Towards a model for structured mass movements

Geosci. Model Dev., 14, 1841–1864, 2021https://doi.org/10.5194/gmd-14-1841-2021© Author(s) 2021. This work is distributed underthe Creative Commons Attribution 4.0 License.

Towards a model for structured mass movements:the OpenLISEM hazard model 2.0aBastian van den Bout1, Theo van Asch2, Wei Hu2, Chenxiao X. Tang3, Olga Mavrouli1, Victor G. Jetten1, andCees J. van Westen1

1Faculty of Geo-Information Science and Earth Observation, University of Twente, Enschede, the Netherlands2State Key Laboratory of Geohazard Prevention and Geo-Environment Protection, Chengdu University of Technology,Chengdu, China3Institute of Mountain Hazards and Environment, Chinese Academy of Sciences, Chengdu, China

Correspondence: Bastian van den Bout ([email protected])

Received: 9 April 2020 – Discussion started: 25 June 2020Revised: 27 November 2020 – Accepted: 15 December 2020 – Published: 6 April 2021

Abstract. Mass movements such as debris flows and land-slides differ in behaviour due to their material propertiesand internal forces. Models employ generalized multi-phaseflow equations to adaptively describe these complex flowtypes. Such models commonly assume unstructured andfragmented flow, where internal cohesive strength is insignif-icant. In this work, existing work on two-phase mass move-ment equations are extended to include a full stress–strainrelationship that allows for runout of (semi-)structured fluid–solid masses. The work provides both the three-dimensionalequations and depth-averaged simplifications. The equationsare implemented in a hybrid material point method (MPM),which allows for efficient simulation of stress–strain relation-ships on discrete smooth particles. Using this framework, thedeveloped model is compared to several flume experimentsof clay blocks impacting fixed obstacles. Here, both finaldeposit patterns and fractures compare well to simulations.Additionally, numerical tests are performed to showcase therange of dynamical behaviour produced by the model. Im-portant processes such as fracturing, fragmentation and fluidrelease are captured by the model. While this provides an im-portant step towards complete mass movement models, sev-eral new opportunities arise, such as application to fragment-ing mass movements and block slides.

1 Introduction

The Earth’s rock cycle involves sudden release and gravity-driven transport of sloping materials. These mass movementshave a significant global impact in financial damage and ca-sualties (Nadim et al., 2006; Kjekstad and Highland, 2009).Understanding the physical principles at work at their initia-tion and runout phase allows for better mitigation and adapta-tion to the hazard they induce (Corominas et al., 2014). Manyvarieties of gravitationally driven mass movements have beencategorized according to their material physical parametersand type of movement. Examples are slides, flows and falls,consisting of soil, rocks or debris (Varnes, 1987). Major fac-tors in determining the dynamics of mass movement runoutare the composition of the moving material and the internaland external forces during initiation and runout.

Within the cluster of existing mass movement processes,a distinction can be made based on the cohesive of the massduring movement. Post-release, a sloping mass might be un-structured, such as mud flows, where grain–grain cohesivestrength is absent. Alternatively, the mass can be fragmen-tative, such as strongly deforming landslides or fragment-ing of rock avalanches upon particle impact. Finally, thereare coherent or structured mass movements, such as can bethe case for block slides where internal cohesive strengthcan resist deformation for some period (Varnes, 1987). Thegeneral importance of the initially structured nature of massmovement material is observed for a variety of reasons. First,block slides are an important subset of mass movement types

Published by Copernicus Publications on behalf of the European Geosciences Union.

1842 B. van den Bout et al.: Towards a model for structured mass movements

(Hayir, 2003; Beutner and Gerbi, 2005; Reiche, 1937; Tanget al., 2009). This type of mass movement features some co-hesive structure to the dynamic material in the movementphase. Secondly, during movement, the spatial gradients inlocal acceleration induce strain and stress that results in frac-turing. This process, often called fragmentation in relation tostructured mass movements, can be of crucial importance formass movement dynamics (Davies and McSaveney, 2009;Delaney and Evans, 2014; Dufresne et al., 2018; Coromi-nas et al. 2019). The lubricating effect from basal fragmenta-tion can enhance velocities and runout distance significantly(Davies et al., 2006; Tang et al., 2009). Otherwise, fragmen-tation generally influences the rheology of the movement byaltering grain–grain interactions (Zhou et al., 2005). The im-portance of structured material dynamics is further indicatedby engineering studies on rock behaviour and fracture mod-els (Kaklauskas and Ghaboussi, 2001; Ngekpe et al., 2016;Dhanmeher, 2017).

Dynamics of geophysical flows are complex and dependon a variety of forces due to their multi-phase interactions(Hutter et al., 1994). Physically based models attempt to de-scribe the internal and external forces of all of these massmovements in a generalized form (David and Richard, 2011;Pudasaini, 2012; Iverson and George, 2014). This allowsthese models to be applied to a wide variety of cases, whileimproving predictive range. A variety of both one-, two- andthree-dimensional sets of equations exist to describe the ad-vection and forces that determine the dynamics of geophysi-cal flows.

For unstructured (fully fragmented) mass movements, avariety of models exist relating to Mohr–Coulomb mix-ture theory. Such mass movements are described as non-Newtonian granular flows with dominant particle–particle in-teractions, assuming perfect mixing and continuous move-ment. Examples are debris flows and mudslides, while blockslides and rockslides do not fit these criteria. Within thesemodels, the Mohr–Coulomb failure surface is described withzero cohesive strength and only an internal friction angle(Pitman and Le, 2005). Examples that simulated a singlemixed material exist throughout the literature (e.g. Ricken-mann et al., 2006; Julien and O’Brien, 1997; Luna et al.,2012; van Asch et al., 2014). Two-phase models describesolids, fluids, and their interactions; provide additional de-tail; and generalize in important ways (Sheridan et al., 2005;Pitman and Le, 2005; Pudasaini, 2012; Iverson and George,2014; Mergili et al., 2017). Recently, a three-phase model hasbeen developed that includes the interactions between smalland larger solid phases (Pudasaini and Mergili, 2019). Typ-ically, implemented forces include gravitational forces and,depending on the rheology of the equations, drag forces, vis-cous internal forces and a plasticity criterion. The assumptionof zero cohesion in the Mohr–Coulomb material is invalidfor any structured mass movement. Some models do imple-ment a non-Newtonian viscous yield stress based on depth-averaged strain estimations (Boetticher et al., 2017; Fornes

et al., 2017; Pudasaini and Mergili, 2019). However, this ap-proach lacks the process of fragmentation and internal fail-ure.

For structured mass movements, limited approaches areavailable. These movements feature some discrete inter-particle connectivity that allows the moving material to main-tain a elasto-plastic structure. Examples here are block slides,rockslides and some landslides (Aaron and Hungr, 2016).These materials can be described by a Mohr–Coulomb mate-rial with cohesive strength (Spencer, 2012). Aaron and Hungrdeveloped a model for simulation of initially coherent rockavalanches (Aaron and Hungr, 2016) as part of DAN3D Flex.Within their approach, a rigid-block momentum analysis isused to simulate initial movement of the block. After a spec-ified time, the block is assumed to fragment, and a granularflow model using a Voellmy-type rheology is used for fur-ther runout. Their approach thus lacks a physical basis forthe fragmenting behaviour. Additionally, by dissecting therunout process in two stages (discrete block and granularflow), benefits of holistic two-phase generalized runout mod-els are lost. Finally, Greco et al. (2019) presented a runoutmodel for cohesive granular matrix. Their approach simi-larly lacks a description of the fragmentation process. Thus,within current mass movement models, there might be im-provements available from assuming non-fragmented move-ment. This would allow for description of structured massmovement dynamics.

In this paper, a generalized mass movement model is de-veloped to describe runout of an arbitrarily structured two-phase Mohr–Coulomb material. The model extents on recentinnovations in generalized models for Mohr–Coulomb mix-ture flow (Pudasaini, 2012; Pudasaini and Mergili, 2019).Section 2 provides the derivation of the extensive set of equa-tions that describe structured mass movements in a gener-alized manner. Section 3 validates the developed model bycomparison with results from controlled flume runout exper-iments. Additionally, Sect. 3 shows numerical simulation ex-amples that highlight fragmentation behaviour and its influ-ence on runout dynamics. Finally, in Sect. 4 a discussion onthe potential usage of the presented model is provided, to-gether with reflection on important opportunities of improve-ment.

2 A set of mass movement equations incorporatinginternal structure

2.1 Structured mass movements

Gravitational mass flows are triggered when the local driv-ing forces within an often steep section of a slope exceed acritical threshold. The instability of such materials is gener-ally understood to take place along a failure plane (Zhang etal., 2011; Stead and Wolter, 2015). Along this plane, forcesexerted due to gravity and possible seismic accelerations can

Geosci. Model Dev., 14, 1841–1864, 2021 https://doi.org/10.5194/gmd-14-1841-2021

B. van den Bout et al.: Towards a model for structured mass movements 1843

act as a driving force towards the downslope direction, whilea normal force on the terrain induces a resisting force (Xieet al., 2006). When internal stress exceeds specified criteria,commonly described using Mohr–Coulomb theory, fractur-ing occurs, and the material becomes dynamic. Observationsindicate material can initially fracture predominantly at thefailure plane (Tang et al., 2009; Davies et al., 2006). Fullfinite-element modelling of stability confirms no fragmenta-tion occurs at initiation, and runout can start as a structuredmass (Matsui and San, 1992; Griffiths and Lane, 1999).

Once movement is initiated, the material is acceler-ated. Due to spatially non-homogeneous acceleration, eithercaused by a non-homogeneous terrain slope or impact withobstacles, internal stress can build within the moving mass.The stress state can reach a point outside the yield surface,after which some form of deformation occurs (e.g. plastic,brittle, ductile) (Loehnert et al., 2008). In the case of rockor soil material, elastic or plastic deformation is limited, andfracturing occurs at relatively low strain values (Kaklauskasand Ghaboussi, 2001; Dhanmeher, 2017). Rocks and soil ad-ditionally show predominantly brittle fracturing, where strainincrements at maximum stress are small (Bieniawaski, 1967;Price, 2016; Hušek et al., 2016). For soil matrices, cohe-sive bonds between grains originate from causes such ascementing, frictional contacts and root networks (Cohen etal., 2009). Thus, the material breaks along either the grain–grain bonds or on the molecular level. In practice, this pro-cess of fragmentation has frequently been both observedand studied. Cracking models for solids use stress–straindescriptions of continuum mechanics (Menin et al., 2009;Ngekpe et al., 2016). Fracture models frequently use smoothparticle hydrodynamics (SPH) since a Lagrangian, mesh-free solution benefits possible fracturing behaviour (Maureland Combescure, 2008; Xu et al., 2010; Osorno and Steeb,2017). Within the model developed below, knowledge fromfracture-simulating continuum mechanical models is com-bined with finite-element fluid dynamic models.

The Mohr–Coulomb mixture models on which the devel-oped model is based can be found in Pitman and Le (2005),Pudasaini (2012), Iverson and George, (2014), and Pudasainiand Mergili (2019). While these are commonly named debrisflow models, their validity extends beyond this typical cate-gory of mass movement. This is both apparent from modelapplications (Mergili et al., 2018) and theoretical considera-tions (Pudasaini, 2012). A major cause for the usage of debrisflow as a term here is the assumption of unstructured flow,which we are aiming to solve in this work.

2.2 Model description

We define two phases within the flow, solids and fluids, indi-cated by s and f , respectively. A specified fraction of solidswithin this mixture is at any point part of a structured ma-trix. This structured solid phase, indicated by sc, envelopsand confines a fraction of the fluids in the mixture, indicated

by fc. The solids and fluids are defined in terms of the phys-ical properties such as densities (ρf,ρs) and volume frac-tions (αf =

ff+s

,αs =s

f+s). The confined fractions of their

respective phases are indicated as fsc and ffc for the volumefraction of confined solids and fluids, respectively (Eqs. 1, 2and 3).

αs+αf = 1 (1)αs (fsc+ (1− fsc))+αf (ffc+ (1− ffc))= 1 (2)

(fsc+ (1− fsc))= (ffc+ (1− ffc))= 1 (3)

For the solids, internal friction angle (φs) and effective(volume-averaged) material size (ds) are additionally de-fined. We also define αc = αs+ffcαf and αu = (1−ffc)αf toindicate the solids with confined-fluid and free-fluid phases,respectively. These phases have a volume-averaged densityρsc,ρf. We let the velocities of the unconfined fluid phase(αu = (1−ffc)αf) be defined as uu = (uu,vu). We assume ve-locities of the confined phases (αc = αs+ ffcαf) can validlybe assumed to be identical to the velocities of the solid phase,uc = (uc,vc)= us = (us,vs). A schematic depiction of therepresented phases is shown in Fig. 1.

A major assumption is made here concerning the veloc-ities of both the confined and free solids (sc and s), thathave a shared averaged velocity (us). We deliberately limitthe flow description to two phases, opposed to the innovativework of Pudasaini and Mergili (2019) that develop a multi-mechanical three-phase model. This choice is motivated byconsiderations of applicability (reducing the number of re-quired parameters), the infancy of three-phase flow descrip-tions and finally the general observations of the validity ofthis assumption (Ishii, 1975; Ishii and Zuber, 1979; Drew,1983; Jakob et al., 2005; George and Iverson, 2016).

The movement of the flow is described initially by meansof mass and momentum conservation (Eqs. 4 and 5).

∂αc

∂t+∇ · (αcuc)= 0 (4)

∂αu

∂t+∇ · (αuuu)= 0 (5)

Here we add the individual forces based on the work ofPudasaini and Hutter (2003), Pitman and Le (2005), Puda-saini (2012), Pudasaini and Fischer (2016), and Pudasainiand Mergili (2019) (Eqs. 6 and 7).

∂

∂t(αcρcuc)+∇ · (αcρcuc⊗uc)= αcρcf

−∇ ·αcTc+pc∇αc

+MDG+Mvm (6)∂

∂t(αuρfuu)+∇ · (αuρfuu⊗uu)= αuρff

−∇ ·αuTu+pf∇αu

−MDG−Mvm (7)

https://doi.org/10.5194/gmd-14-1841-2021 Geosci. Model Dev., 14, 1841–1864, 2021


Figure 1. A schematic depiction of the flow contents. Both structured and unstructured solids are present. Fluids can be either free or confinedby the structured solids.

Here f is the body force (a part of which is gravity), MDG isthe drag force, Mvm is the virtual mass force, and TcandTuare the stress tensors for solids with confined fluids andunconfined phases, respectively. The virtual mass force de-scribed the additional work required by differential accelera-tion of the phases. The drag force describes the drag along theinterfacial boundary of fluids and solids. The body force de-scribes external forces such as gravitational acceleration andboundary forces. Finally, the stress tensors describe the in-ternal forces arising from strain and viscous processes. Boththe confined and unconfined phases in the mixture are subjectto stress tensors (Tc, and Tu), for which the gradient acts as amomentum source. Additionally, we follow Pudasaini (2012)and add a buoyancy force (pc∇αc and pf∇αu).

2.2.1 Stress tensors describing internal structure

Based on known two-phase mixture theory, the internal andexternal forces acting on the moving material are now set up.This results in several unknowns, such as the stress tensors(Tc and Tu, described by the constitutive equation), the bodyforce (f), the drag force (MDG) and the virtual mass force(Mvm). This section will first describe the derivation of thestress tensors. These describe the internal stress and viscouseffects. To describe structured movements, these require afull stress–strain relationship, which is not present in earliergeneralized mass movements models. Afterwards, existingderivation of the body, drag and virtual mass force are alteredto conform the new constitutive equation.

Our first step in defining the momentum source termsin Eqs. (6) and (7) is the definition of the fluid and solidstress tensors. Current models typically follow the assump-tions made by Pitman and Le (2005), who indicate that “theproportionality (and alignment) of the tangential and normalforces that is imposed as a basal boundary condition is as-sumed to hold throughout the thin flowing layer of material. . . following Rankine (1857), an earth–pressure relation isassumed for diagonal stress components.” Here, the earth–

pressure relationship is a vertically averaged analytical so-lution for lateral forces exerted by an earth wall. Thus, un-structured columns of moving mixtures are assumed. Here,we aim to use the full Mohr–Coulomb relations. Describ-ing the internal stress of soil and rock matrices is com-monly achieved by elasto-plastic simulations of the mate-rial’s stress–strain relationship. Since we aim to model a fullstress description, the stress tensor is equal to the elasto-plastic stress tensor (Eq. 8).

Tc = σ (8)

Here σ is the elasto-plastic stress tensor for solids. The stresscan be divided into the deviatoric and non-deviatoric contri-butions (Eq. 9). The non-deviatoric part acts normally on anyplane element (in the manner in which a hydrostatic pressureacts equally in all directions). Note that we switch to ten-sor notation when describing the stress–strain relationship.Thus, superscripts (α and β) represent the indices of basisvectors (x, y, or z axis in Euclidian space), and obtain tensorelements. Additionally, the Einstein convention is followed(automatic summation of non-defined repeated indices in asingle term).

σαβ = sαβ +13σ γ γ δαβ (9)

Here s is the deviatoric stress tensor and δαβ = [α = β] is theKronecker delta.

Here, we define the elasto-plastic stress (σ ) based ona generalized Hooke-type law in tensor notation (Eqs. 10and 11), where plastic strain occurs when the stress statereaches the yield criterion (Spencer, 2012; Necas, andHlavácek, 2017; Bui et al., 2008).

εαβ

elastic =sαβ

2G+

1− 2νE

σmδαβ (10)

εαβ

plastic = λ∂g

∂σαβ(11)



Here εelastic is the elastic strain tensor, εplastic is the plasticstrain tensor, σm is the mean stress rate tensor, ν is Poisson’sratio, E is the elastic Young’s modulus,G is the shear modu-lus, s is the deviatoric shear stress rate tensor, λ is the plasticmultiplier rate and g is the plastic potential function. Addi-tionally, the strain rate is defined from velocity gradients asEq. (12).

εαβ

total = εαβ

elastic+ εαβ

plastic =12

(∂uαc∂xβ−∂u

βc

∂xα

)(12)

By solving Eqs. (9), (10) and (11) for σ , a stress–strain rela-tionship can be obtained (Eq. 13) (Bui et al., 2008).

σαβ = 2Geγ γ δαβ +Kεγ γ δαβ − λ[(K −

2G3

)∂g

∂σmn δmnδαβ + 2G

∂g

∂σαβ

](13)

Here e is the deviatoric strain rate (eαβ = εγ γ − 13 εαβδαβ ), ψ

is the dilatancy angle, and K is the elastic bulk modulus andthe material parameters defined from E and ν (Eq. 14).

K =E

3(1− 2ν),G=

E

2(1+ ν)(14)

Fracturing or failure occurs when the stress state reachesthe yield surface, after which plastic deformation occurs.The rate of change of the plastic multiplier specifies themagnitude of plastic loading and must ensure a new stressstate conforms to the conditions of the yield criterion. Bymeans of substituting Eq. (13) in the consistency condition( ∂f

∂σαβdσαβ = 0), the plastic multiplier rate can be defined

(Eq. 15) (Bui et al., 2008).

λ=2Gεαβ ∂f

∂σαβ+

(K − 2G

3

)εγ γ

∂f

∂σαβσαβδαβ

2G ∂f∂σmn

∂g∂σmn +

(K − 2G

3

)∂f∂σmn δmn ∂g

∂σmn δmn(15)

The yield criteria specifies a surface in the stress state spacethat the stress state can not pass and at which plastic deforma-tion occurs. A variety of yield criteria exist, such as Mohr–Coulomb, Von Mises, Drucker–Prager and Tresca (Spencer,2012). Here, we employ the Drucker–Prager model fittedto Mohr–Coulomb material parameters for its accuracy insimulating rock and soil behaviour and numerical stability(Spencer, 2012; Bui et al., 2008) (Eqs. 16 and 17).

f (I1,J2)=√J2+αφI1− kc = 0 (16)

g (I1,J2)=√J2+αφI1 sin(ψ) (17)

Here I1 and J2 are tensor invariants (Eqs. 18 and 19).

I1 = σxx+ σ yy + σ zz (18)

J2 =12sαβsαβ (19)

Here the Mohr–Coulomb material parameters are used to es-timate the Drucker–Prager parameters (Eq. 20).

αφ =tan(φ)√

9+ 12tan2φ,kc =

3c√9+ 12tan2φ

(20)

Using the definitions of the yield surface and stress–strainrelationship, combining Eqs. (13), (15), (16) and (17), the re-lationship for the stress rate can be obtained (Eqs. 21 and 22).

σ = 2Geαβ +Kεγ γ δαβ − λ[

9K sinψδαβ +G√J2sαβ

](21)

λ=3αKεγ γ +

(G√J2

)sαβ εαβ

27αφK sinψ +G(22)

In order to allow for the description of large deformation, theJaumann stress rate can be used, which is a stress-rate that isindependent from a frame of reference (Eq. 23).

˙σ = σαγ ωβγ + σ γβ ωαγ + 2Geαβ

+Kεγ γ δαβ − λ

[9K sinψ δαβ +

G√J2sαβ

](23)

Here ω is the spin rate tensor, as defined by Eq. (24).

ωαβ =12

(∂vα

∂xβ−∂vβ

∂xα

)(24)

Due to the strain within the confined material, the density ofthe confined solid phase (ρc) evolves dynamically accordingto Eq. (25).

ρc = fscρsεv0

εv+ (1− fsc)ρs+ ffcρf (25)

Here εv is the total volume strain, εv ≈ ε1+ε2+ε3, εi is oneof the principal components of the strain tensor. Since we aimto simulate brittle materials, where volume strain remainsrelatively low, we assume that changes in density are smallcompared to the original density of the material ( ∂ρc

∂t� ρc).

2.2.2 Fragmentation

Brittle fracturing is a processes commonly understood to takeplace once a material internal stress has reached the yieldsurface, and plastic deformation has been sufficient to passthe ultimate strength point (Maurel and Cumescure, 2008;Hušek et al., 2016). A variety of approaches to fracturing ex-ist within the literature (Ma et al., 2014; Osomo and Steeb,2017). Finite element method (FEM) models use strain-based approaches (Loehnert et al., 2008). For SPH imple-mentations, as will be presented in this work, distance-basedapproaches have provided good results (Maurel and Cumbes-cure, 2008). Other works have used strain-based fracture cri-teria (Xu et al., 2010) . Additionally, dynamic degradationof strength parameters have been implemented (Grady and



Kipp, 1980; De Vuyst and Vignjevic, 2013; Williams, 2019).Comparisons with observed fracture behaviour has indicatedthe predictive value of these schemes (Xu et al., 2010; Hušeket al., 2016). We combine the various approaches to best fitthe dynamical multi-phase mass movement model that is de-veloped. Following Grady and Kipp (1980), we simulate adegradation of strength parameters. Our material consists ofa soil and rock matrix. We assume fracturing occurs along theinter-granular or inter-rock contacts and bonds (see also Co-hen et al., 2009). Thus, cohesive strength is lost for any frac-tured contacts. We simulate degradation of cohesive strengthaccording to a volume strain criteria. When the stress statelies on the yield surface (the set of critical stress states withinthe six-dimensional stress-space), during plastic deforma-tion, strain is assumed to contribute to the fracturing of thegranular or rock material. A critical volume strain is takenas a material property, and the breaking of cohesive bondsoccurs based on the relative volume strain. Following Gradyand Kipp (1980) and De Vuyst and Vignjevic (2013), we as-sume that the degradation behaviour of the strength param-eter is distributed according to a probability density distri-bution. Commonly, a Weibull distribution is used (Williams,2019). Here, for simplicity we use a uniform distribution ofcohesive strength between 0 and 2c0, although any other dis-tribution can be substituted. Thus, the expression governingcohesive strength becomes Eq. (26).

∂c

∂t=

−c012

(εvεv0

)εc

f (I1,J2)≥ 0,c > 00 otherwise

(26)

Here c0 is the initial cohesive strength of the material, εv0 isthe initial volume,

(εvεv0

)is the fractional volumetric strain

rate and εc is the critical fractional volume strain for fractur-ing.

2.2.3 Water partitioning

During the movement of the mixed mass, the solids canthus be present as a structured matrix. Within such a ma-trix, a fluid volume can be contained (e.g. as originatingfrom a groundwater content in the original landslide ma-terial). These fluids are typically described as groundwaterflow following Darcy’s law, which poses a linear relationshipbetween pressure gradients and flow velocity through a soilmatrix. In our case, we assumed the relative velocity of waterflow within the granular solid matrix as very small comparedto both solid velocities and the velocities of the free fluids. Asan initial condition of the material, some fraction of the wa-ter is contained within the soil matrix (ffc). Additionally, forloss of cohesive structure within the solid phase, we trans-fer the related fraction of fluids contained within that solid

structure to the free fluids.

∂ffc

∂t=−

∂ (1− ffc)

∂t

=

{−ffc

c0c

max(0.0,εv)εf


(27)

∂fsc

∂t=−

∂ (1− fsc)

∂t

=

{−fsc

c0c

max(0.0,εv)εf


(28)

Beyond changes in ffc through fracturing of structured solidmaterials, no dynamics are simulated for influx or outflux offluids from the solid matrix. The initial volume fraction offluids in the solid matrix defined by fffc and sfsc remainsconstant throughout the simulation. The validity of this as-sumption can be based on the slow typical fluid velocities ina solid matrix relative to fragmented mixed fluid–solid flowvelocities (Kern, 1995; Saxton and Rawls, 2006). While theaddition of evolving saturation would extend the validity ofthe model, it would require implementation of pre-transferfunctions for evolving material properties, which is beyondthe scope of this work. An important note on the pointsmade above is the manner in which fluids are re-partitionedafter fragmentation. All fluids in fragmented solids are re-leased, but this does not equate to free movement of the flu-ids or a disconnection from the solids that confined them.Instead, the equations continue to connect the solids and flu-ids through drag, viscous and virtual mass forces. Finally, thedensity of the fragmented solids is assumed to be the initiallyset solid density. Any strain-induced density changes are as-sumed small relative to the initial solid density (ρc

ρs� 1).

2.2.4 Fluid stresses

The fluid stress tensor is determined by the pressure and theviscous terms (Eqs. 29 and 30). Confined solids are assumedto be saturated and constant during the flow.

Tu = pfI+ τ f (29)

τ f = ηf[∇uu+ (∇uc)

t]

−ηf

αuA(αu)(∇αc (uu−uc)+ (uc−uu)∇αc ) (30)

Here I is the identity tensor, τ f is the viscous stress tensor forfluids , pf is the fluid pressure, ηf is the dynamic viscosity ofthe fluids and A is the mobility of the fluids at the interfacewith the solids that acts as a phenomenological parameter(Pudasaini, 2012).

The fluid pressure acts only on the free fluids here, asthe confined fluids are moved together with the solids. InEq. (30), the second term is related to the non-Newtonian vis-cous force induced by gradients in solid concentration. Theeffect as described by Pudasaini (2012) is induced by a solidconcentration gradient. In the case of unconfined fluids and



unstructured solids (fsc = 0,ffc = 0); this force is identicalto the description in the unstructured equations. Within ourflow description, we see no direct reason to eliminate or alterthis force with a variation in the fraction of confined fluidsor structured solids. We only consider the interface betweensolids and free fluids as an agent that induces this effect, andtherefore the gradient of the gradient of the solids and con-fined fluids (∇(αs+ffcαf)=∇αc) is used instead of the totalsolid phase (∇αs).

2.2.5 Drag force and virtual mass

Our description of the drag force follows the work of Pu-dasaini (2012), Pudasaini et al. (2018), where a generalizedtwo-phase drag model is introduced and enhanced. We splittheir work into a contribution from the fraction of structuredsolids (fsc) and unconfined fluids (1− ffc) (Eq. 31).

CDG =fscαcαu (ρc− ρf)g

UT,c (G (Re))+ Sp(uu−uc) |uu−uc|

j−1

+(1− fsc)αcαu (ρs− ρf)g

UT,uc(PF

(Rep

)+ (1−P)G (Re)

)+ Sp

(uu−uc) |uu−uc|j−1 (31)

Here UT,c is the terminal or settling velocity of the struc-tured solids, UT,uc is the terminal velocity of the unconfinedsolids, P is a factor that combines solid- and fluid-like contri-butions to the drag force, G is the solid-like drag contribution,F is the fluid-like drag contribution, and Sp is the smoothingfunction (Eqs. 32 and 34). The exponent j indicates the typeof drag: linear (j = 0) or quadratic (j = 1).

Within the drag, the following functions are defined:

F =γ

180

(αf

αs

)3

Rep,G= αM(Rep)−1f . (32)

Sp = (Pαc+

1−Pαu

)K, (33)

K = |αcuc+αuuu| ≈ 10ms−1, (34)

where M is a parameter that varies between 2.4 and 4.65based on the Reynolds number (Pitman and Le, 2005). Thefactor P that combines solid- and fluid-like contributions tothe drag is dependent on the volumetric solid content in the

unconfined and unstructured materials(P =

(αs(1−fsc)αf(1−ffc)

)m)with m≈ 1. Additionally, we assume the factor P is zerofor drag originating from the structured solids. As stated byPudasaini and Mergili (2019), “as limiting cases: P suit-ably models solid particles moving through a fluid”. In ourmodel, the drag force acts on the unconfined fluid momen-tum (uucαf(1− ffc)). For interactions between unconfinedfluids and structured solids, larger blocks of solid structuresare moving through fluids that contains solids of smaller size.

Virtual mass is similarly implemented based on the workof Pudasaini (2012) and Pudasaini and Mergili (2019)

(Eq. 35). The adapted implementation considers the solidstogether with confined fluids to move through a free-fluidphase.

CVMG = αcρu

(12

(1+ 2αc

αu

))((∂uu

∂t+ uu · ∇uu

)−

(∂uc

∂t+ uc · ∇uc

))(35)

Here CDG =12

(1+2αcαu

)is the drag coefficient.

2.2.6 Boundary conditions

Finally, following the work of Iverson and Denlinger (2001),Pitman and Le (2005), and Pudasaini (2012), a boundarycondition is applied to the surface elements that contact theflow (Eq. 36).

|S| =N tan(φ) (36)

Here N is the normal pressure on the surface element and Sis the shear stress.

2.3 Depth-averaging

The majority of the depth-averaging in this work is analo-gous to the work of Pitman and Le (2005), Pudasaini (2012),and Pudasini and Mergili (2019). Depth-averaging throughintegration over the vertical extent of the flow can be donebased on several useful and often-used assumptions, e.g.1h

∫ h0 xdh= x, for the velocities (uu and uc); solid, fluid, and

confined fractions (αf, αs, ffc and fsc); and material proper-ties (ρu, φ and c). Besides these similarities and an identicalderivation of depth-averaged continuity equations, three ma-jor differences arise.

i. Fluid pressure. Previous implementations of general-ized two-phase debris flow equations have commonlyassumed hydrostatic pressure ( ∂p

∂z= gz) (Pitman and

Le, 2005; Pudasaini, 2012; Abe and Konagai, 2016).Here we follow this assumption for the fluid pressureat the base and solid pressure for unstructured material(Eqs. 37 and 38).

Pbs,u =−(1− γ )αsgzh (37)

Pbu =−gzh (38)

Here γ = ρfρs

is the density ratio (not to be confused witha tensor index when used in superscript) (–).

However, larger blocks of structure material can havecontact with the basal topography. Due to density dif-ferences, larger blocks of solid structures are likely tomove along the base (Pailhia and Pouliquen, 2009; Iver-son and George, 2014). If these blocks are saturated,water pressure propagates through the solid matrix andhydrostatic pressure is retained. However, in cases of



an unsaturated solid matrix that connects to the base,hydrostatic pressure is not present there. We introducea basal fluid pressure propagation factor B(θeff,dsc, . . .)

that describes the fraction of fluid pressure propagatedthrough a solid matrix (with θeff as the effective satu-ration, and as dsc the average size of structured solidmatrix blocks). This results in a basal pressure equal toEq. (39).

Pbc =−(1− fsc)(1− γ )(1− fsc)αs

(1− ffc)αfgzh

− fsc (1− γ )B(fsc)αs

(ffc)αfgzh (39)

The basal pressure propagation factor (B) should the-oretically depend mostly on saturation level, similarlyto the pedotransfer function, as a full saturation meansperfect propagation of pressure through the mixture, andlow saturation equates to minimal pressure propagation(Saxton and Rawls, 2006). Additionally, it should de-pend on pedotransfer functions and the size distributionof structured solid matrices within the mixture. For lowsaturation levels, it can be assumed that no fluid pres-sure is retained. Combined with an assumed soil ma-trix height being identical to the total mixture height,this results in B = 0. Assuming saturation of a struc-ture’s solids results in a full propagation of pressuresand B = 1.

ii. Stress–strain relationship. Depth-averaging the stress–strain relationship in Eqs. (22) and (23) requires a verti-cal solution for the internal stress. First, we assume anynon-normal vertical terms are zero (Eq. 40). Commonly,Rankine’s earth pressure coefficients are used to expressthe lateral earth pressure by assuming vertical stress tobe induced by the basal solid pressure (Eqs. 41 and 42)(Pitman and Le, 2005; Pudasaini, 2012; Abe and Kona-gai, 2016).

σ zx = σ zy = σ yz = σ xz = 0 (40)

σ zz =12Pbs ,σ

zzb = Pbs (41)

Ka =1− sin(φ)1+ sin(φ)

,Kp =1− sin(φ)1+ sin(φ)

(42)

Here we enhance this with Bell’s extension for cohe-sive soils (Eq. 45) (Xu et al., 2019). This lateral normal-directed stress term is added to the full stress–strain so-lution.

σxx =Kσzzb− 2c√K +

1h

∫ h

0σxxdh (43)

Finally, the gradient in pressure of the lateral interfacesbetween the mixture is added as a depth-averaged accel-eration term (Eq. 44).

Sxc = αc(1h

(∂ (hσ xx)

∂x+∂ (hσ yx)

∂y

))+ . . . (44)

iii. Depth-averaging other terms. While the majority ofterms allow for depth-averaging as it was proposed byPudasaini (2012), an exception arises. Depth-averagingof the vertical viscosity terms is required. The non-Newtonian viscous terms for the fluid phase were de-rived assuming a vertical profile in the volumetric solidphase content. Here, we alter the derivation to use thisassumption only for the non-structured solids, as op-posed to the structured solids where ∂αs

∂z= 0.∫ s

b

∂

∂z

(∂αs∂z

(uu− uc)

)dz=

[∂αs∂z

(uu− uc)

]s

b

= (uu− uc)

[∂αs∂z

]s

b

= (uu− uc)

[∂αs∂z

]s

b

=(uu− uc)(1− fsc)ζαs

h(45)

Here ζ is the shape factor for the vertical distributionof solids (Pudasaini, 2012). Additionally, the momen-tum balance of Pudasaini (2012) ignores any deviatoricstress (τxy = 0), following Savage and Hutter (1989)and Pudasaini and Hutter (2007). Previously this termhas been included by Iverson and Denlinger (2001), Pit-man and Le (2005), and Abe and Kanogai (2016). Herewe include these terms since a full stress–strain relation-ship is included.

2.3.1 Basal frictions

Additionally we add the Darcy–Weisbach friction, which is aChézy-type friction law for the fluid phase that provides drag(Delestre et al., 2014). This ensures that without solid phasea clear fluid does lose momentum due to friction from basalshear. This was successfully done in Bout et al. (2018) andwas similarly assumed in Pudasaini and Fischer (2016) forfluid basal shear stress.

Sf =g

n2uu |uu|

h43

(46)

Here n is Manning’s surface roughness coefficient.

2.3.2 Depth-averaged equations

The following set of equations is thus finally achieved fordepth-averaged flow over sloping terrain (Eqs. 47–71).

∂h

∂t+∂

∂x[h(αuuu+αcuc)]+

∂

∂y[h(αuuu+αcuc)]

= R− I (47)∂αch

∂t+∂αchuc

∂x+∂αchvc

∂y= 0 (48)

∂αuh

∂t+∂αuhuu

∂x+∂αuhvu

∂y= R− I (49)



∂

∂t

[αch(uc− γcCVM(uu− uc))

]+∂

∂x

[αch

(u2

c − γcCVM

(u2

u− u2c

))]+∂

∂y

[αch(ucvc− γC (uuvu− ucvc))

]= hSxc (50)

∂

∂t

[αch(vc− γcCVM(vu− vc))

]+∂

∂x

[αch(usvs− γcCVM (uuvu− ucvc))

]+∂

∂y

[αch

(v2

c − γCVM

(v2

u − v2c

))]= hSyc (51)

∂

∂t

[αuh

(uu−

αc

αuCVM(uu− uc)

)]+∂

∂x

[αuh

(u2

u−αc

αuCVM

(u2

u− u2c

)+βxuh

2

)]+∂

∂y

[αuh(uuvu− γcCVM (uuvu− ucvc))

]= hSxu − Iuu (52)

∂

∂t

[αuh

(vu−

αc

αuCVM(vu− vc)

)]+∂

∂x

[αuh

(uuvu−

αc

αuCVM (uuvu− ucvc)

)]+∂

∂y

[αuh

(v2

u − γcCVM

(v2

u − v2c

)+βyuh

2

)]= hSyu − Ivu (53)

Sxc = αc

[gx +

1h

(∂ (hσ xx)

∂x+∂ (hσ yx)

∂y

)−Pbc(

uc

|uc|tanφ+ ε

∂b

∂x)

]− εαcγcpbu

[∂h

∂x+∂b

∂x

]+CDG (uu− uc) |uu−uc|

J−1 (54)

Syc = αc

[gy +

1h

(∂ (hσ xy)

∂x+∂ (hσ yy)

∂y

)−Pbc(

vs

|us|tanφ+ ε

∂b

∂y)

]− εαcγcpbu

[∂h

∂y+∂b

∂y

]+CDG (vu− vc) |vu− vc|J−1 (55)

Sxu = αu

[gx −

12Pbuh

αu

∂αc

∂x+Pbu

∂b

∂x−

Aηu

αu(2∂2uu

∂x2 +∂2vu

∂xy+∂2uu

∂y2 −Xuu

ε2h2

)+Aηu

αu

(2∂

∂x

(∂

∂x(uu− uc)

)+∂

∂y

(∂αc

∂x(vu− vc)

+∂αu

∂y(uu− uc)

))−

Aηuζαs(1− fsc)(uu− uc)

αuh2

−g

n2uu |uu|

h43

]−

1γcCDG (uu− uc) |uu−uc|

J−1 (56)

Syu = αu

[gy −

12Pbuh

αf

∂αc

∂y+Pbu

∂b

∂y

−Aηu

αu

(2∂2uf

∂y2 +∂2vf

∂xy+∂2uf

∂x2 −Xufε2h2

)

+Aηu

αc

(2∂

∂y

(∂

∂y(vu− vc)

)+∂

∂x

(∂αc

∂y(uu− uc)+

∂αc

∂x(vu− vc)

))−Aηuζαs(1− fsc)(vu− vc)

αuh2 −g

n2vu |uu|

h43

]−

1γcCDG (vu− vc) |uu−uc|

J−1 (57)

Pbc =−(1− fsc)(1− γ )(1− fsc)αs

(1− ffc)αfgzh

− fsc (1− γ )(fsc)αs

(ffc)αfgzh (58)

Pbu =−gzh (59)

γc =ρu

ρc,γ =

ρf

ρs(60)

CDG =fscαcαu (ρc− ρf)g

UT,c (G (Re))+ Sp

+(1− fsc)αcαu (ρs− ρf)g

UT,uc(PF

(Rep

)+ (1−P)G (Re)

)+ Sp

(61)

Sp = (Pαc+

1−Pαu

)K (62)

K = |αcuc+αuuu| (63)

F =γ

180

(αf

αs

)3

ReP,G= αM(Rep)−1f ,

Rep =ρfdUt

ηf,NR =

√gLHρf

αfηf,

NRA =

√gLHρf

Aηf(64)

CVM =

(12

(1+ 2αc

αu

))(65)

˙σ = σαγ ωβγ + σ γβ ωαγ + 2Geαβ

+Kεγ γ δαβ − λ

[9K sinψ δαβ +

G√J2sαβ

](66)

λ=3αKεγ γ +

(G√J2

)sαβ εαβ

27αφK sinψ +G(67)

K =E

3(1− 2ν),G=

E

2(1+ ν)(68)

σαβ = sαβ +13σ γ γ δαβ (69)

εαβ =12

(∂vα

∂xβ−∂vβ

∂xα

)ωαβ =

12

(∂vα

∂xβ−∂vβ

∂xα

)(70)



αφ =tan(φ)√

9+ 12tan2φkc =

3c√9+ 12tan2φ

(71)

Here X is the shape factor for vertical shearing of the fluid(X ≈ 3 in Iverson and Denlinger, 2001), R is the precipita-tion rate and I is the infiltration rate.

2.3.3 Closing the equations

Viscosity is estimated using the empirical expression fromJulien and O’Brien (1997), which relates dynamic viscosityto the solid concentration of the fluid (Eq. 72).

η = αeβαs (72)

Here α is the first viscosity parameter and β the second vis-cosity parameter.

Finally, the settling velocity of small (d<100 µm) grainsis estimated by Stokes equations for a homogeneous spherein water. For larger grains (>1 mm), the equation byZanke (1977) is used (Eq. 73).

UT = 10ηρf

2

d

√√√√√1+

0.01((ρs− ρf)ρf

gd3)

ηρf

− 1

(73)

In which UT is the settling (or terminal) velocity of a solidgrain, η is the dynamic viscosity of the fluid, ρf is the densityof the fluid, ρs is the density of the solids and d is the graindiameter (m).

2.4 Implementation in the material point methodnumerical scheme

Implementing the presented set of equations into a numeri-cal scheme requires considerations of that scheme’s limita-tions and strengths (Stomakhin et al., 2013). Fluid dynam-ics are almost exclusively solved using an Eulerian finite-element solution (Delestre et al., 2014; Bout et al., 2018).The diffusive advection part of such scheme typically doesnot degrade the quality of modelling results. Solid material,however, is commonly simulated with higher accuracy us-ing an Lagrangian finite-element method or discrete-elementmethod (Maurel and Cumbescure, 2008; Stomakhin et al.,2013). Such schemes more easily allow for the material tomaintain its physical properties during movement. Addition-ally, advection in these schemes does not artificially diffusethe material since the material itself is discretized, instead ofthe space (grid) on which the equations are solved. In ourcase, the material point method (MPM) provides an appro-priate tool to implement the set of presented equations (Buiet al., 2008; Maurel and Cumbescure, 2008; Stomakhin etal., 2013). Numerous existing modelling studies have imple-mented in this method (Pastor et al., 2014; Abe and Kanogai,2016). Here, we use the MPM method to create a two-phasescheme. This allows the usage of finite-element aspects for

the fluid dynamics, which are so successfully described bythe that method (particularly for water in larger areas; seeBout et al., 2018).

2.4.1 Mathematical framework

The mathematic framework of smooth particles solves dif-ferential equations using discretized volumes of mass repre-sented by kernel functions (Libersky and Petschek, 1991; Buiet al., 2008; Stomakhin et al., 2013). Here, we use the cubicspline kernel as used by Monaghan (2000) (Eq. 74).

W (r,h)=

10

7πh2

(1− 3

2q2+

34q

3)

0≤ |q| ≥ 210

28πh2 (2− q)3 1≤ |q|< 2

0 |q| ≥ 2 |q < 0

(74)

Here r is the distance, h is the kernel size and q is the nor-malized distance (q = r

h).

Using this function mathematical operators can be defined.The average is calculated using a weighted sum of particlevalues (Eq. 75), while the derivative depends on the functionvalues and the derivative of the kernel by means of the chainrule (Eq. 76) (Libersky and Petschek, 1991; Bui et al., 2008).

〈f (x)〉 =∑N

j=1

mj

ρjf(xj)W(x− xj ,h) (75)⟨

∂f (x)

∂x

⟩=

∑N

j=1

mj

ρjf(xj) ∂Wij

∂xi(76)

HereWij =W(xi−xj ,h) is the weight of particle j to parti-cle I , and r =

∣∣xi − xj ∣∣ is the distance between two particles.The derivative of the weight function is defined by Eq. (77).

∂Wij

∂xi=xi − xj

r

∂Wij

∂r(77)

Using these tools, the momentum equations for the parti-cles can be defined (Eqs. 78–84). Here, we follow Mon-aghan (2000) and Bui et al. (2008) for the definition of artifi-cial numerical forces related to stability. Additionally, stress-based forces are calculated on the particle level, while othermomentum source terms are solved on a Eulerian grid withspacing h (identical to the kernel size).

dvαidt=

1mi

(Fg +Fgrid

)+

∑N

j=1mj

(σαβi

ρ2i

+σαβj

ρ2j

+F nijRαβij

+5ij δαβ) ∂Wij

∂xβi

(78)

εαβ =12

(∑N

j=1

mj

ρj

(vαj −V

αI

) ∂Wij

∂xβi

+

∑N

j=1

mj

ρj

(vβj −V

βI

) ∂Wij

∂xαi

)(79)



ωαβ =12

(∑N

j=1

mj

ρj

(vαj −V

αI

) ∂Wij

∂xβi

−

∑N

j=1

mj

ρj

(vβj −V

βI

) ∂Wij

∂xαi

)(80)

dσαβdt= σ

αγ

i ωβγ

i + σγβ

i ωαγ

i

+ 2Gi eαβi +Ki ε

γ γ δαβi

− λi

[9Ki sinψiδαβ +

Gi√J2isαβi

](81)

λi =

3αKεγ γi +(

Gi√j2i

)sαβi εi

αβ

27αφKi sinψi +Gi(82)

Here i and j are indices indicating the particle, 5ij is an ar-tificial viscous force as defined by Eqs. (83) and (84), andF nijR

αβij is an artificial stress term as defined by Eqs. (85)

and (86).

5ij =

{α5usoundij φij+β5φ

2

ρijvij · xij < 0

0 vij · xij ≥ 0(83)

φij =hijvijxij∣∣xij ∣∣2+ 0.01h2

ij

,xij = xi − xj ,

vij = vi − vj ,hij =12

(hi +hj

)(84)

F nijRαβij =

[Wij

W (d0,h)

]n(R

αβi +R

αβj ) (85)

Rγ γ

i =−ε0σ

γ γ

i

ρ2i

(86)

Here ε0 is a small parameter ranging from 0 to 1, α5 andβ5 are constants in the artificial viscous force (often chosenclose to 1), and usound is the speed of sound in the material.

The conversion from particles to gridded values and viceversa depends on a grid basis function that weighs the in-fluence of particle values for a grid centre. Here, a functionderived from dyadic products of one-dimensional cubic B-splines is used as has been done previously by Steffen etal. (2008) and Stomakhin et al. (2013) (Eq. 84).

N (x)=N(xx)·N

(xy),

N (x)=

12 |x|

3− x2+

23 0≤ |x| ≥ 2

−16 |x|

3+ x2− 2 |x| + 4

3 1≤ |x|< 20 |x| ≥ 2|x = 0

(87)

2.4.2 Particle placement

Particle placement is typically done in a constant pattern, asinitial conditions have some constant density. The simplestapproach is a regular square or triangular network, with parti-cles on the corners of the network. Here, we use an approachthat is more adaptable to spatially varying initial flow height.

Figure 2. Example of a kernel function used as integration domainfor mathematical operations.

The R2 sequence approaches, with a regular quasi-randomsequence, a set of evenly distributed points within a square(Roberts, 2020) (Eq. 85).

xn = nαmod1,α =

(1cp,

1c2

p

)(88)

Here xn is the relative location of the nth particle

within a grid cell, and cp =(

9+√

6918

) 13+

(9−√

6918

) 13≈

1.32471795572 is the plastic constant.The number of particles placed for a particular flow height

depends on the particle volume VI, which is taken as a globalconstant during the simulation.

3 Flume experiments

3.1 Flume setup

In order to validate the presented model, several controlledexperiments were performed and reproduced using the devel-oped equations. The flume setup consists of a steep incline,followed by a near-flat runout plane (Fig. 3). A massive ob-stacle is placed on the separation point of the two planes.This blocks the path of two-fifths of the width of the movingmaterial. For the exact dimensions of both the flume partsand the obstacle, see Fig. 3.

Two tests were performed whereby a cohesive granularmatrix was released at the upper part of the flume setup.Both of these volumes had dimensions of 0.2× 0.3× 0.25 m(height, length, width). For both of these materials, amixture of high-organic-content silty-clay soils where used.The material’s strength parameters were obtained usingtri-axial testing (cohesion, internal friction angle Young’smodulus and Poisson ratio). The first set of materialsproperties were c = 26.7 kPa and φ = 28◦. The second setof materials properties were c = 18.3 kPa and φ = 27◦.For both of the events, pre- and post-release elevationmodels were made using photogrammetry. The modelwas set up to replicate the situations using the measured



Figure 3. Example particle distributions using the R2 sequence, note that while not all particles are equidistant, the method producesdistributed particle patterns that adapt well to varying density.

Figure 4. The dimensions of the flume experiment setup used in this work.

input parameters. Numerical settings were chosen as fol-lows: {αs = 0.5,αf = 0.5,fsc = 1.0,ffc = 1.0,ρf = 1000,ρs = 2400,E = 12 · 106 Pa,K = 23 · 106 Pa,ψ = 0,α5 =1,β5 = 1,X,ζ,j = 2,usound = 600, dx = 10,VI =

0m/s,h= 10,n= 0.1,α = 1,β = 10,M = 2.4,B =0,NR = 15000,NRA = 30}. Calibration was performedby means of input variation. The solid fraction and elasticand bulk modulus were varied between 20 % and 200 % oftheir original values with increments of 10 %. Accuracy wasassessed based on the percentage accuracy of the deposition(comparison of the modelled vs. the observed presence ofmaterial).

3.2 Results

Both the mapped extent of the material after flume experi-ments, as the simulation results are shown in Fig. 5. Cali-brated values for the simulations are {αs = 0.45, E = 21.6 ·106 Pa,K = 13.8 · 106 Pa }.

As soon as the block of material impacts the obstacle,stress increases as the moving object is deformed. This stressquickly propagates through the object. Within the scenariowith lower cohesive strength, as soon as the stress reachedbeyond the yield strength, degradation of strength parame-

ters took place. In the results, a fracture line developed alongthe corner of the obstacle into the length direction of themoving mass. Eventually, this fracture developed to half thelength of the moving body and severe deformation resulted.As was observed from the tests, the first material experienceda critical fracture, while the second test resulted in moderatedeformation near the impact location. Generally, the resultscompare well with the observed patterns, although the ex-act shape of the fracture is not replicated. Several reasonsmight be the cause of the moderately accurate fracture pat-terns. Other studies used a more controlled setup where un-certainties in applied stress and material properties were re-duced. Furthermore, the homogeneity of the material used inthe tests can not completely assumed. Realistically, minor al-terations in compression used to create the clay blocks haveleft spatial variation in density, cohesion and other strengthparameters.



Figure 5. A comparison of the final deposits of the simulations and the mapped final deposits and cracks within the material: (from left toright) photogrammetry mosaic, comparison of simulation results to mapped flume experiment, strain, final strength fraction remaining.

4 Numerical tests

4.1 Numerical setup

In order to further investigate some of the behaviours of themodel and highlight the novel types of mass movement dy-namics that the model implements, several numerical testshave been performed. The setup of these tests is shown inFig. 6.

Numerical settings were chosen for three differentblocks with equal volumes but distinct properties. Co-hesive strength and the bulk modulus were varied (seeFig. 6). Remaining parameters were chosen as follows:{αs = 0.5,αf = 0.5,fsc = 1.0,ffc = 1.0,ρf = 1000kgm−3,ρs = 2400kgm−3,E = 1e12Pa,ψ = 0,α5 = 1,β5 =1,X,ζ,j = 2,usound = 600ms−1,dx = 10m,VI =

0m/s,h= 10m,n= 0.1,α = 1,β = 10,M = 2.4,B =0,NR = 15000,NRA = 30}.

4.2 Results

Several time slices for the described numerical scenarios areshown in Figs. 7 and 8.

Fractures develop in the mass movements based on accel-eration differences and cohesive strength. For scenario 2A,the stress state does not reach beyond the yield surface, andall material is moved as a single block. Scenario 2B, whichfeatures lowered cohesive strength, fractures and the massesseparate based on the acceleration caused by slopes.

Fracturing behaviour can occur in MPM schemes due tonumerical limitations inherent in the usage of a limited in-tegration domain. Here, validation of real physically based

fracturing is present in the remaining cohesive fraction. Thisvalue only reduces in the case of plastic yield, where increas-ing strain degrades strength parameters according to our pro-posed criteria. Numerical fractures would thus have a cohe-sive fraction of 1. In all simulated scenarios, such numericalissues were not observed.

Fragmentation occurs due to spatial variation in acceler-ation in the case of scenario 3A and 3B. For scenario 3A,the yield surface is not reached and the original structureof the mass is maintained during movement. For 3C, frag-mentation is induced by lateral pressure and buoyancy forcesalone. Scenario 3B experiences slight fragmentation at theedges of the mass but predominantly fragments when reach-ing the valley, after which part of the material is acceleratedto count to the velocity of the mass. For all the shown simu-lations, fragmentation does not lead to significant phase sep-aration since virtual mass and drag forces converge the sepa-rate phase velocities to their mixture-averaged velocity. Thestrength of these forces partly depends on the parameters; ef-fects of more immediate phase separation could be studied ifother parameters are used as input.

5 Discussion

A variety of existing landslide models simulate the behaviourof lateral connected material through a non-linear, non-Newtonian viscous relationship (von Boetticher et al., 2017;Fornes et al., 2017; Pudasaini and Mergili, 2019; Greco etal., 2019). These relationships include a yield stress and areusually regularized to prevent singularities from occurring.While this approach is incredibly powerful, it is fundamen-



Figure 6. The dimensions of the numerical experiment setups used in this work: setup 1 (left) and setup 2 (right).

Figure 7. Several time slices for numerical scenarios 2A, 2B and2C. See Fig. 6 for the dimensions and terrain setup.

tally different from the work proposed here. These viscousapproaches do not distinguish between elastic or plastic de-formation and typically ignore deformations if stress is insuf-ficient. Additionally, fracturing is not implemented in thesemodels. The approach taken in this work attempts to sim-ulate a full stress–strain relationship with Mohr–Coulombtype yield surface. This provides new types of behaviour andcan be combined with non-Newtonian viscous approaches asmentioned above. A major downside to the presented workis the steep increase in computational time required to main-tain an accurate and stable simulation. Commonly, a near 100

times increase in computational time has been observed dur-ing the development of the presented model.

The presented model shows a good likeness to flume ex-periments, and numerical tests highlight behaviour that iscommonly observed for landslide movements. There are,however, inherent scaling issues and the material used inthe flume experiments is unlikely to form larger landslidemasses. The measured physical strength parameters of thematerial used in the flume experiments would not allow forsustained structured movement at larger scales. There is thusthe need for more real-scale validation cases. The applicationof the presented type of model is most directly noticeablefor block-type landslide movements that have fragmented ei-ther upon impact with some obstacle or during the transitionphase. Of importance here is that the moment of fragmen-tation is often not reported in studies on fast-moving land-slides, potentially due to the complexities involved in know-ing the details of this behaviour from post-event evidence.Validation would therefore have to occur in cases where de-posits are not fully fragmented, indicating that this processwas ongoing during the whole movement duration. The spa-tial extent of initiation and deposition would then allow val-idation of the model. Another major opportunity for vali-dation of the novel aspects of the model is the full three-dimensional application to landslides that were reported tohave lubrication effects due to fragmentation of the lowerfraction of flow due to shear.

An important point of consideration in the developmentof complex multi-process generalized models is the applica-bility. As a detailed investigative research tool, these mod-els provide a basic scenario of usage. However, both for re-search and beyond for applicability in disaster risk reductiondecision support, the benefit drawn from these models de-pends on the practical requirement for parameterization andthe computational demands for simulation. With an increas-ing complexity in the description of multi-process mechanicscomes the requirement for more measured or estimated phys-ical parameters. Inspection of the presented method showsthat in principle, a minor amount of new parameters are in-



Figure 8. Several time slices for numerical scenarios 3A ,3B and3C. See Fig. 6 for the dimensions and terrain setup.

troduced. The cohesive strength, a major focus of the model,becomes highly important depending on the type of move-ment being investigated. Additionally, the bulk and elasticmodulus are required. These three parameters are commonsimulation parameters in geotechnical research and can beobtained from common tests on sampled material (Alsalmanet al., 2015). Finally, the basal pressure propagation parame-ter (B) is introduced. However, within this work the value ofthis parameter is chosen to have a constant value of 1. As a re-sult, the model does require additional parameters, althoughthese are relatively easy to obtain with accuracy.

There are a variety of aspects of the model that could besignificantly improved. Here, we list several major opportu-nities for future research.

1. Groundwater mechanics. The presented model allowsfor the a solid or granular matrix to be present within theflow. We have assumed the flows in and out of these ma-trices are small enough that they can be ignored. In re-ality, there is a fluid flux in and out of structured solids.This could occur both due to pressure differences anddue to stress and strain of the structured solids. Im-plementing this kind of mechanic requires a dynamic,solid-properties-dependent, soil water retention curve(Van Looy et al., 2017). An example of MPM soil me-chanics with dynamic groundwater implementation canbe found in Bandera et al. (2016).

2. Implementing entrainment and deposition. Currentequations for entrainment (erosion with major grain-grain interactions) are limited to unstructured mix-ture flows (Iverson, 2012; Iverson and Ouyang, 2015;Cuomo et al., 2016; Pudasaini and Fischer, 2016). Ex-tending these models to include a contribution fromstructured solids would be required to implement en-trainment in the presented work.

3. Separation of phases. A major assumption in the pre-sented work is that the velocities of structured solids,free solids and confined fluids are all equal. In reality,there might be separation of structured- and free-solidphases. Additionally, we already discussed the possibil-ity of influx and outflux of confined fluids from the solidmatrix. Recent innovations in three-phase mixture flowsmight be used to extend the presented work to a three-,four- or five-phase model by separating free solids andconfined fluids or adding a Bingham viscous solid–fluidphase (Pudasaini and Mergili, 2019). However, whilethis would implement an additional process, it wouldsignificantly increase the complexity of the equations(in an exponential manner with relation to the numberof phases) and the numerical solutions, which could hin-der practical applicability.

4. Application to large slow-moving landslides. Whenconfined fluids would act as a distinct phase guided bythe mechanics of water flow in granular matrix, ground-water pressures and movement through the structuredsolids could be described. This might enable the modelto perform detailed deformation and groundwater simu-lation of large slow-moving landslides.

5. Numerical improvements. Numerical techniques forparticle-based discretized methods (SPH, MPM) havebeen proposed in the literature. A common issue is nu-merical fracturing of materials when particle strain in-creases beyond the length of the kernel function. Due tothis, the connection between particles is lost, and frac-turing occurs as an artefact of the numerical method.This issue is partly solved by the artificial stress termthat is also used by Bui et al. (2008). Additionally, a ge-ometric subdivide, as used by Xu et al. (2012) and Li



et al. (2015), could counter these artificial fractures. Im-plementing this technique does require additional workto maintain mass and momentum conservation.

6. Three-dimensional solutions. In a variety of scenarios,the assumptions made in the depth-averaged applica-tion of flow models are invalid. A common example isthe impact of mass movements into lakes or other largewater bodies. In such cases, the vertical velocity andconcentration variables are not well described by theirdepth-averaged counterparts. Additionally, the lubrica-tion effect of basal fragmentation of landslides due toshear can not be described without velocity profiles anda vertical stress solution. A full three-dimensional ap-plication would therefore have the potential to increaseunderstanding of these important processes.

6 Conclusions

We have presented a novel generalized mass movementmodel that can describe both unstructured mixture flows andstructured movements of Mohr–Coulomb-type material. Thepresented equations are part of the continuous developmentof the OpenLISEM hazard model, an open-source tool forphysically based multi-hazard simulations. The model buildson the works of Pudasaini (2012) and Bui et al. (2008) todevelop a single holistic set of equations. The model wasimplemented in a GPU-based material point method (MPM)Code. The equations were validated on flume experimentsand numerical tests that highlight the new movement dynam-ics possible with the presented model. The integration of co-hesive structure and a full stress–strain relationship for thestructured solids allows for movement of block-type slides asa single whole. Interactions with terrain, other flow massesor obstacles lead to elasto-plastic deformation and eventu-ally fragmentation. This type of self-alteration of flow prop-erties is novel with mass movement models. Although thepresented equations can provide additional detail for spe-cific mass movement types, applicability of the model forreal events need to be investigated as computational costs aresignificantly increased.

The presented simulation both validates the basic be-haviour of the model and highlights the types of flow dynam-ics made possible by the presented equations. The model’sdependency on cohesive strength and internal friction an-gles matches the flume experiments. The numerical ex-amples show commonly described behaviour for landslidemovements. Although the simulations compare well to theflume experiments, validation is required for real-scale ap-plication to various types of mass movements. Additionally,the presented equations still lack descriptions of processesthat might become important. Separating the fluid and solidphases (as in Pudasaini and Mergili, 2019) could improveflow dynamics and phase separation. With added groundwa-ter mechanics, such as those in Bandera et al. (2016), slow-moving landslide simulations might be described.



Appendix A: List of symbols

h flow heights solid phasef fluid phasesc structured solid phasefc confined fluid phaseρf density of fluidsρs density of solidsαf volumetric fluid-phase fractionαs volumetric solid-phase fractionfsc fraction of solids that is structured (confining)ffc fraction of fluids that is confinedαc volumetric fraction of solids, structured solids and confined fluidsαu volumetric fraction of free fluids (unconfined phase).ρsc volume-averaged density of the solids and confined fluidsuu velocity of the unconfined phase (free fluids)uc velocity of the solids, confining solids and confined fluidsus velocity of the solidsf body forceMDG drag forceMvm virtual mass forceTc stress tensor for the solids, confining solids and confined fluidsTu stress tensor for the free-fluid phaseσ stress tensors deviatoric shear stress rate tensorδ Kronecker deltaεplastic plastic strain rateεelastic elastic strain rateλ plastic multiplier rateg plastic potential functionεtotal total strain ratee deviatoric strain rateν Poisson’s ratioE elastic Young’s modulusG shear modulusK bulk elastic modulusf (I1,J2) yield surface or yield criteriong (I1,J2) plastic potential functionψ dilatancy angleI1 first stress invariantJ2 second stress invariantαφ first Drucker–Prager material constantkc second Drucker–Prager material constantω spin rate tensorεv0 initial volumetric strainεv volumetric strainc0 initial cohesionτ f fluid Cauchy stress tensorpf fluid pressureηf fluids dynamic viscosityA mobility of the fluid at the interfaceCDG drag coefficientUT,c settling velocity of the solids, structured solids and confined fluids



UT,uc settling velocity of the unstructured solidsF drag contribution from solid-like dragG drag contribution from fluid-like dragSp smoothing functionK absolute total mass fluxM(Rep

)empirical function weakly dependent on the Reynolds number

P partitioning parameter for the fluid- and solid-like contributions to dragm an exponent for PCVMG virtual mass coefficient|S| norm of the shear forceN normal force on a plane elementg gravitational accelerationPbs,u basal pressure from the unconfined phasePbu basal pressure from the free fluidsPbc basal pressure from the solids, structured solids and confined fluidsB pressure propagation factor for structured solidsKa active lateral earth–pressure coefficientKp passive lateral earth–pressure coefficientζ shape factor for the vertical gradient in solid concentrationn Manning’s surface roughness coefficientX shape factor for the vertical fluid velocity profileRep particle Reynolds numberNR Reynolds numberNRA interfacial Reynolds numberH typical height of the flowL typical length of the flowα first viscosity parameterβ second viscosity parameterd grain diameterW kernel weight functionr distanceh kernel width (not to be confused with the flow height)q normalized particle distance5ij an artificial viscosity termF nijR

αβij an artificial stress term

ε0 a constant parameter for the artificial stress termα5 and β5 constants in the artificial viscous forceusound speed of sound in the materialN (x) grid kernel functioncp plastic coefficient



Appendix B: Stress remapping

If the state of the stress tensor lies beyond the yield surface,either due to degradation of strength parameters or buildingnumerical errors, a correction must be applied. We imple-ment the correction scheme used by Bui et al. (2008). Thisscheme considers two primary ways in which the stress canhave an undesired state: tension cracking and imperfectlyplastic stress.

B1 Tension cracking

In the case of tension cracking, the stress state has movedbeyond the apex of the yield surface, as described by Chenand Mizuno (1990). The employed solution in this case is tore-map the stress tensor along the I1 axis to be at this apex.The apex is provided by the following yield function:

−αφI1+ kc < 0. (B1)

To solve for this condition, the non-deviatoric stress state isincreased (since I1−

kcαφ

is negative) to lie perpendicular tothe apex point on the I1 axis as follows:

˜σ γ γ = rsγ γ −13

(I1−

kc

αφ

). (B2)

B2 Imperfect plastic stress

Imperfect plastic stress described the state where the stresstensor lies above the apex but beyond the yield criterion andthus have more stress than is supported by the failure criteriathat is set. This criterion is simply the yield surface itself(Eq. B3).

−αφI1+ kc <√J2 (B3)

For this state, re-mapping is done by scaling of the J2 value(Eqs. B4, B5 and B6).

r =−αφI1+ kc√J2

(B4)

˜σ γ γ = rsγ γ +13I1 (B5)

˜σ xy = rsxy, ˜σ xy = rsxz ˜σ xy = rsyz (B6)

Appendix C: Software implementation

The model presented in this article is part of the continueddevelopment of the OpenLISEM modelling tools. The mostrecent set of equations were implemented in the open-sourcealpha version of OpenLISEM hazard model 2.0a. Here wedescribe the details of the implementation of the model intosoftware.

C1 Hybrid MPM

We utilize the MPM framework to be able to discretize partof the equations on a Eulerian regular grid and part of theequations on the Lagrangian particles. Our distinct take onthis method is the representation of the fluid phase entirelyas a finite-element solution, while solids are simulated asdiscrete particle volumes. This allows the model to use themajor benefits that are present when depth-averaged fluidflow is simulated in a grid. Both numerical efficiency andhigh-accuracy coupling with hydrology are lacking in parti-cle methods. For the solid phase, non-dissipative advection,fracturing and stiffness is a major benefit of the MPM ap-proach. Since our model assumed that confined fluids sharetheir velocity with the solids, we advect the confined fluidsas part of the particles. Total fluid volume is then calculatedfrom the free fluids in the finite-element data and the griddedparticle data. A flow chart of the software setup is providedin Fig. 6.



Figure C1. The sub-steps taken by the software to complete a single step of numerical integration.

C2 Finite-element solution

We use a regular Cartesian grid to describe the modellingdomain. Terrain and cell-boundary-based variables are re-produced using the Monotonic Upstream-centered Schemefor Conservation Laws (MUSCL) piecewise linear recon-struction (Delestre et al., 2014). For each cell boundary, aleft and right estimation of acceleration terms, velocity up-dates, and new discharges is made. The left estimates use left-reconstructed variables, while the right estimates use right-reconstructed variables. The final average flux through theboundary determines the actual mass and momentum trans-fer. Local acceleration is averaged from the right estimate ofthe left boundary and left estimate of the right boundary. Anadditional benefit of the used scheme is the automatic esti-mation of continuous and discontinuous terrain. The piece-wise linear reconstructions do not guarantee smooth terrain;for sharp locally variable terrain, pressure terms from verti-cal walls arise that block momentum. These terms allow forbetter estimation of momentum loss by barriers but can beturned off if required for the simulated scenario.

Figure C2. Piecewise linear reconstruction is used by the MUSCLscheme to estimate values of flow heights, velocities and terrain atcell boundaries.



C3 GPU acceleration using OpenCL/OpenGL

In order to create a more efficient setup, both the finite-element and particle interactions are performed on the GPU.We utilize the OpenCL API to compile kernels written inC-style language. These kernels are compiled at the start ofthe simulation and thereby allow for easy customization byusers. While the usage of OpenCL 1.1 forces the usage ofsingle precision floating point numbers, it allows for a widerrange of GPU types to be supported. Finite-element solutionson the GPU are straightforward, as maps are a basic data stor-age type for graphical processing units. Particles are stored assingle precision floating point arrays. Within the frameworkof MPM, iteration of particles within a kernel is required foreach time step and particle. This effectively means O

(n2)

operations are required. Significant efficiency improvementsare obtained by pre-calculation sorting. Particles are sortedbased on their location within the finite-element grid. Basedon the ID of the grid cell, a bitonic mergesort is performed.This sorting algorithm works seamlessly on parallel architec-ture and operates as O

(nlog2 (n)

)(Batcher, 1968). Follow-

ing this, a raster is allocated to store the first indexed occur-rence within the sorted list of particles of that grid cell. Sincethe kernel used for the presented work extends at most to afull width of two grid cells, we must iterate over all particlespresent in nine neighbouring grid cells.

Figure C3. By limiting the kernel and sorting particles before cal-culation, only the distance of particles in neighbouring cells need tobe checked, significantly reducing computational load, particularlyfor larger datasets.

A final benefit to the usage of OpenCL is direct access tosimulation variables for visualization in OpenGL using theOpenGL/OpenCL interoperability functionality. The built-inviewing window of OpenLISEM hazard 2.0a directly usesthe data to draw particles, shapefiles and grid data using cus-tomizable shaders written in the OpenGL shader language.



Code and data availability. All code and data used within thiswork are made open-source as part of the continuous developmentof the OpenLISEM hazard model under the GNU General PublicLicence v3.0. The code and the data are hosted on GitHub(https://github.com/bastianvandenbout/OpenLISEM-Hazard-2.0-Pre-Release, https://doi.org/10.17026/dans-xz4-2tut) (van denBout, 2020). Both binaries and a copy of the source code are alsoavailable on SourceForge, where the manual and compilation guidecan similarly be found (https://sourceforge.net/projects/lisem/,van den Bout, 2021). Finally, more information can be foundat the blog of Bastian van den Bout and Victor G. Jetten(https://blog.utwente.nl/lisem/, van den Bout and Jetten, 2020)

The software and its user interface are written for Windows, butplatform-independent libraries are used and compilation can be per-formed on other platforms.

Hardware requirements for the usage of the model are a 64-bitoperating system that can compile all required external libraries (seethe manual for a full list and description), a graphical processingunit conforming to at least the OpenCL 1.2 standard, and supportfor both OpenGL 4.2 and OpenGL/OpenCL interoperability. Addi-tionally, an approximate 500 MB of hard drive space and 750 MBof memory must also be available.

Author contributions. During this work, model derivation and con-ceptual work was performed by BVDB, VGJ, CJVW, TVA and OM.The presented flume experiments were carried out by CXT, WH andBVDB. Finally, manuscript production, data analysis, and numeri-cal implementation of the methods and equations were carried outby BVDB.

Competing interests. The authors declare that they have no conflictof interest.

Review statement. This paper was edited by Andrew Wickert andreviewed by two anonymous referees.

References

Aaron, J. and Hungr, O.: Dynamic simulation of the motion ofpartially-coherent landslides, Eng. Geol., 205, 1–11, 2016.

Abe, K. and Konagai, K.: Numerical simulation for runout processof debris flow using depth-averaged material point method, SoilsFound., 56, 869–888, 2016.

Alsalman, M., Myers, M., and Sharf-Aldin, M.: Comparison ofmultistage to single stage triaxial tests, in: 49th US Rock Me-chanics/Geomechanics Symposium, Francisco, California, 28June 2015, ARMA-2015-767, 2015.

Bandara, S., Ferrari, A., and Laloui, L.: Modelling landslides in un-saturated slopes subjected to rainfall infiltration using materialpoint method, Int. J. Numer. Anal. Met., 40, 1358–1380, 2016.

Batcher, K. E.: Sorting networks and their applications, in: Proceed-ings of the spring joint computer conference, Atlantic City, NewJersey, 30 April–2 May 1968, 307–314, 1968.

Beutner, E. C. and Gerbi, G. P.: Catastrophic emplacement of theHeart Mountain block slide, Wyoming and Montana, USA, Geol.Soc. Am. Bull., 117, 724–735, 2005.

Bieniawski, Z. T.: Mechanism of brittle fracture of rock: part I –theory of the fracture process, Int. J. Rock. Mech. Min., 4, 395–406, https://doi.org/10.1016/0148-9062(67)90030-7, 1967.

Bout, B., Lombardo, L., van Westen, C. J., and Jetten, V. G.: Inte-gration of two-phase solid fluid equations in a catchment modelfor flashfloods, debris flows and shallow slope failures, Environ.Modell. Softw., 105, 1–16, 2018.

Bui, H. H., Fukagawa, R., Sako, K., and Ohno, S.: Lagrangianmeshfree particles method (SPH) for large deformation and fail-ure flows of geomaterial using elastic–plastic soil constitutivemodel, Int. J. Numer. Anal. Met., 32, 1537–1570, 2008.

Chen, W.-F. and Mizuno, E.: Nonlinear analysis in soil mechanics,Elsevier, Amsterdam, 1990.

Cohen, D., Lehmann, P., and Or, D.: Fiber bundle modelfor multiscale modeling of hydromechanical triggering ofshallow landslides, Water resources research, 45, W10436,https://doi.org/10.1029/2009WR007889, 2009.

Corominas, J., van Westen, C., Frattini, P., Cascini, L., Malet, J.-P.,Fotopoulou, S., Catani, F., Van Den Eeckhaut, M., Mavrouli, O.,and Agliardi, F.: Recommendations for the quantitative analysisof landslide risk, Bull. Eng. Geol. Environ., 73, 209–263, 2014.

Corominas, J., Matas, G., and Ruiz-Carulla, R.: Quantitative analy-sis of risk from fragmental rockfalls, Landslides, 16, 5–21, 2019.

Cuomo, S., Pastor, M., Capobianco, V., and Cascini, L.: Modellingthe space–time evolution of bed entrainment for flow-like land-slides, Eng. Geol., 212, 10–20, 2016.

David, L. G. and Richard, M.: A two-phase debris-flow model thatincludes coupled evolution of volume fractions, granular dila-tancy, and pore-fluid pressure, Ital. J. Eng. Geol. Environ., 43,415–424, 2011.

Davies, T. and McSaveney, M.: The role of rock fragmentation inthe motion of large landslides, Eng. Geol., 109, 67–79, 2009.

Davies, T., McSaveney, M., and Beetham, R.: Rapid block glides:slide-surface fragmentation in New Zealand’s Waikaremoanalandslide, Q. J. Eng. Geol. Hydroge., 39, 115–129, 2006.

Delaney, K. B. and Evans, S. G.: The 1997 Mount Munday land-slide (British Columbia) and the behaviour of rock avalancheson glacier surfaces, Landslides, 11, 1019–1036, 2014.

Delestre, O., Cordier, S., Darboux, F., Du, M., James, F., La-guerre, C., Lucas, C., and Planchon, O.: FullSWOF: A softwarefor overland flow simulation, in: Advances in hydroinformatics,Springer, Singapore, 221–231, 2014.

De Vuyst, T. and Vignjevic, R.: Total Lagrangian SPH modelling ofnecking and fracture in electromagnetically driven rings, Int. J.Fracture, 180, 53–70, 2013.

Dhanmeher, S.: Crack pattern observations to finite element sim-ulation: An exploratory study for detailed assessment of rein-forced concrete structures, Delft University, Delft, the Nether-lands, 2017.

Drew, D. A.: Mathematical modeling of two-phase flow, Annu. Rev.Fluid Mech., 15, 261–291, 1983.

Dufresne, A., Geertsema, M., Shugar, D., Koppes, M., Higman,B., Haeussler, P., Stark, C., Venditti, J., Bonno, D., and Larsen,C.: Sedimentology and geomorphology of a large tsunamigeniclandslide, Taan Fiord, Alaska, Sediment. Geol., 364, 302–318,2018.


https://github.com/bastianvandenbout/OpenLISEM-Hazard-2.0-Pre-Release

https://github.com/bastianvandenbout/OpenLISEM-Hazard-2.0-Pre-Release

https://doi.org/10.17026/dans-xz4-2tut

https://sourceforge.net/projects/lisem/

https://blog.utwente.nl/lisem/

https://doi.org/10.1016/0148-9062(67)90030-7

https://doi.org/10.1029/2009WR007889


Fornes, P., Bihs, H., Thakur, V. K. S., and Nordal, S.: Implementa-tion of non-Newtonian rheology for Debris Flow simulation withREEF3D, in: E-proceedings of the 37th IAHR World Congress,Kuala Lumpur, Malaysia, 13–18 August 2017, 2017.

Grady, D. E. and Kipp, M. E.: Continuum modelling of explosivefracture in oil shale, Int. J. Rock Mechanics Min., 17, 147–157,1980.

Greco, M., Di Cristo, C., Iervolino, M., and Vacca, A.: Numeri-cal simulation of mud-flows impacting structures, J. Mt. Sci., 16,364–382, 2019.

Griffiths, D. and Lane, P.: Slope stability analysis by finite elements,Geotechnique, 49, 387–403, 1999.

Hayir, A.: The effects of variable speeds of a submarine block slideon near-field tsunami amplitudes, Ocean Eng., 30, 2329–2342,2003.

Hušek, M., Kala, J., Hokeš, F., and Král, P.: Influence of SPH reg-ularity and parameters in dynamic fracture phenomena, Proc.Eng., 161, 489–496, 2016.

Hutter, K., Svendsen, B., and Rickenmann, D.: Debris flow model-ing: a review, Continuum Mech. Therm., 8, 1–35, 1994.

Ishii, M.: Thermo-fluid dynamic theory of two-phaseflow, NASA Sti/recon Technical Report A, 75, 29657,https://doi.org/10.1007/978-1-4419-7985-8, 1975.

Ishii, M. and Zuber, N.: Drag coefficient and relative velocity inbubbly, droplet or particulate flows, AICHE J., 25, 843–855,1979.

Iverson, R. M.: Elementary theory of bed-sediment entrainmentby debris flows and avalanches, J. Geophys. Res.-Earth, 117,F03006, https://doi.org/10.1029/2011JF002189, 2012.

Iverson, R. M. and Denlinger, R. P.: Flow of variably fluidized gran-ular masses across three-dimensional terrain: 1. Coulomb mix-ture theory, J. Geophys. Res.-Solid Ea., 106, 537–552, 2001.

Iverson, R. M. and George, D. L.: A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I.Physical basis, P. Roy. Soc. A Math. Phy., 470, 20130819,https://doi.org/10.1098/rspa.2013.0819, 2014.

Iverson, R. M. and George, D. L.: Modelling landslide liquefaction,mobility bifurcation and the dynamics of the 2014 Oso disaster,Géotechnique, 66, 175–187, 2016.

Iverson, R. M. and Ouyang, C.: Entrainment of bed material byEarth-surface mass flows: Review and reformulation of depth-integrated theory, Rev. Geophys., 53, 27–58, 2015.

Jakob, M., Hungr, O., and Jakob, D. M.: Debris-flow hazards andrelated phenomena, Springer, Berlin, Heidelberg, 2005.

Kaklauskas, G. and Ghaboussi, J.: Stress-strain relations for crackedtensile concrete from RC beam tests, J. Struc. Eng., 127, 64–73,2001.

Kern, J. S.: Evaluation of soil water retention models based on basicsoil physical properties, Soil Sc. Soc. Am. J., 59, 1134–1141,1995.

Kjekstad, O. and Highland, L.: Economic and social impactsof landslides, in: Landslides–disaster risk reduction, Springer-Verlag Berlin Heidelberg, 573–587, 2009.

Li, C., Wang, C., and Qin, H.: Novel adaptive SPH with geometricsubdivision for brittle fracture animation of anisotropic materi-als, Visual Comput., 31, 937–946, 2015.

Libersky, L. D. and Petschek, A. G.: Smooth particle hydrodynam-ics with strength of materials, in: Advances in the free-Lagrangemethod including contributions on adaptive gridding and the

smooth particle hydrodynamics method, Springer, Berlin, Hei-delberg, 248–257, 1991.

Loehnert, S. and Mueller-Hoeppe, D. S.: Multiscale methods forfracturing solids, in: IUTAM symposium on theoretical, compu-tational and modelling aspects of inelastic media, Springer, Dor-drecht, the Netherlands, pp. 79–87, 2008.

Luna, B. Q., Remaître, A., Van Asch, T. W., Malet, J.-P., and VanWesten, C.: Analysis of debris flow behavior with a one dimen-sional run-out model incorporating entrainment, Eng. Geol., 128,63–75, 2012.

Ma, G., Wang, Q., Yi, X., and Wang, X.: A modified SPH methodfor dynamic failure simulation of heterogeneous material, Math.Probl. Eng., 2014, 808359, https://doi.org/10.1155/2014/808359,2014.

Matsui, T. and San, K.-C.: Finite element slope stability analysisby shear strength reduction technique, Soils Found., 32, 59–70,1992.

Maurel, B. and Combescure, A.: An SPH shell formulation for plas-ticity and fracture analysis in explicit dynamics, Int. J. Num.Meth. Eng., 76, 949–971, 2008.

Menin, R., Trautwein, L. M., and Bittencourt, T. N.: Smeared crackmodels for reinforced concrete beams by finite element method,Revista IBRACON de Estruturas e Materiais, 2, 166–200, 2009.

Mergili, M., Fischer, J.-T., Krenn, J., and Pudasaini, S. P.: r.avaflowv1, an advanced open-source computational framework for thepropagation and interaction of two-phase mass flows, Geosci.Model Dev., 10, 553–569, https://doi.org/10.5194/gmd-10-553-2017, 2017.

Mergili, M., Emmer, A., Juricová, A., Cochachin, A., Fischer, J.T., Huggel, C., and Pudasaini, S. P.: How well can we simulatecomplex hydro-geomorphic process chains? The 2012 multi-lakeoutburst flood in the Santa Cruz Valley (Cordillera Blanca, Perú),Earth Surf. Proc. Land., 43, 1373–1389, 2018.

Monaghan, J. J.: SPH without a tensile instability, J. Comput. Phys.,159, 290–311, 2000.

Nadim, F., Kjekstad, O., Peduzzi, P., Herold, C., and Jaedicke, C.:Global landslide and avalanche hotspots, Landslides, 3, 159–173,2006.

Necas, J. and Hlavácek, I.: Mathematical theory of elas-tic and elasto-plastic bodies: an introduction, Elsevier,https://doi.org/10.1016/B978-0-444-99754-8.50008-4, 2017.

Ngekpe, B., Ode, T., and Eluozo, S.: Application of total-straincrack model in finite element analysis for punching shear at edgeconnection, Int. J. Res. Eng. Social Sci., 6, 1–9, 2016.

Julien, P. Y. and O’Brien, J. S.: Selected notes on debrisflow dynamics, in: Recent Developments on Debris Flows.Lecture Notes in Earth Sciences, , edited by: Armanini,A. and Michiue, M., vol 64, Springer, Berlin, Heidelberg,https://doi.org/10.1007/BFb0117766, 1997.

Osorno, M. and Steeb, H.: Coupled SPH and Phase Field methodfor hydraulic fracturing, Proc. Appl. Math. Mech., 17, 533–534,https://doi.org/10.1002/pamm.201710236, 2017.

Pailha, M. and Pouliquen, O.: A two-phase flow description of theinitiation of underwater granular avalanches, J. Fluid Mech., 633,115, https://doi.org/10.1017/S0022112009007460, 2009.

Pastor, M., Blanc, T., Haddad, B., Petrone, S., Sanchez Morles, M.,Drempetic, V., Issler, D., Crosta, G. B., Cascini, L., Sorbino,G., and Cuomo, S.: Application of a SPH depth-integrated


https://doi.org/10.1007/978-1-4419-7985-8

https://doi.org/10.1029/2011JF002189

https://doi.org/10.1098/rspa.2013.0819

https://doi.org/10.1155/2014/808359

https://doi.org/10.5194/gmd-10-553-2017

https://doi.org/10.5194/gmd-10-553-2017

https://doi.org/10.1016/B978-0-444-99754-8.50008-4

https://doi.org/10.1007/BFb0117766

https://doi.org/10.1002/pamm.201710236

https://doi.org/10.1017/S0022112009007460


model to landslide run-out analysis, Landslides, 11, 793–812,https://doi.org/10.1007/s10346-014-0484-y, 2014.

Pitman, E. B. and Le, L.: A two-fluid model for avalanche and de-bris flows, Philos. Trans. A. Math. Phys. Eng. Sci., 363, 1573–1601, https://doi.org/10.1098/rsta.2005.1596, 2005.

Price, N. J.: Fault and joint development: in brittle and semi-brittlerock, Elsevier, https://doi.org/https://doi.org/10.1016/C2013-0-05410-2, 2016.

Pudasaini, S. P.: A general two-phase debris flowmodel, J. Geophys. Res.-Earth, 117, F03010,https://doi.org/10.1029/2011JF002186, 2012.

Pudasaini, S. P. and Fischer, J.-T.: A mechanical erosion model fortwo-phase mass flows, arXiv preprint, arXiv:1610.01806, 2016.

Pudasaini, S. P. and Hutter, K.: Rapid shear flows of dry granularmasses down curved and twisted channels, J. Fluid Mech., 495,193–208, 2003.

Pudasaini, S. P. and Hutter, K.: Avalanche dynamics: dynam-ics of rapid flows of dense granular avalanches, Springer Sci-ence & Business Media, Springer-Verlag Berlin Heidelberg,https://doi.org/10.1007/978-3-540-32687-8, 2007.

Pudasaini, S. P. and Mergili, M.: A multi-phase mass flow model, J.Geophys. Res.-Earth, 124, 2920–2942, 2019.

Pudasaini, S. P., Hajra, S. G., Kandel, S., and Khattri,K. B.: Analytical solutions to a nonlinear diffusion–advection equation, ZAMM-Z Angew. Math. Phys., 69,150, https://doi.org/10.1007/s00033-018-1042-6, 2018.

Rankine, W. J. M.: II. On the stability of loose earth, Philos. T. R.Soc. Lond., 147, 9–27, 1857.

Reiche, P.: The Toreva-Block: A distinctive landslide type, J. Geol.,45, 538–548, 1937.

Rickenmann, D., Laigle, D., McArdell, B., and Hübl, J.: Compari-son of 2D debris-flow simulation models with field events, Com-put. Geosci., 10, 241–264, 2006.

Roberts, M.: Evenly Distributing Points in a Tri-angle, available at: http://extremelearning.com.au/evenly-distributing-points-in-a-triangle/ (last access:3 March 2021), 2020.

Savage, S. B. and Hutter, K.: The motion of a finite mass of granularmaterial down a rough incline, J. Fluid Mech., 199, 177–215,1989.

Saxton, K. E. and Rawls, W. J.: Soil water characteristic estimatesby texture and organic matter for hydrologic solutions, Soil Sci.Soc. Am. J., 70, 1569–1578, 2006.

Sheridan, M. F., Stinton, A. J., Patra, A., Pitman, E., Bauer, A., andNichita, C.: Evaluating Titan2D mass-flow model using the 1963Little Tahoma peak avalanches, Mount Rainier, Washington, J.Volcanol. Geotherm. Res., 139, 89–102, 2005.

Spencer, A. J. M.: Continuum mechanics, Courier Corporation,Dover Publications, ISBN-13 9780486139470, 2012.

Stead, D. and Wolter, A.: A critical review of rock slope fail-ure mechanisms: the importance of structural geology, J. Struct.Geol., 74, 1–23, 2015.

Steffen, M., Kirby, R. M., and Berzins, M.: Analysis and reductionof quadrature errors in the material point method (MPM), Int. J.Num. Meth. Eng., 76, 922–948, 2008.

Stomakhin, A., Schroeder, C., Chai, L., Teran, J., and Selle, A.: Amaterial point method for snow simulation, ACM Transactionson Graphics (TOG), 32, 1–10, 2013.

Tang, C.-L., Hu, J.-C., Lin, M.-L., Angelier, J., Lu, C.-Y., Chan, Y.-C., and Chu, H.-T.: The Tsaoling landslide triggered by the Chi-Chi earthquake, Taiwan: insights from a discrete element simu-lation, Eng. Geol., 106, 1–19, 2009.

Van Asch, T. W., Tang, C., Alkema, D., Zhu, J., and Zhou, W.: Anintegrated model to assess critical rainfall thresholds for run-outdistances of debris flows, Nat. Haz., 70, 299–311, 2014.

van den Bout, B.: OpenLISEM Hazard 2.0 alpha, DANS, Facultyof Geo-Information Science and Earth Observation (ITC), Uni-versity of Twente https://doi.org/10.17026/dans-xz4-2tut, 2020.

van den Bout, B.: openLISEM, available at: https://sourceforge.net/projects/lisem/ (last access: 22 March 2021), SourceForge, 2021.

van den Bout, B and Jetten, V. G.: openLISEM a spatial model forrunoff, floods and erosion, available at: https://blog.utwente.nl/lisem/ (last access: 22 March 2021), 2020.

Van Looy, K., Bouma, J., Herbst, M., Koestel, J., Minasny, B.,Mishra, U., Montzka, C., Nemes, A., Pachepsky, Y. A., andPadarian, J.: Pedotransfer functions in Earth system science:challenges and perspectives, Rev. Geophys., 55, 1199–1256,2017.

Varnes, D. J.: Slope movement types and processes, Special Rep.,176, 11–33, 1978.

von Boetticher, A., Turowski, J. M., McArdell, B. W., Rick-enmann, D., Hürlimann, M., Scheidl, C., and Kirchner, J.W.: DebrisInterMixing-2.3: a finite volume solver for three-dimensional debris-flow simulations with two calibration pa-rameters – Part 2: Model validation with experiments, Geosci.Model Dev., 10, 3963–3978, https://doi.org/10.5194/gmd-10-3963-2017, 2017.

Williams, J. R.: Application of SPH to coupled fluid-solid problemsin the petroleum industry, Videos of Plenary Lectures presentedat the IV International Conference on Particle-Based Methods,Fundamentals and Applications (PARTICLES 2015), TechnicalUniversity of Catalonia (UPC), Barcelona, Spain, 28–30 Septem-ber 2015, 2019.

Xie, M., Esaki, T., and Cai, M.: GIS-based implementation ofthree-dimensional limit equilibrium approach of slope stability,J. Geotech. Geoenviron. Eng., 132, 656–660, 2006.

Xu, F., Zhao, Y., Li, Y., and Kikuchi, M.: Study of numerical andphysical fracture with SPH method, Acta Mech. Solida Sinica,23, 49–56, 2010.

Xu, R., Kang, L., and Tian, H.: A g-octree based fast collision de-tection for large-scale particle systems, in: 2012 InternationalConference on Computer Science and Electronics Engineering,IEEE, 3, 269–273), 2012.

Xu, S. Y., Lawal, A. I., Shamsabadi, A., and Taciroglu, E.: Estima-tion of static earth pressures for a sloping cohesive backfill usingextended Rankine theory with a composite log-spiral failure sur-face, Acta Geotech., 14, 579–594, 2019.

Zhang, L., Zhang, J., Zhang, L., and Tang, W.: Stability analysis ofrainfall-induced slope failure: a review, P. I. Civil Eng.-Geotec.,164, 299–316, 2011.

Zhou, F., Molinari, J.-F., and Ramesh, K.: A cohesive model basedfragmentation analysis: effects of strain rate and initial defectsdistribution, Int. J. Solid. Struct., 42, 5181–5207, 2005.


https://doi.org/10.1007/s10346-014-0484-y

https://doi.org/10.1098/rsta.2005.1596

https://doi.org/https://doi.org/10.1016/C2013-0-05410-2

https://doi.org/https://doi.org/10.1016/C2013-0-05410-2

https://doi.org/10.1029/2011JF002186

https://doi.org/10.1007/978-3-540-32687-8

https://doi.org/10.1007/s00033-018-1042-6

http://extremelearning.com.au/evenly-distributing-points-in-a-triangle/

http://extremelearning.com.au/evenly-distributing-points-in-a-triangle/

https://doi.org/10.17026/dans-xz4-2tut





https://doi.org/10.5194/gmd-10-3963-2017

https://doi.org/10.5194/gmd-10-3963-2017

Towards a model for structured mass movements

Documents