Infiltration effects on a two-dimensional molecular dynamics model of landslides
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- Martelloni, G. & Bagnoli, F. Nat Hazards (2014) 73: 37. doi:10.1007/s11069-013-0944-z
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We propose a two-dimensional computational model for deep landslides triggered by rainfall, based on interacting particles or grains. The model describes a vertical section of a fictitious granular material along a slope, in order to study the behavior of a wide-thickness landslide. The triggering of the landslide is caused by the exceeding of two conditions: a threshold speed and a condition on the static friction of the particles, the latter based on the Mohr–Coulomb failure criterion (Coulomb in Mem Acad R Div Sav 7:343–387, 1776; Mohr in Abhandlungen aus dem Gebiete der Technischen Mechanik. Ernst, Berlin, 1914). The interparticle interactions are represented as a potential that, in the absence of suitable experimental data and due to the arbitrariness of the grain dimension, is modeled similarly to the Lennard-Jones’ one (Lennard-Jones in Proc R Soc Lond A 106(738):463–477, 1924), i.e., with an attractive and a repulsive part. For the updating of the particle positions, we use a molecular dynamics method, which is quite suitable for this type of systems (Herrmann and Luding in Continuum Mech Thermodyn 10:189–231, 1998). An infiltration scheme is introduced for modeling the increasing pore pressure due to the rainfall. Finally, we also introduce the viscosity in the dynamical equations of motion. The statistical characterization and dynamical behavior of the results of simulations are quite satisfactory relative to real landslides: We obtain a power law distribution of landslide triggering times, and the velocity patterns are typical of real cases, including the acceleration progression. Therefore, we can claim that this type of modeling can represent a new method to simulate landslides triggered by rainfall.
KeywordsLandslide Infiltration model Molecular dynamics Computational technique
Landslides are extreme and recurrent events in mountainous areas, often with many implications for urban environments, and consequently on the stricken population, with human casualties and economical losses (Van Asch et al. 2007). Major changes in the natural environment may result, depending on the extent of the phenomenon. “A major threat is induced by all types of slope movements (e.g., falls, topples, slides, lateral spreads, flows)… which represent one of the most destructive natural hazards on Earth” (Brabb 1991). For these reasons, landsliding represents a challenging problem in earth science.
The landslide triggering is often caused by an intense and/or long rain. In particular, shallow landslides are triggered by short intense rainfalls (Campbell 1975; Crosta and Frattini 2007), while deep landslides are connected with prolonged and less intense rainfall events (Bonnard and Noverraz 2001). Thanks to the rapid development of computers and advanced numerical methods, physical-based models have been developed to predict the landslide triggering and to evaluate the run-out.
Two fundamental approaches have been proposed to assess the dependence of landslide triggering on rainfall measurements. The first one relies on deterministic models (based on infiltration models and geotechnical considerations), while the second one defines statistically the average rainfall threshold above which the triggering of one or more landslides is possible (Segoni et al. 2009; Martelloni et al. 2012a; Rosi et al. 2012). Regarding the propagation of a landslide, most of the numerical methods exploited a continuum approach, i.e., an Eulerian description of a suitable fluid (Crosta et al. 2003; Patra et al. 2005). Other modeling approaches are based on cellular automata, which is again an Eulerian description of a discretized fluid, with which one may more easily include nonlinear terms. Examples are represented by the SCIDDICA cellular automata-based model family: SCIDDICA S4c is a release for the simulation of flow-type landslides (D’Ambrosio et al. 2007), SCIDDICA S3-hex is specifically developed for simulating debris flow (D’Ambrosio et al. 2003), while SCIDDICA-SS2 is conceived in order to simulate combined subaerial-subaqueous flowlike landslides (Avolio et al. 2008). Another approach, based on cellular automata and more linked to rheological characteristics of the flow, was presented by Iovine and Mangraviti (2009).
A relatively less common approach is the Lagrangian one, based on discrete-particle methods, in which the material forming the slope (and the landslide) is represented as an ensemble of interacting elements, called particles or grains. The discrete element method (DEM) is used to model granular materials, debris flow and flowlike landslides (Cundall and Strack (1979); Iordanoff et al. 2010), and it has been used successfully in modeling geophysical applications (Cleary and Campbell 1993; Campbell et al. 1995). Moreover, Potapov et al. (2001) presented a study combining smoothed particle hydrodynamics (SPH) and DEM to model flows containing a viscous fluid and solid particles.
Another Lagrangian method is the molecular dynamics (MD) one, closely related to DEM. This latter method is generally distinguished by the inclusion of rotational degrees of freedom as well as stateful contacts with grains often showing complicated geometries. However, the inclusion of a detailed description of the elementary components and of their interactions (Thorton and Yin 1991) sets a limit to the maximum number of the elements in simulations, for a given computational power.
The DEM approach allows the investigation of the flow details, including the variations in the velocities and the fluctuations or the solid fractions. From a theoretical point of view, constitutive relations and transport properties have been developed for dry granular flows (Campbell 1997). Obviously, the accuracy of the simulation has to be compared with the available experimental data. In the case of laboratory experiments, very accurate data can be obtained, but this is not possible for real landslides.
These arguments motivated us to reduce the complexity of the model as much as possible, examining whether this choice is compatible with the behavior of real landslides. In previous works, we proposed a shallow landslide modeling (Massaro et al. 2011; Martelloni et al. 2012b). In this paper, we present the integration of an infiltration mechanism into a MD model for the starting and prosecution of particle movement along a slope, after a triggering induced by rainfalls. This model is designed for the study of deep landslides.
The inclusion of the rainfall effects, i.e., the modeling of the fluid that filters in a porous material, finally triggering the landslide, is a challenging problem. Our idea is to integrate the Iverson model of infiltration (Iverson 2000) with the MD approach, by considering the infiltration at the particle level. We use a failure criterion of the Mohr–Coulomb type to assess the triggering of the local landslides. We also introduce in the model some stochastic variations to take into account the variability of the slope in terms of the water infiltration and frictional behavior. The resulting numerical method, similar to that of molecular dynamics, is based on the use of an interaction potential between the particles, analogous to the Lennard-Jones one. As we shall see in the following sections, by means of this type of force, we can also simulate the compressed state of the particles. At present, we do not pretend to be able to simulate a real landslide or debris flow. Our goal is rather to explore new alternative approaches useful for this kind of problems.
2 The model
2.1 Infiltration modeling and triggering mechanism
In a previous work (Martelloni et al. 2012b), we proposed a model for shallow landslides triggered by rainfall. This model is coarse-grained, based on fictitious particles, using a molecular dynamic approach for the updating of their positions. In the previous version, we considered only one particle layer. Due to the quick response to rainfall of shallow landslides, we did not introduce there an infiltration model to integrate the triggering dynamics. Indeed, the failure conditions of shallow soils “are not necessarily determined by the development of positive pore pressures on a potential slip plane. Failure conditions can also occur when, at a critical depth, which is determined by the cohesion of the soil material and the slope angle, the moisture content in the soil becomes close to saturation, resulting in a considerable reduction of soil strength” (Van Asch et al. 1999). Obviously, in the case of deep landslides, this choice cannot be made. Therefore, we extended the model by including the crucial role of increasing pore pressure due to the rain infiltration, which is the main actor of the triggering mechanism (Van Asch et al. 1999). In particular in deep-seated landslides, failure deformations have been reported at depths of up to 250 m (Petley and Allison 1998), and this type of landslide shows a complex response to rainfall. The landslide triggering is mainly influenced by moderate but exceptionally prolonged (up to 6 months) rainfall period (Ibsen and Casagli 2004). At present, we use a reformulation of the Iverson infiltration model (Iverson 2000), adapted to the molecular dynamics approach according to the failure criterion of Mohr–Coulomb.
At the time t = 0, the material is dry, i.e., the particles exhibit initially an initial pore pressure equal to zero. Starting from time t = 0, a constant rain is simulated considering the infiltration along the direction normal to the slope: We adopt a simple scheme, considering only positive pore pressures at particle positions. These pore pressures should be simply interpreted as a perturbation of the rest state of each grain. In other words, according to our scheme, the pore pressure function p(z, t) is to be interpreted as a time–space-dependent scalar field. Only in this sense it is possible to have positive pore pressures at the top of the soil (Figs. 8, 9). Other effects, such as the limited infiltration capacity (the water infiltration capacity of a soil decreases with time and may become limiting) and the unsaturated soil strength (suction stress), are not taken into account.
The local pore pressure p is related to the absorbed water wi(t), which can be expressed as the ratio between p and the hydraulic conductivity Kz per unit area S and unit time.
2.2 Dynamic conditions and updating algorithm
Initially, the particles are arranged on a regular grid, i.e., at the instant t = 0, each mass is placed in the nodes of a regular rectangular grid, and therefore, every particle interacts with the eight particles placed in the nearest and next-to-nearest nodes of the regular grid (Fig. 5). At each time step, the interactions are recalculated for each object within a given interaction range. This technique is used in molecular dynamics and congruent with principle of action and reaction (Fig. 5).
For the updating of the positions and velocities, we have adopted the Verlet algorithm, which is simple and very stable and allows a good numerical approximation in the case of energy conservation. Moreover, as the forces are calculated once for each time step, this computational updating method does not require a large computational power.
In the case of a uniform rainfall, it is simple to theoretically deduce the time of local triggering, i.e., the time of the first particle detachment. However, since the sliding masses could stop after a first detachment, the triggering of single particle cannot represent the definition of the whole landslide triggering. A better definition in this sense is based on the motion of center of mass of the global system or the center of mass of all particles in motion (Martelloni et al. 2012b). In the next section, we see that it is possible to use a Fukuzono method (Fukuzono 1985) to predict the failure time for our simulated system.
3 Results of model simulations
The transition of the distribution of the mean kinetic energy increments is also observed in the same simulation at different times. By computing this distribution in the stick phase, we always observe a Gaussian distribution (and never a power law) even for a vanishing viscosity coefficient υ = 0. In the slip phase, we observe a power law also for high values of the viscosity.
We measure the distribution of stick durations, i.e., the distribution of the time intervals between two subsequent events of the detachment occurrences, and the corresponding distribution of slip durations. A power law distribution of these durations is observed in all simulations (in Fig. 12 the result for υ = 0). This result is consistent with our simulations for shallow landslides (Martelloni et al. 2012b).
Fitting parameters concerning the mean kinetic energy increment distribution varying the coefficient of viscosity υ; parameters of fit goodness and parameters of obtained distribution
0 (Power law)
Fitting parameters concerning the local time triggering distribution varying the coefficient of viscosity υ, parameters of fit goodness and parameters of obtained power law distribution; from brackets in the fourth column, for υ = 0, the parameters of correspondent distribution of slip duration are also reported
0.01 (Power law)
0.0025 (Power law)
0 (Power law)
4 Sensitivity analysis of the model
Then, we consider the parameters that influence the velocity of the system, and we perform some simulations considering the mean velocity of the simulated landslide at a certain instant during the motion.
5 Discussion and conclusions
Although the model proposed in this paper is still quite schematic, our results encourage the investigations in this direction. The results are consistent with the behavior of real landslides induced by rainfall, and interesting behaviors emerge from the dynamical and statistical points of view. Emerging phenomena such as fractures, detachments and arching can be observed (see also Martelloni et al. 2012b). In particular, the model reproduces well the energy and time distribution of avalanches, analogously to the Gutenberg–Richter and Omori power law distributions for earthquakes (Gutenberg and Richter 1956; Omori 1895). We observe a power law distribution also for the number of particles in motion. We note that other natural hazards (landslides, earthquakes and forest fires) also exhibit similar distributions (Malamud et al. 2004; Turcotte 1997), characteristic of self-organized critical systems (Turcotte and Malamud 2004).
From the statistical point of view, we observe an interesting characteristic of this type of systems, i.e., a transition in the distribution of the mean kinetic energy increments from a Gaussian to a power law after decreasing the viscosity coefficient up to zero. This behavior is compatible with the corresponding velocity increasing, i.e., such a crossover in the distribution means that we pass from a relative slow movement to a relative fast one. This transition is found also in the same simulation at different time instants, for a fixed value of the viscosity coefficient (even for zero value of this parameter). Explicitly, we observe a Gaussian distribution of kinetic energy increments in an initial phase of movement (all particles having similar velocities), while, continuing the simulation, a power law distribution appears, due to particles with higher velocity. Actually, we observe a characteristic velocity and an energy pattern typical of a stick-and-slip dynamics, similar to the behavior of real landslides (Sornette et al. 2004).
In comparison with real landscapes, we observe a relatively smooth crossover from primary to secondary creep (i.e., the variation in average speed of the moving portion, see Fig. 6). We have to consider, however, that the timescale of our model is arbitrary and that this transition may be influenced by the size of the system.
We also show that it is possible to apply the method of the inverse surface displacement velocity for predicting the failure time (Fukuzono 1985). Finally, we achieve a complete sensibility analysis of the model parameters considering also the fluctuations necessary to take into account the variability of the soil.
We thank the Ente Cassa di Risparmio di Firenze for its support under the contract Studio dei fenomeni di innesco e propagazione di frane in relazione ad eventi di pioggia e/o terremoti per mezzo di modelli matematici ed esperimenti di laboratorio su mezzi granulari.