f[half_row ])

data_f_am.append(rows_f[half_row]-rows_0[half_row ])

data_s.append(rows_s[half_row ])

elif proftype == "median":

data_f.append(np.nanmedian(rows_f , axis =0))

data_f_am.append(np.nanmedian(rows_f , axis =0)-np.nanmedian(rows_0 ,axis =0))

data_s.append(np.nanmedian(rows_s , axis =0))

data_f_std.append(np.nanstd(rows_f , axis =0))

data_f_am_std.append(np.nanstd(rows_f_am , axis =0))

data_s_std.append(np.nanstd(rows_s , axis =0))

data_f = np.array(data_f)

data_f_std = np.array(data_f_std)

data_f_am = np.array(data_f_am)

data_f_am_std = np.array(data_f_am_std)

data_s = np.array(data_s)

data_s_std = np.array(data_s_std)

max_f = np.nanmax(data_f)

if asc_type ==’h’:

max_f_am = np.nanmax(data_f_am [:,:int(np.shape(data_f_am)[1]/2) ]) #to avoid the

code to calculate the maximum at the end of the channel

else:

max_f_am = np.nanmax(data_f_am)

max_s = np.nanmax(data_s)

ind_max_f = np.where(data_f == max_f)

ind_max_f_am = np.where(data_f_am == max_f_am)

if np.isnan(max_f):

max_f_std = np.nan

else:

max_f_std = data_f_std[ind_max_f [0][0] , ind_max_f [1][0]]

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if np.isnan(max_f_am):

max_f_am_std = np.nan

ind_max_f_am_time = 0

ind_max_f_am_x = 0

else:

max_f_am_std = data_f_am_std[ind_max_f_am [0][0] , ind_max_f_am [1][0]]

ind_max_f_am_time = ind_max_f_am [0][0]

ind_max_f_am_x = ind_max_f_am [1][0]

’’’PLOTS ’’’

if allprof_for_each_time:

for i in range(int(np.shape(data_f_am)[0])):

fig , ax = plt.subplots(figsize=figsize ,dpi=dpi)

x = np.linspace(0, len(data_f_am[i])*cellsize , len(data_f_am[i]))

ax.plot(x, data_f_am[i], ’-’, color=’orange ’, lw=2, alpha =0.5)

plt.fill_between(x, data_f_am[i]-data_f_am_std[i], data_f_am[i]+

data_f_am_std[i],alpha =0.3, color=’orange ’)

#ax.axis(’equal ’)

plt.xlabel(’x (m)’, fontsize=fs_label)

plt.ylabel(’$a_m$ (m)’, fontsize=fs_label)

plt.ylim ((-0.5, 0.5))

plt.xlim((6, 12))

#plt.title(’Height at time ’ + str(time[i]))

#plt.grid(True)

plt.setp(ax.get_xticklabels (), fontsize=fs_ticks)

plt.savefig(directory + r"\\" + today.strftime("%Y%m%d") + "_" + params +

"_lab_h_prof_" + flowtype +

proftype + "_allrows_for_time_" + str(time[i]* timestep) + r".pdf")

plt.close(fig)

if volplot:

fig , ax = plt.subplots(figsize=figsize ,dpi=dpi)

ax.plot([t*timestep for t in time], vol_s , ’.-’, lw=2, color=’black ’)

#ax.axis(’equal ’)

plt.xlabel(’t (s)’)

plt.ylabel(r’$V (m^{3})$’)

plt.ylim((0, max(vol_s)*1.1))

#plt.xlim((0, max(x)))

#plt.tight_layout ()

plt.setp(ax.get_xticklabels (), fontsize=fs_ticks)

plt.savefig(directory + today.strftime("%Y%m%d") + "_" + params + "

_volume_change.pdf")

plt.close(fig)

if plot:

for i in range(int(np.shape(data_f)[0])):

x = np.linspace(0, len(data_elev)*cellsize , len(data_elev))

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’’’textbox ’’’

params2 = params.split("_")

textstr = ’\n’.join((r’$t=%.2f’ % (time[i]*timestep , ) + ’s$’,

r’$(h_{f})_{0}=’ + str(np.around(float(params2 [0])

*100 ,1)) + ’cm$’,

r’$\rho_{s}=’ + params2 [1] + r’kg/m^{3}$’,

r’$\phi_{s}=’ + params2 [2] + r’\degree$ ’,

r’$\rho_{f}=’ + params2 [3] + r’kg/m^{3}$’,

r’$a_{m}’ + ’=’ + str(np.around(max_f_am *100 ,1)) + r’

cm$’ + r’ at ’ + str(np.around(ind_max_f_am_time*ts ,2)) + r’s and ’ +str(np.around(

ind_max_f_am_x*cellsize -x[ind],2)) + r’m’,

r’$V_{s}=%.2f’ %( vol_s[i]*1000 , ) + r’dm^{3}$’))

plt.ioff()

fig , ax = plt.subplots(figsize=figsize ,dpi=dpi)

h_f = list(map(add , data_elev , data_f[i]))

h_s = list(map(add , data_elev , data_s[i]))

ymax = max([max_f , max_s ])

ax.plot(x-x[ind], data_elev , ’-’, lw=2, color=’grey’, alpha =0.5)

ax.plot(x-x[ind], h_f , ’-’, lw=1, color=’royalblue ’, label=’Fluid ’)

ax.plot(x-x[ind], h_s , ’-’, lw=1.5, color=’peru’, label=’Solids ’)

ax.axis(’equal’)

plt.xlabel(’x (m)’, fontsize=fs_label)

plt.ylabel(’height (m)’, fontsize=fs_label)

#plt.ylim((0, ymax))

plt.xlim((0-x[ind], 47.46-x[ind]))

#plt.title(’Height at time ’ + str(time[i]))

#plt.grid(True)

plt.legend(loc=’upper right’, fontsize=fs_legend)

#plt.legend(loc =(0.9 ,0.6))

plt.setp(ax.get_xticklabels (), fontsize=fs_ticks)

plt.setp(ax.get_yticklabels (), fontsize=fs_ticks)

axins = zoomed_inset_axes(ax , 2.5, loc=’center ’, bbox_to_anchor =(0.6 ,0.7),

bbox_transform=plt.gcf().transFigure) #zoom 2.5

axins.plot(x-x[ind], data_elev , ’-’, lw=2, color=’grey’, alpha =0.5)

axins.plot(x-x[ind], h_f , ’-’, lw=1, color=’royalblue ’, label=’Fluid’)

axins.plot(x-x[ind], h_s , ’-’, lw=1.5, color=’peru’, label=’Solids ’)

axins.axis(’equal ’)

#axins.grid(True , alpha =0.5)

mark_inset(ax , axins , loc1=2, loc2=4, fc="none", ec="0.5")

if flowtype == ’dry’ or flowtype == ’saturated ’ or flowtype == ’sat’:

ax.text (0.75, 0.35, textstr , transform=ax.transAxes , fontsize =10,

verticalalignment=’top’)

x1 , x2 , y1 , y2 = 5.5-x[ind], 12.5-x[ind], -0.1, 2.5 #subaerial

elif flowtype == ’submerged ’:

ax.text (0.75, 0.30, textstr , transform=ax.transAxes , fontsize =10,

verticalalignment=’top’)

x1 , x2 , y1 , y2 = 1.3-x[ind], 2.5-x[ind], -0.05, 0.75 #submerged

axins.set_xlim(x1, x2)

axins.set_ylim(y1, y2)

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if pdf:

plt.savefig(directory + r"\\" + today.strftime("%Y%m%d") + "_" +

params + "_lab_h_prof_" + flowtype +

proftype + str(time[i]).zfill (4) + ".pdf")

plt.savefig(directory + r"\\" + today.strftime("%Y%m%d") + "_" + params +

"_lab_h_prof_" + flowtype +

proftype + str(time[i]).zfill (4) + ".png")

plt.close(fig)

filenames = [directory + r"\\" + today.strftime("%Y%m%d") + "_" + params + "

_lab_h_prof_" + flowtype +

proftype + str(t).zfill (4) + ".png" for t in time]

writer = imageio.get_writer(directory + r"\\" + today.strftime("%Y%m%d") + "_"

+ params + "_lab_h_prof_" + flowtype +

proftype + r".mp4", fps=fps)

for filename in filenames:

writer.append_data(imageio.imread(filename))

writer.close ()

if allprof:

fig , ax = plt.subplots(figsize=figsize ,dpi=dpi)

evenly_spaced_interval = np.linspace(0, 1, len(data_f_am))

colrs = [cm.Oranges(x) for x in evenly_spaced_interval]

for i in range(int(np.shape(data_f_am)[0])):

x = np.linspace(0, len(data_f_am[i])*cellsize , len(data_f_am[i]))

ax.plot(x, data_f_am[i], ’-’, lw=2,color=colrs[i])

plt.fill_between(x, data_f_am[i]-data_f_am_std[i], data_f_am[i]+

data_f_am_std[i],alpha =0.3, color=colrs[i])

#ax.axis(’equal ’)

plt.xlabel(’x (m)’, fontsize=fs_label)

plt.ylabel(’$a_m$ (m)’, fontsize=fs_label)

plt.ylim ((-0.5, 0.5))

plt.xlim((6, 12))

#plt.title(’Height at time ’ + str(time[i]))

#plt.grid(True)

plt.savefig(directory + r"\\" + today.strftime("%Y%m%d") + "_" + params + "

_lab_h_prof_" + flowtype +

proftype + "_allprof.pdf")

plt.close(fig)

return [max_f_am , max_s , vol_s , vol_f , max_f , max_f_am_std , max_f_std , data_f ,

data_f_am , data_f_std , data_f_am_std , data_s , data_s_std , ind_max_f_am_time ,

ind_max_f_am_x]

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