Exploring the submarine landslide is challenging due to the invisibility nature and the complex soil-water interaction and large deformation throughout its runout process. The purpose of this paper is to investigate the ability of the coupled material point method (MPM) in modeling the soil flows under water. A sand-column collapse experiment is performed fully under water initially, with the results used to benchmark the MPM analysis. Thereafter, the whole failure process of a real submarine landslide in the South Mediterranean sea is simulated using MPM. The results show that MPM can be a reliable tool in capturing the postfailure behaviors of the submarine landslide. The failure mode of the landslide is flow-type, with an initial translational slide moving to a diffusive one eventually.
Submarine landslides are the most common and frequent geological hazards in the ocean [
Evolutions of sliding distance and velocity during a landslide runout process are necessary for disaster prevention and mitigation [
Material point method (MPM), originating from the fluid mechanics, was first applied to the solid mechanics in 1994 [
Due to the fact that the submarine landslides occur fully under water, their invisibility makes it hard to determine the runout distances as opposed to the landslides. The main objective of this paper is to investigate the ability of MPM to simulate submarine landslides in order to quantify the calamity caused by it. A series of parametric analyses are conducted as well in order to gain a better understanding of the influencing factors to the slide consequences.
MPM is one of the mesh-free methods, which is described by a Lagrange particle system and Euler grid. Although MPM utilizes a background mesh, it is still regarded as a mesh-free method because no polygonal tessellation is involved in the initial discretization of the material. A schematic diagram of MPM is shown in Figure
Schematic diagram of MPM.
At every time step, the material information is first mapped to the background mesh using the shape functions, which is derived from the position of each material point with regard to the mesh. After gradient terms are calculated and governing equations are solved on the background mesh, the updated information will be mapped back to the particles. At last the distorted mesh is abandoned and a new one is initialized.
Because of this feature, MPM avoids difficulty caused by mesh distortion, interface tracking, and nonlinear convection, making it outstand other numerical methods in simulating geotechnical problems involving large displacement. Meanwhile MPM can adopt history-dependent constitutive model. The governing equations and spatial discretization of MPM can be referred to the principle parts of literature [
Based on Biot’s theory [ Saturated soil is considered to be made up of solid framework and water phases, both of which are described in a Lagrangian formulation and assumed as a continuous medium. The solid framework is considered as incompressible. An effective stress model is adopted for the solid framework. There is no mass exchange between the solid framework and water phases. Variation of water density is negligible. Variation of porosity and volume of solid framework with time is considered.
Porosity of soil can be updated from equilibrium equation of solid framework:
Porosity of soil can be obtained with formula (
Momentum balance equation of soil framework can be expressed as
Momentum balance equation of liquid part can be expressed as
Introducing formulas (
Introducing formulas (
Combining formulas (
The calculation procedures in each computational cycle are described by the following steps: The material information is projected to the background mesh using shape functions. Internal forces of the water phase and mixture parts are evaluated in the nodes. The momentum balance equations of solid framework and water phase are solved and nodal accelerations are calculated. With accelerations obtained in step (b), the momentum balance equation of the mixture part is solved. Velocities of particles and nodes are updated with the forward Euler scheme, while the displacements are updated with the backward Euler scheme. The mass balance equations are solved and the material information is updated. The updated information is mapped back to material points. The distorted mesh is abandoned and a new background mesh is initialized for next step.
The collapse process of a sand column includes some key stages similar to the runout process of a landslide [
A small-scale in-door experiment was conducted to serve as a benchmark, as shown in Figure
Morphology of the accumulation. (a) Before the collapse. (b) After the collapse.
Once the experiment starts, the baffle is quickly pulled up, rendering the sand-column collapses to the right direction, and forms a granular flow in fluids. After the system is stabilized, the sliding distance and the morphology of the sample in its final configuration are measured.
The material used here is silty sand, with the material properties summarized as shown in Table
Material parameters for the sand.
Initial porosity | Buoyant density (kg·m−3) | Internal friction angle (°) | Young’s modulus (kPa) | Poisson’s ratio |
---|---|---|---|---|
0.4 | 1368 | 33 | 2000 | 0.3 |
In order to verify the reliability of the MPM model, the whole failure process is simulated with the dynamic explicit MPM software Anura3D [
The deformation of the sand column at different times is shown in Figure
Displacement morphology (cm). (a) Time = 0 s. (b) Time = 2 s. (c) Time = 4 s. (d) Time = 5 s.
Both the sliding distance and the morphology in the final configuration are compared between the numerical solutions and laboratory results. The total displacement of the simulation model is 19.1 cm, which is similar to the experiment, with the sliding distance being measured as 19.2 cm, as shown in Figure
Morphology of the accumulation. (a) Experimental result. (b) Simulation result.
Submarine landslides usually end up in a longer runout distance and affect a wider range of areas than the ashore landslides. In order to have a better understanding of the kinematics of submarine landslides, a real case of submarine escarpment failure in the Southern Mediterranean is selected in this study. The Mediterranean Sea is a land-locked basin that is called a “natural geological laboratory” due to a variety of sedimentary and tectonic environments coexisting in the same place. Submarine landslides are ubiquitous in this region [
The detected slope morphology of the submarine landslide is shown in Figure
Detected slope morphology (courtesy of Dong et al. [
Idealization of the slide transect (modified following Dong et al. [
Several predisposing factors contributed to the destabilization of this slope. The most obvious one is the steep slope itself, which can be a result of a relatively high sedimentation rate [
A single-layer two-phase material point model based on the Mohr-Coulomb failure criterion for soil is used. A total stress analysis is utilized due to the undrained situations. Both physical and mechanical parameters of the slide necessary for the simulation are shown in Table
Material parameters for simulation.
Initial porosity | Buoyant density (kg·m−3) | Cohesion (kPa) | Young’s modulus (kPa) | Poisson’s ratio | Friction coefficient |
---|---|---|---|---|---|
0.3 | 525 | 15 | 3000 | 0.49 | 0.052 |
After a convergence study, a mesh spacing of 10 m for this simulation is selected. A total of 2613 material points are generated for the numerical model, which are distributed uniformly in the background grid, with 10 material points per cell in the slide body and 4 material points per cell in the bedrock.
The whole simulation process is shown in Figure
Displacement morphology. (a) Time = 0 s. (b) Time = 10 s. (c) Time = 20 s. (d) Time = 30 s. (e) Time = 40 s. (f) Time = 50 s. (g) Time = 57 s.
Likewise the sand-column collapse example, the sliding distance and the morphology in its final configuration are compared to verify the simulation results [
Landslide morphology.
As shown in Figure
Velocity versus time for the 3 monitoring points.
Displacement versus time for the 3 monitoring points.
The mechanism of slope failure is of great importance for assessing geologic hazards caused by submarine landslide. In this section, the underlying mechanism will be discussed with the above simulation results. So far, knowledge of mechanism involved in submarine landslide generation remains poorly known. One common practice in the analysis is to borrow some established theories from subaerial landslides.
Submarine landslides are initiated when the shear stress downslope exceeds the shear strength of the material forming the slope. A failure criterion that represents the material’s physical characteristics is required to define the shear strength. The most general choice here is the Mohr-Coulomb failure criterion:
Three types of shear stress downslope are significant with regard to submarine landslides: gravity, storm-wave-induced stress, and seismically induced stress. Detailed site-specific analyses must be performed in order to evaluate the postfailure behaviors of submarine slopes considering three types of stress, which is beyond the scope of our study. As it is a preliminary study, only gravity loading is considered in our simplified model. The seismic loading can generate vertical and normal accelerations and induce normal and shear stress in the slope. The storm-wave loading is much like that caused by an earthquake, which induces shear stress that varies cyclically between the crests and troughs, and may alternate in direction. These two types of loading are complex and cannot be described easily with a few parameters. However, it is certain that the presence of them will cause a reduction in the stability of submarine slopes [
From Figures
With the development of deformation, a sliding body is formed in the slope and it begins to slide along the seabed. The sliding body moves forward essentially as a block, with the height reducing very slowly. This stage lasts for about 13 seconds, and the velocity of each part of the landslide increases rapidly until it reaches the maximum velocity in the sliding process. The displacement and velocity curves are consistent with the experiment results by Olivares and Damiano [
Then the kinetic energy of the sliding body dissipates continuously due to the friction resistance of the interface exceeding the driving force caused by gravity. This stage lasts for a relatively long period (about 35 s). Although the velocity is decreasing, there is still considerable displacement in this stage. The displacement curve changes from upper concave to lower concave. Finally the movement of the landslide has almost stopped and the displacement curves of each point tend to a certain value. According to Whitman’s terminology [
Orthogonal experiments are used for parametric analysis in this section [
Six influential factors, namely, initial porosity, Young’s modulus, buoyant density, cohesion, friction coefficient, and slope angle, are chosen for the sensitivity analysis. Based on the principle of orthogonal experiments, the following orthogonal experimental table with 6 factors and 5 value levels is designed, leading to a total of 25 MPM analyses. The maximum sliding distance and velocity are recorded, with the results summarized and shown in Table
Orthogonal experiment table.
Scheme | Influential factors | Sliding distance (m) | Maximum velocity (m/s) | |||||
---|---|---|---|---|---|---|---|---|
Initialporosity | Young’s modulus (kPa) | Buoyant density (kg/m3) | Friction coefficient | Cohesion (kPa) | Slope angle (°) | |||
1 | 0.23 | 2250 | 393.75 | 0.038 | 11.25 | 10.31 | 539.0 | 12.2 |
2 | 0.23 | 2625 | 459.38 | 0.044 | 13.13 | 12.03 | 472.6 | 13.1 |
3 | 0.23 | 3000 | 525 | 0.052 | 15 | 13.75 | 420.3 | 14.52 |
4 | 0.23 | 3375 | 590.63 | 0.056 | 16.88 | 15.47 | 424.5 | 14.9 |
5 | 0.23 | 3750 | 656.25 | 0.062 | 18.75 | 17.19 | 434.0 | 15.6 |
6 | 0.26 | 2250 | 459.38 | 0.052 | 16.88 | 17.19 | 528.3 | 16.2 |
7 | 0.26 | 2625 | 525 | 0.056 | 18.75 | 10.31 | 272.5 | 9.8 |
8 | 0.26 | 3000 | 590.3 | 0.062 | 11.25 | 12.03 | 283.7 | 11.9 |
9 | 0.26 | 3375 | 656.25 | 0.038 | 13.13 | 13.75 | 737.9 | 15.3 |
10 | 0.26 | 3750 | 393.75 | 0.044 | 15 | 15.47 | 563.1 | 16.0 |
11 | 0.3 | 2250 | 525 | 0.062 | 13.13 | 15.47 | 370.4 | 14.1 |
12 | 0.3 | 2625 | 590.63 | 0.038 | 15 | 17.19 | 929.8 | 17.5 |
13 | 0.3 | 3000 | 656.25 | 0.044 | 16.88 | 10.31 | 402.8 | 11.2 |
14 | 0.3 | 3375 | 393.75 | 0.052 | 18.75 | 12.03 | 355.0 | 12.2 |
15 | 0.3 | 3750 | 459.38 | 0.056 | 11.25 | 13.75 | 375.1 | 14.7 |
16 | 0.34 | 2250 | 590.63 | 0.044 | 18.75 | 13.75 | 561.5 | 14.3 |
17 | 0.34 | 2625 | 656.25 | 0.052 | 11.25 | 15.47 | 500.7 | 15.2 |
18 | 0.34 | 3000 | 393.75 | 0.056 | 13.13 | 17.19 | 477.7 | 16.2 |
19 | 0.34 | 3375 | 459.38 | 0.062 | 15 | 10.31 | 232.8 | 9.3 |
20 | 0.34 | 3750 | 525 | 0.038 | 16.88 | 12.03 | 602.6 | 13.5 |
21 | 0.38 | 2250 | 656.25 | 0.056 | 15 | 12.03 | 328.2 | 11.7 |
22 | 0.38 | 2625 | 393.75 | 0.062 | 16.88 | 13.75 | 326.7 | 14.2 |
23 | 0.38 | 3000 | 459.38 | 0.038 | 18.75 | 15.47 | 727.7 | 15.85 |
24 | 0.38 | 3375 | 525 | 0.044 | 11.25 | 17.19 | 712.9 | 17.2 |
25 | 0.38 | 3750 | 590.63 | 0.052 | 13.13 | 10.31 | 304.5 | 10.3 |
After completing 25 numerical experiments and collecting all the data, the range analysis and variance analysis are conducted to study the influential degree of 6 factors on the target indices. The statistical results of the range analysis and variance analysis are shown in Tables
Results of range analysis.
Statistics | Initial porosity | Young’s modulus | Buoyant density | Friction coefficient | Cohesion | Slope angle | |
---|---|---|---|---|---|---|---|
Sliding distance (m) | 486.62 | 462.88 | 452.3 | 707.4 | 505.28 | 287.9 | |
480.0 | 455.86 | 467.3 | 542.58 | 482.28 | 350.32 | ||
477.1 | 500.46 | 476.18 | 422.2 | 472.62 | 517.28 | ||
475.06 | 492.62 | 480.72 | 375.6 | 470.14 | 605.26 | ||
458.52 | 465.48 | 500.8 | 329.52 | 456.98 | 616.54 | ||
Range | 28.1 | 44.6 | 48.5 | 377.88 | 58.3 | 328.64 | |
Maximum velocity (m/s) | 13.7 | 14.02 | 14.18 | 14.87 | 13.55 | 9.78 | |
13.84 | 13.96 | 13.84 | 14.36 | 13.8 | 10.56 | ||
13.85 | 13.95 | 13.83 | 13.7 | 13.82 | 15.21 | ||
13.94 | 13.78 | 13.8 | 13.46 | 14.0 | 16.54 | ||
14.08 | 13.7 | 13.76 | 13.02 | 14.24 | 17.32 | ||
Range | 0.38 | 0.32 | 0.42 | 1.85 | 0.69 | 7.54 |
Results of variance analysis.
Factor | Deviation square sum | Degree of freedom | |||
---|---|---|---|---|---|
Sliding distance (m) | Initial porosity | 2174.85 | 4 | 1.00 | 6.39 |
Young’s modulus | 7807.41 | 4 | 3.59 | 6.39 | |
Buoyant density | 8366.36 | 4 | 3.85 | 6.39 | |
Friction coefficient | 462041.94 | 4 | 212.45 | 6.39 | |
Cohesion | 10086.12 | 4 | 4.64 | 6.39 | |
Slope angle | 446696.46 | 4 | 205.39 | 6.39 | |
Maximum velocity (m/s) | Initial porosity | 0.39 | 4 | 1.07 | 6.39 |
Young’s modulus | 0.37 | 4 | 1.00 | 6.39 | |
Buoyant density | 0.49 | 4 | 1.35 | 6.39 | |
Friction coefficient | 10.79 | 4 | 29.49 | 6.39 | |
Cohesion | 1.31 | 4 | 3.59 | 6.39 | |
Slope angle | 242.55 | 4 | 662.71 | 6.39 |
The result of parametric analysis shows that the postfailure behavior of submarine landslides depends on the combined influences of different factors. The parameters can be classified into two types: the inner and the outer factors. The inner factors are the physical and mechanical properties of a slope. They can affect the initial stability of a submarine slope and the runout mechanism after a landslide is triggered by external factors. A sliding mass with high initial porosity will generate intense excess pore pressure, which can greatly reduce the effective stress as well as the shear strength, since they follow a linear relation. From the parametric analysis, a slope with high initial porosity and low cohesion and buoyant density often ends up in a higher velocity and longer runout distance as its shear strength decreases significantly and can easily change from a block sliding into a flow-like movement.
The outer factors include the slope angle and friction coefficient, which are exactly the most significant two influential factors to the slide. The slope angle represents the geomorphology of the slope when the slide initially happens, and the friction coefficient refers to the pathway resistance to the slide. During the sliding process, the initial gravitational potential would be transferred to the kinetic energy due to the particle movement and the strain energy, and part of dissipation due to the friction occurred in between the slope base and the floor. The slope angle can affect the initial morphology, and thereby a higher slope angle may represent a higher gravitational potential, from which more kinetic energy could be transferred, that is, higher velocity. Likewise, the friction would have very few effects on fluid flow with high Reynolds when its velocity is high. As the velocity decreases, the influence of friction becomes more dominant. Moreover, a higher friction coefficient corresponds to a higher energy dissipation rate, which may influence the time the friction takes to have an effect. Therefore, the friction appears to control the sliding distance the most.
By utilizing a hydromechanical coupled MPM formulation, two cases of soil flows fully under water have been analyzed, in which the ability and adequacy of MPM in modeling the submarine landslides are demonstrated. As for the South Mediterranean landslide case, the simulation results match well with the real situation and it can be concluded that the failure mode of the slide is a disintegrative one. The seawater and gravity loading increase the differences in the stress levels of the initial nonuniform strain field and triggered the landslide. Plastic deformation first occurs at the foot of the slope, and the deformation zone becomes larger and extends into the slope body. Due to the sudden increase in excess pore pressure, the block sliding gradually evolves into a flow-like movement with the sliding body elongating and its thickness reducing, ending up with a high maximum velocity of 14.52 m/s and final runout distance of 420.3 m. Parametric analysis results show that the postfailure mechanism of submarine landslides depends on the combined influences of different parameters. The slope angle may influence the maximum sliding velocity in the greatest extent; meanwhile, for the sliding distance, the friction coefficient in between the seabed and sliding materials may have the biggest influence.
Our study proves that MPM is suitable for simulating large deformation and multiphase problems such as submarine landslides by employing both the Lagrangian and Eulerian descriptions. Its ability to reproduce the entire runout process provides an insight into analyzing the postfailure mechanism of submarine slopes, which helps people better assess the risk caused by a potential landslide in offshore or abyssal areas. Our work presents a preliminary application of MPM in studying submarine landslides. However, a fully coupled MPM model still requires further development for refinement. More complex factors should be taken into consideration such as the interaction between the flowing mass and submarine structures to ensure accurate simulation results in geotechnical engineering problems.
The data (including all the figures, tables, and MPM simulation results) used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge the financial supports of the National Key R&D program of China (Grant no. 2018YFC1505104), the CAS Pioneer Hundred Talents Program, the Natural Science Foundation of Jiangsu Province (Grant no. BK20181182), and the Science & Technology Program of Suzhou (Grant no. SYG201613). They also thank Professor Dong Youkou from the China University of Geosciences for kindly providing them with geological information of the submarine landslide in the Southern Mediterranean.