A TPDP-MPM-based approach to understanding the evolution mechanism of landslide-induced disaster chain

Wenjie Du, Qian Sheng, Xiaodong Fu,*, Jian Chen,c, Yongqiang Zhou

a State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences, Wuhan,430071,China

b School of Engineering Science, University of Chinese Academy of Sciences, Beijing,100049, China

c China-Pakistan Joint Research Center on Earth Sciences, CAS-HEC, Islamabad, Pakistan

Keywords:Disaster chain Landslide-induced surge Material point method (MPM)Energy evolution U-shaped valley

ABSTRACT With complex topographic and hydrological characteristics, the landslide-induced surge disaster chain readily develops in mountainous and gorge areas, posing a huge challenge for infrastructure construction. This landslide-induced surge disaster chain involves a complex fluid-solid coupling between the landslide mass and a water body and exhibits complex energy conversion and dissipation characteristics,which is challenging to deal with using traditional finite element analysis. In this study, the energy evolution characteristics in the whole process of the disaster chain were first investigated, and the momentum-conservation equations for different stages were established. Then, the two-phase doublepoint material point method(TPDP-MPM)was used to model the landslide-induced surge disaster chain,and an experiment involving block-induced surge was modeled and simulated to validate this method.Finally,three generalized models were established for the landslide-induced surge process in a U-shaped valley, including subaerial, partly submerged, and submarine scenarios. The interaction mechanism between the landslide mass and the water body in the disaster chain was revealed by defining the system energy conversion ratio and the mechanism of evolution of the disaster chain from the perspective of energy. The results help further evaluate the secondary disasters, given the submerged position of the landslide mass.

1. Introduction

With complex topographic and hydrological characteristics,mountainous and gorge areas provide a hazard-inducing environment for disaster chains,posing a significant threat to the safety of residents and the infrastructure construction. A disaster chain refers to the phenomenon whereby other disasters occur one after another or at a certain removal after a primary disaster. The landslide-induced surge disaster chain is a widespread form of disaster chain in mountainous and gorge areas (Wu et al., 2019),coastal areas (Mohammed, 2010; Ruffini et al., 2019; Fan et al.,2022), and reservoir areas (Huang et al., 2017, 2020). In southwestern China,with frequent tectonic activity and well-developed surface water systems, deep valleys are densely distributed since the river cuts at the bedrock as it flows downhill,which leads to the development of a landslide-induced surge disaster chain. Taking the Wu Gorge as an example, it is known for its continuous, steep bank slopes and deep-cut valleys, as shown in Fig.1. In this area,both the Hongyanzi landslide (left bank of Daning River) and Gongjiafang landslide (left bank of Yangtze River) generated giant river surges after sliding and these resulted in casualties and the destruction of facilities (Wang et al., 2019a).

Usually,the material sources of a disaster chain are multi-phase media,and its evolution is highly complicated.Benefitting from the continuous improvement of high-performance computing, numerical simulation has become an important means for the analysis and assessment of geological disasters. Emerging numerical methods such as smoothed particle hydrodynamics(SPH)(Manenti et al., 2015; Xenakis et al., 2017; Yeylaghi et al., 2017), Discontinuous deformation analysis(DDA)(Fu et al.,2017,2020),and discrete element method-smoothed particle hydrodynamics (DEM-SPH)coupling model(Tan and Chen,2017;Wang and Li,2017;Xu,2020)were applied in the simulation of geological disasters(chains).The material point method (MPM) that combines the best aspects of both Lagrangian and Eulerian formulations, traditional grid-based methods and meshless methods, have been widely used to model landslides (Dong et al., 2015, 2021; Li et al., 2016; Xu et al., 2019;Yerro et al., 2019; Conte et al., 2020; Ying et al., 2020; Lei et al.,2021), fluid (Kularathna et al., 2017; Zhang et al., 2017), and fluidsolid coupling problems (Bandara et al., 2016; Wang et al., 2018;Liang et al., 2020; Liu et al.,2021).

Fig.1. Two landslides that occurred in the Wu Gorge(modified from Xiao et al.,2018).

Regarding the landslide-induced surge disaster chain, the water participates in the whole process of the disaster chain in different forms: free water (rivers, reservoirs, and oceans) and pore water(infiltrating the landslide mass); in other words, there is a complex“fluid-solid”coupling process between the landslide mass and water body,which is difficult to model precisely with the above methods.The two-phase double-point MPM(TPDP-MPM)program can model the landslide and water body by two sets of particles (Abe et al.,2014; Bandara and Soga, 2015; Pradhana, 2017). Under this premise, the two forms of the water be uniformly expressed, and the complex coupling process between the landslide mass and the water body can be modeled at the same time. Fig. 2 shows the modeling process of a landslide-induced surge disaster chain based on the TPDP-MPM.

In this study,the evolution mechanism of the landslide-induced surge disaster chain was examined. To achieve this goal, a TPDPMPM program for simulating the whole process of landslideinduced surge disaster chain was developed. The effectiveness of the TPDP-MPM program in solving this complex coupling problem was verified by modeling and simulating the physical model testing of a block-induced surge process on an experimental scale. Three generalized models of the landslide-induced surge disaster chain in U-shaped valley were established, and the influences of the inundated position of the landslide mass on the generation of the secondary disasters were investigated. Finally, the evolution mechanism of the landslide-induced surge disaster chain was ascertained from an energy point of view.

2. Methodology

2.1. Momentum conservation in the landslide-induced surge disaster chain

The evolution of a landslide-induced surge disaster chain includes a landslide initiation stage, a sliding stage, and a stage whereupon the landslide enters the river. As shown in Fig. 3, the momentum conservation equations corresponding to different stages of the disaster chain evolution process are different. At the landslide initiation and sliding stages,the external forces exerted on the system include gravity, seismic force, friction, etc. The momentum conservation equations governing the behavior of the landslide body were integrated over elemental volume dV and area dA:

where b denotes the body force per unit mass acting on the continuum body Ω,v is the velocity vector at the current step,ρsis the solid density,τ represents the external tractions on a given surface boundary Γ, and D/Dt is the material derivative operator.

When the landslide enters the river, the coupling of two-phase medium exists, which means that the momentum exchange between the solid-phase and liquid-phase should be considered.Thus, the momentum exchange term should be supplemented in the momentum conservation equations:

where dαrepresents the exchange of momentum between phases.

Based on the momentum conservation equation,the differential equations of motion of two phases (Eq. (3)) and mixture (Eq. (4))were derived:

Fig. 2. Modeling of landslide-surge disaster chain.

Fig. 3. Momentum conservation in the landslide surge disaster chain.

2.2. The TPDP-MPM for simulating landslide-induced surge disaster chain

As for the TPDP-MPM,the solid and fluid phases are modeled by two sets of particles:solid particles and fluid particles(Fig.4).Each set of particles carries the physical information pertaining to the corresponding phase.The aforementioned sets of particles share a common background mesh. A grid is deemed to be in a saturated state if mapped with both fluid particles and solid particles at the same step, as marked by the red dashed grid (Fig. 4). Here we introduce the solid phase velocity-fluid phase velocity formulation(vs-vf) of the TPDP-MPM regarding to the coupling simulation(Jassim et al.,2013).

The coupling behavior of saturated porous media can be formulated by the conservation of momentum equation, which is given by Eq.(3).By following the formulation of Jassim et al.(2013),which is concerned primarily with gravity-driven flows, the momentum exchange terms dsand dffor fluid and solid,respectively,are assumed to be

where cE= nρfg/^k denotes the drag coefficient, in which n is the porosity of the sand, ^k represents the permeability of the material forming the landslide mass, and g = 9.8 m/s2; ρfis the fluid density; and φfdenotes the water volume fraction. On right-hand side, the first term denotes the viscous forces, and the second term is the so-called “buoyancy term” in mixture theories.

By introducing the mass matrix M and momentum exchanged matrix D, the coupled system can be finally discretized as

Fig. 4. The TPDP-MPM formulation (modified from Fern et al., 2019).

The mass of particles was assumed to be fixed during the simulation to ensure the conservation of mass (Yerro, 2015). For further details, the reader is referred to Abe et al. (2014), Bandara and Soga (2015) and Pradhana (2017).

In addition to the balance equations,the constitutive equations that describe the mechanical behavior of the landslide mass and water body are introduced to model the coupled system.The fluid phase was considered as quasi-incompressible by introducing an equation of state (Becker and Teschner, 2007):

where B is a pressure-related item, γ is a term that reflects the incompressibility of fluid,and Jfis the ratio of the current to initial volume of fluid particles.Besides,the movement of the solid phase was governed by an elastoplastic constitutive model with a Drucker-Prager yield criterion (Pradhana, 2017). It should be mentioned that the pressure of fluid phase is a function of the potential Ψfas

2.3. Numerical procedures within a computational cycle

A discretization in time is necessary for solving the momentum balance equations.To this end,an explicit time integration scheme has been chosen. With the advantages of both Lagrangian and Eulerian descriptions, the algorithm of TPDP-MPM is described by Fig. 5:

(1) Modeling of the landslide mass and water body by two sets of particles and construction of a structural background mesh;

(2) Initializing the grid nodes of each phase,and the information pertaining to the grid nodes is mapped from the particles;

(3) Solving the momentum equations of each phase on the background grid (Eq. (6)). For grids in a saturated state,momentum exchange is conducted to couple the two phases;

(4) Mapping information from grids to particles; and

(5) Resetting the background mesh.

2.4. Quantitative evaluation of the disaster chain using TPDP-MPM

The TPDP-MPM can simulate the whole process of the landslideinduced surge disaster chain. However, to quantify the disaster chain, it is difficult to establish a relationship between two set of particles. Energy, as a broader physical quantity applicable to all particles, is the key to link the primary and secondary disasters.Taking the disaster chain as an open system,the particles of disaster chain will be uniformly expressed, and the chain generation process is quantitatively described from the perspective of energy.

Fig. 5. Algorithm of TPDP-MPM.

In the simulation, we focus on the energy evolution of the system and output the energy information on particles. For the MPM and other particle methods, the energy evolution of the landslide mass and water body can be quantitatively described since the physical information such as mass,velocity and energy are inherent to the particles. On this basis, the evolution mechanism of the landslide-induced surge process was investigated from an energy point of view.

This research focuses on the conversion of energy between the landslide mass and water body when the landslide reaches, and passes below, the water level. Various forms of energy, including the kinetic energy of the landslide mass, the kinetic energy and potential energy of the water body, and that energy dissipated in the entire process,can be assumed to come from the initial kinetic energy plus the work done by gravity on the landslide mass.Various energy forms of particles are recorded during the calculation, including the kinetic energy of the two-phase particles and the potential energy of the two-phase particles as

where htrepresent the current elevation of particles and h0is the initial elevation.

3. Benchmark example

Heinrich (1992) conducted an experiment on waves generated by a submarine block. The submarine block could slide along the slope with an inclination of 45°,and a buffer was set on the bottom to stop the block, as illustrated in Fig. 6.

The velocity of the block was given according to the time history of measured vertical displacement of the block(Wang and Li,2017):

The computational domain was discretized into 23,825 fluid particles. We considered the submarine block as a rigid and impermeable moving boundary and sliding at a velocity given by Eq. (12). In this study, the block was treated as frictionless and perfectly inelastic boundary.The explicit forward Euler scheme and a time step of Δt = 1.11×10-5s were adopted and the input parameters are listed in Table 1.

The wave profile at 0.5 s,1 s and 1.5 s were recorded in Heinrich(1992) and marked as red squares in Fig. 7. Besides, the wave profiles simulated by the TPDP-MPM at 0.5 s, 1 s and 1.5 s after the sliding of the submerge block are illustrated in Fig.7,where we can see that the numerical results obtained using the TPDP-MPM are similar to those obtained in Heinrich’s experiment.The reliability of the proposed boundary treatment was also verified.

Fig. 6. Heinrich (1992)’s experiment on submarine block sliding.

Table 1 Input parameters for simulating benchmark example.

4. Landslide-induced surge disaster chain in a U-shaped valley

4.1. Generalized model and numerical implementation

Here, we focus on accumulation landslides composed of soilrock mixtures in the U-shaped valley with the shear outlet that extends into the bottom of the valley,and even lies under the water level, which is the typical landslide-induced surge disaster chain developed in mountainous and gorge areas. Typical cases of landslide-induced surge disaster chain were counted and classified based on the relative position of the landslide mass and water body(see Table 2).These typical cases show that the secondary disaster(surge)enlarges the influential zone such disasters in both time and space.In general,the valleys can be simplified by their shapes,such as “V-shaped”, “U-shaped”, or “box-shaped” valleys (Lin and Oguchi, 2006). The following sections show further discussion on the propagation behavior of a surge (secondary disaster) in the Ushaped valley and the evolution mechanism of the interaction between the landslide mass and the water body.

Fig.7. Comparison between numerical and experimental results:(a)0.5 s,(b)1 s,and(c) 1.5 s.

Table 2 Typical landslide-induced surge disaster chain events.

Fig. 8. Generalized model of subaerial, partly submerged and submarine landslide.

According to the typical cases of disasters in Table 2, the landslide-induced surge disaster chain is classified into three categories based on the initial position of the landslide mass relative to the water body, as shown in Fig. 8 (Mohammed, 2010):

(1) Subaerial landslides involve the subaerial landslide motion,impact of the landslide on the water surface,and submarine runout and deposition. In some high-velocity impact scenarios,the formation of an impact crater separates the water surface from the sliding mass;

(2) Partially-submerged landslides involve all three phases;

(3) Submarine landslides are fully submerged and the waves generally begin with a trough due to the motion of the sliding mass.

Based on the three generalized models (subaerial, partially submerged and submarine), numerical models were established,and the landslide mass was discretized into 1316 soil particles,and the river was discretized into 14,118 water particles. A structural background grid with mesh size of 0.02 m was adopted.

The landslide started by giving an initial velocity to the landslide particles. The energy conversion in the landslide-induced surge process was studied while ignoring the influence of the frictional effect of sliding surface on the system energy dissipation. Besides,the parallelized explicit scheme was adopted with the time step of Δt = 2 ×10-5s (Mieremet et al., 2015), and the other input parameters are listed in Table 3 (Du et al., 2021). It is noted that the typical deformation and strength parameters of soil-rock mixturesare adopted as the main geomaterial of accumulation landslide(Xu et al., 2011; Zhang et al., 2019; Fu et al., 2021; Zhang, 2021). A slightly lower friction angle is used for numerical simulation to maximize the released energy for the following disaster chain analysis. The local damping was adopted in this simulation with a damping ratio of 0.03, considering the inherent dissipative property of soil material.

Table 3 Input parameters for simulating landslide-induced surge process.

4.2. Transverse propagation of surge

Taking the subaerial landslide-induced surge disaster chain as an example, the velocity vector field of the water body during the initiation and propagation stage is shown in Fig. 9. The transverse propagation process of a landslide-induced surge in a U-shaped valley is described below.

After the landslide mass enters the water body, the impulsive wave propagates towards the opposite bank slope. The flow velocity of the returned water on the landslide bank(Fig.9a and b)is higher, but most of the water rushes to the opposite bank along with the first wave;thereafter,the impulsive wave propagates and climbs back and forth between the opposite bank and the landslide bank. Both the wave height and the climbing height on the banks are reduced (Chen and Yeh, 2014), with the collision of multiple waves. The kinetic energy of the water and the climbing height of the waves on both sides of the bank slope are significantly decreased, with the continuous dissipation of the system energy,and the system is then stabilized.

5. Evolution mechanism of the disaster chain from the perspective of energy

5.1. Energy evolution along the disaster chain

Fig. 10 shows the energy evolution diagram of a landslideinduced surge disaster chain system. The initial energy of the disaster chain system includes the initial potential energy, kinetic energy,and energy input generated by landslide triggering factors.Along with the landslide movement,part of the energy is dissipated in various forms (sound, heat, and crushing of blocks). When the disaster chain develops to the stage involving the interaction between two-phase medium,the coupling process between landslide and water body is accompanied by energy transfer and dissipation(Mackenzie et al., 2010; Zhou et al., 2022). As the energy is converted from the landslide to the water body, a secondary disaster(i.e. a surge) forms.

According to Eqs. (10) and (11), Fig. 11 demonstrates the evolution of water energy, including kinetic energy, potential energy,and total energy (the area beneath the curve represents the energy). According to Figs. 9 and 11, the landslide-induced surge disaster chain in a U-shaped valley can be divided into four stages:

Fig. 9. Transverse propagation behaviors of surge in a U-shaped valley.

(1) Wave initiation stage: After the landslide mass intrudes the water body, the wave is lifted under the impact of the landslide (Fig. 9a).

(2) Wave run-up stage:The impulsive wave run-up on both the opposite and landslide banks and the potential energy reaches their peak values (Fig. 9b, Range I in Fig.11).

(3) Subsiding stage: The potential energy of the water body decreases under the action of gravity, coupled with the energy dissipation during the infiltration process and the collision between wave crests, the total energy of the water body attenuates significantly (Fig. 9c and d, Range II in Fig.11).

(4) Wave oscillation stage: After the surge subsides, the water body propagates back and forth with a lower energy dissipation (wave oscillation stage), the kinetic energy and potential energy of the water body are mutually transformed,and the total energy of the water body attenuates steadily(Fig. 9e-j, Range III in Fig.11).

Fig.10. The energy evolution of landslide surge disaster chain.

5.2. Evolution mechanism of a landslide-induced surge disaster chain

The energy conversion ratio from landslide to river owing to the interaction between landslide mass and water body is a key parameter in the risk assessment of a landslide-induced surge disaster chain. According to the definition of Wang et al. (2019b),the energy conversion ratio in the landslide-induced surge process denotes the ratio of energy of water body (i.e. the sum of the potential energy Pwtand kinetic energy Kwt)to the mechanical energy loss of the landslide(i.e.the sum of the initial kinetic energy of the landslide Ks0and the gravitational work done by the sliding body Pstminus the kinetic energy of the landslide Kst):

The energy conversion ratio curves of the three generalized models during the initial stage of surge are shown in Fig.12.Since the frictional effect of the system is not considered in this study,the range of the system energy conversion ratio is between 20% and 68%, which is higher than that obtained from the model experiments (Fritz, 2002).

For the subaerial landslide model, the landslide mass lifts the water body after entering the water body and induces surges. The kinetic energy and potential energy of the water body increase rapidly (the so-called wave run-up stage). It should be mentioned that, as the potential energy of water body reaches its peak, the energy conversion ratio reaches its maximum. The system energy conversion ratio reaches a peak value of 60% at 0.4 s. As the potential energy of the water body is greatly reduced under the action of gravity, the surge enters the subsiding stage. Along with the energy dissipation in the two-phase coupling process (Mackenzie et al., 2010), the system energy conversion ratio starts to decrease. At the end of the subsiding stage, the system energy conversion ratio of the subaerial landslide model is maintained at about 38%.

Fig.11. Energy evolution of the water body.

Fig.12. Energy conversion ratio of three models.

For the partly submerged landslide and submarine landslide models,the proportion of landslides in the saturated state is higher than that in the subaerial landslide model due to the difference in the initial submerged position of the landslide mass,and the energy conversion and dissipation between the two phases increases. At the wave run-up stage, the system energy conversion ratio of the submarine landslide model increases sharply. It reaches a peak value of 68%:the energy of the landslide mass is converted into the energy of the water body. At the subsiding stage, due to the high proportional contribution of the saturated landslide mass, the energy dissipation in the coupling process of the partially-submerged landslide and submarine landslide models is more significant,resulting in the energy conversion ratios of the system being lower than that of subaerial landslide model(maintained at 35%and 25%,respectively).

In conclusion, as the initial inundated depth of the landslide increases, the maximum energy conversion ratio of the system increases, while the stable conversion ratio decreases. In other words,the initial inundated depth of the landslide determines the interaction channel between the landslide mass and the water body, which controls the energy conversion and dissipation behavior of the system.

6. Conclusions

In this study, the whole process of landslide-induced surge disaster chain was quantitatively evaluated by the TPDP-MPM from an energy point of view.On this basis,three generalized models of a landslide-induced surge disaster chain in a U-shaped valley were established and simulated. The main conclusions are summarized below:

(1) By establishing the momentum conservation equations for different stages, the numerical analysis method for the landslide-induced surge disaster chain was proposed based on TPDP-MPM, and the energy evolution characteristics along the entire disaster chain were analyzed.

(2) A landslide-induced surge could be divided into four stages:wave initiation,wave run-up,subsiding and wave oscillation.The landslide-induced surge process was quantitatively evaluated from the perspective of energy. Regarding the generalized models, the ratio of energy conversion from landslide mass to water body ranges between 20% and 68%.

(3) The peak of energy conversion ratio curve can divide the secondary disaster into two stages: wave run-up and subside. At the pre- and post-peak stages, the evolution of energy conversion ratio with the initial inundated depth of landslide follows the opposite trend. The initial position of the landslide determines the interaction channel between the landslide and the water body,which controls the energy conversion and dissipation behavior of the system.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work is financially supported by the National Natural Science Foundation of China (Grant Nos. 52179117 and U21A20159),and the Youth Innovation Promotion Association of Chinese Academy of Sciences (CAS) (Grant No. 2021325). A special acknowledgement should be expressed to the China-Pakistan Joint Research Center on Earth Sciences that supported the implementation of this study.