Natural Hazards

, Volume 73, Issue 1, pp 37–62

Infiltration effects on a two-dimensional molecular dynamics model of landslides

Original Paper

DOI: 10.1007/s11069-013-0944-z

Cite this article as:
Martelloni, G. & Bagnoli, F. Nat Hazards (2014) 73: 37. doi:10.1007/s11069-013-0944-z

Abstract

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.

Keywords

Landslide Infiltration model Molecular dynamics Computational technique 

1 Introduction

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.

The idea is to use the one-dimensional infiltration equation along the z coordinate of the reference system (xz) along the slope (Fig. 1):
$$\frac{\partial p(z,t)}{\partial t} = K \cdot \frac{{\partial^{2} p(z,t)}}{{\partial z^{2} }}$$
(1)
where p(z, t) is the pore pressure at depth z and time t, while K is the diffusion coefficient along the Z-direction. This quantity in principle depends on slope angle α, which is, however, kept constant in our simulations.
Fig. 1

Reference system (xz) of the slope modeled with particles arranged in a regular grid according to disposition in horizontal and vertical layer

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.

For each vertical layer nj (Fig. 1), we assume a different infiltration process (small stochastic variations) along x-axes of the slope (Fig. 2). According to Eq. (1), the solution is given by the rainfall input multiplied by the response function, i.e.,
$$\left\{ {\begin{array}{*{20}c} {p(z,t^{*} ) = \frac{{I_{z} }}{{K_{z} }} \cdot R^{*} } \hfill \\ {R^{*} = \left\{ {\begin{array}{*{20}c} {R(t^{*} ),} \hfill & {t^{*} \le T^{*} } \hfill \\ {R(t^{*} ) - R(t^{*} - T^{*} ),} \hfill & {t^{*} > T^{*} } \hfill \\ \end{array} } \right.} \hfill \\ {R\left( {t^{*} } \right) = \sqrt {{{t^{*} } \mathord{\left/ {\vphantom {{t^{*} } \pi }} \right. \kern-0pt} \pi }} \cdot \exp \left( {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {t^{*} }}} \right. \kern-0pt} {t^{*} }}} \right) - erfc\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt {t^{*} } }}} \right. \kern-0pt} {\sqrt {t^{*} } }}} \right)} \hfill \\ {t^{*} = \frac{t}{{{{Z^{2} } \mathord{\left/ {\vphantom {{Z^{2} } {(K)}}} \right. \kern-0pt} {K}}}};\quad T^{*} = \frac{T}{{{{Z^{2} } \mathord{\left/ {\vphantom {{Z^{2} } {K}}} \right. \kern-0pt} {K}}}};\quad Z = \frac{z}{\cos (\alpha )}} \hfill \\ \end{array} } \right.$$
(2)
where \(t^{*}\) and \(T^{*}\) are, respectively, the normalized time and the normalized rainfall duration as explained by Iverson (2000), but rewritten in the coordinate system of the slope z (Z is the direction of gravity). Iz (derived explicitly in the following) and Kz are, respectively, the average infiltration rate and the hydraulic conductivity in the slope-normal direction. Finally, note that Eq. (2) does not include the initial pressure distribution, differently from the Iverson model, since in our simulation we start from a dry configuration of the system. However, in Sect. 4 we show the results in the presence of an initial wet material.
Fig. 2

Pore pressure response versus simulation time for each position x for various horizontal layers ni. The differences, between the curves, are due to stochastic variations

The first step, for the integration of the infiltration model into our numerical scheme, is to appropriately relate the simulated rainfall with the water content of the particles that constitute our fictitious soil. Let Iz be the discrete infiltration rate, i.e.,
$$I_{z} = K_{z} \cdot \frac{\varDelta h}{\varDelta z}$$
(3)
In Eq. (3) we assume that Δh is the rainfall increment at a generic instant t and thickness z, while Δz is the initial distance between the centers of mass of two adjoining particles. It is now possible to deduce Iz in terms of a mass ratio from simple considerations on the density ρ,
$$\rho = \frac{{{\text{d}}m}}{{{\text{d}}V}} = \frac{{{\text{d}}m}}{{S \cdot {\text{d}}h}}$$
(4)
where S is the unit area, m the mass and V the volume. Let us consider the density of water ρw and the density of the fictitious material (i.e., the particles) ρp. From Eq. (4) we obtain the infinitesimal heights dhw and dhp:
$$\left\{ {\begin{array}{*{20}c} {\rho_{w} = \frac{{{\text{d}}m_{w} }}{{S_{w} \cdot {\text{d}}h_{w} }} \Rightarrow {\text{d}}h_{w} = \frac{{{\text{d}}m_{w} }}{{S_{w} \cdot \rho_{w} }}} \hfill \\ {\rho_{p} = \frac{{{\text{d}}m_{p} }}{{S_{p} \cdot {\text{d}}h_{p} }} \Rightarrow {\text{d}}h_{p} = \frac{{{\text{d}}m_{p} }}{{S_{p} \cdot \rho_{p} }}} \hfill \\ \end{array} } \right.$$
(5)
In our case, since the dimensions of particles are relatively small, it is possible to assume, to a good approximation,
$$S_{w} \cdot \rho_{w} = S_{p} \cdot \rho_{p}$$
(6)
Consequently, the ratio between dhw and dhp gives our discrete infiltration rate Iz:
$$\left\{ {\frac{{{\text{d}}h_{w} }}{{{\text{d}}h_{p} }} = \frac{{{\text{d}}m_{w} }}{{{\text{d}}m_{p} }} \Rightarrow \frac{{\varDelta h_{w} }}{{\varDelta h_{p} }} = \frac{{\varDelta m_{w} }}{{\varDelta m_{p} }} \Rightarrow I_{z} = K_{z} \cdot \frac{{\varDelta m_{w} }}{{\varDelta m_{p} }}} \right.$$
(7)
Hence, we can simulate the rainfall in terms of water content, and using the response function \(R^{*}\), we can take into account the absorbed water wi(t) in time and space (the sum in time of Δmw elements) at thickness z, i.e., at each level of the particle layers. We then define, for each particle, a maximum porosity beyond which we have local saturation.

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.

The components of the gravity force on each particle i along the slope reference system are
$${\mathbf{F}}_{{{\mathbf{gi}}}} = \left\{ {g \cdot \sin (\alpha ) \cdot M_{i} ,\quad - g \cdot \cos (\alpha ) \cdot M_{i} } \right\}$$
(8)
where Mi = mi + wi(t) is the total particle mass, i.e., the sum of the dry mass mi, variable from particle to particle, and the cumulative amount of absorbed water wi(t); g is the gravitational acceleration; and α is the angle of the slope.
While in other approaches the interaction forces between grains are computed considering the dimensions of particles, for instance using the Hertz-Mindlin model (Goren et al. 2011), in our case the interaction force Fij (which acts on particle i due to particle j, considering pointlike particles) is defined through a potential, analogous to the Lennard-Jones one,
$$\begin{aligned} {\mathbf{F}}_{{{\mathbf{ij}}}} & = - {\mathbf{F}}_{{{\mathbf{ij}}}} = - \left[ {k_{1} \left( {\frac{{r_{ij} }}{L}} \right)^{ - 2} - k_{2} \left( {\frac{{r_{ij} }}{L}} \right)^{ - 1} } \right] \cdot {\hat{\mathbf{r}}}_{{{\mathbf{ij}}}} \\ r_{ij} & = \left| {{\mathbf{r}}_{{{\mathbf{ij}}}} } \right| = \sqrt {(x_{j} - x_{i} )^{2} + (y_{j} - y_{i} )^{2} } \\ \end{aligned}$$
(9)
where rij is the distance between the two centers of mass, k1 and k2 are constants (k1 = k2 in “classical” Lennard-Jones potential), \({\hat{\mathbf{r}}}_{ij}\) is the unit vector relative to the force, and L is the equilibrium distance (Fig. 3). If k1 = k2 we have the equilibrium at distance L = 1 (Fig. 4); with the same configuration and setting k1 ≠ k2, it is possible to simulate, starting from t = 0, a compressed stress state of the particles (Fig. 4). We have chosen the 2–1 power terms in the force (Eq. 9), considering that higher powers can easily lead to high compressive energies and therefore to high accelerations and velocities of the particles (Martelloni et al. 2012b), a situation very far from the reality.
Fig. 3

Schematic description of the interaction force considering the distance between two particles relating to the equilibrium distance L

Fig. 4

Interaction force for the equilibrium distance L = 1 and for the equilibrium distance L > 1 (simulated initial compressed stress state)

The triggering mechanism is modeled after the Mohr–Coulomb law (Coulomb 1776; Mohr 1914) in the form of an effective stress (Terzaghi 1943),
$$\tau_{f} = (\sigma - p) \cdot \tan \phi^{{\prime }} + c^{{\prime }}$$
(10)
where τf is the shear stress at failure, σ the normal stress, ϕ′ the friction angle and c′ the cohesion term. As the Mohr–Coulomb failure criterion is a simple friction law, short of the term of cohesion, it can be easily adapted to our case, rewriting Eq. (10) as
$$\left\{ {\begin{array}{*{20}c} {\tau_{\text{f}} = F_{s} + c^{\prime } } \hfill \\ {F_{s} = \left[ {M_{i} (z,t) \cdot g \cdot \cos (\alpha ) - p(z,t)} \right] \cdot \tan \phi^{\prime } } \hfill \\ \end{array} } \right.$$
(11)
Finally, we express the local triggering, i.e., the triggering at the particle level, using a failure criterion and considering a speed threshold vd for the static–dynamic transition. In synthesis, for each particle i,
$$\left\{ {\begin{array}{*{20}c} {\left| {{\mathbf{F}}_{{\mathbf{i}}} } \right| < F_{si} + c_{i}^{{\prime }} } \hfill \\ {\left| {{\mathbf{v}}_{{\mathbf{i}}} } \right| < v_{d} } \hfill \\ \end{array} } \right.$$
(12)
$${\mathbf{F}}_{{\mathbf{i}}} = {\mathbf{F}}_{{{\mathbf{gi}}}} + \sum\limits_{{{\mathbf{j}} = 1}}^{{{\mathbf{j}} = n_{k} }} {{\mathbf{F}}_{{{\mathbf{ij}}}} }$$
(13)
where |Fi| represents the module of the active forces, i.e., the force of gravity Fgi plus the force resulting from the potential, the latter being the sum of the terms in Eq. (13). In this equation, nk denotes the total number of particles that interact (being inside the interaction radius) with particle i. At the initial instant, nk = 8 (Fig. 5). The term |vi| in Eq. (12) is the module of the speed, and Fsi comes from Eq (11). The double control on the force and velocity in Eq. (12) expresses the criterions for the starting and the stopping conditions of the particles.
Fig. 5

Interaction scheme of particles within the interaction radius. (Left) At time t = 0, particles are arranged on a regular grid. (Right) Recomputation of the interactions for each mass within the assigned range

2.2 Dynamic conditions and updating algorithm

Equation (12) is valid in dynamical conditions, as they represent, in synthesis, a control on the state of motion of the particles. Once a particle is moving, we consider also a dynamic friction (the force direction is opposed to the velocity one), expressed for each particle i by:
$${\mathbf{F}}_{{{\mathbf{di}}}} = (m_{i} + w_{i} (t))g\cos (\alpha ) \cdot (\mu_{d} \cdot \exp ( - w_{0} t) + \mu_{{d{\text{low}}}} \cdot (1 - \exp ( - w_{0} t))) \cdot ( - {\hat{\mathbf{v}}})$$
(14)
The force Fdi depends on two friction terms, characterized by coefficients μd and μdlow: μd for t = 0 (time of rainfall start) and μdlow for t  , with μd > μdlow. In synthesis, the effect of rainfall is to decrease the friction of the particles during time (through the constant velocity w0 of the exponential). Moreover, the friction coefficients μd and μdlow vary randomly (with a small dispersion) with the position, modeling the roughness between the particles. This friction law is inspired by Jop et al. (2006) and has been adopted for simplicity.

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).

The total force F acting on a moving mass is given by the sum of the active forces, the dynamic friction force and a viscosity term with coefficient υ,
$${\mathbf{F}} = {\mathbf{F}}_{{\mathbf{i}}} + {\mathbf{F}}_{{{\mathbf{di}}}} + {\mathbf{F}}_{{\gamma {\mathbf{i}}}} = {\mathbf{F}}_{{{\mathbf{gi}}}} + \sum\limits_{{{\mathbf{j}} = 1}}^{{{\mathbf{j}} = n_{k} }} {{\mathbf{F}}_{{{\mathbf{ij}}}} } + {\mathbf{F}}_{{{\mathbf{di}}}} - \upsilon \cdot {\mathbf{v}}_{{\mathbf{i}}}$$
(15)
In this case, the value nk in the sum of Eq. (15) can be less or greater than 8, due to the possible compression or dilatation effects during the motion of masses.

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.

The velocity Verlet algorithm (Verlet 1967), for the updating of positions r and velocities v of each particle after a time interval Δt, reads, respectively, for the first (Eq. 16) and second order (Eq. 17)
$$\left\{ {\begin{array}{*{20}c} {{\mathbf{r}}(t + \varDelta t) = {\mathbf{r}}(t) + {\mathbf{v}}(t)\varDelta t} \hfill \\ {{\mathbf{v}}(t + \varDelta t) = {\mathbf{v}}(t) + \frac{1}{m}{\mathbf{F}}(t)\varDelta t} \hfill \\ \end{array} } \right.$$
(16)
$$\left\{ {\begin{array}{*{20}c} {{\mathbf{r}}(t + \varDelta t) = {\mathbf{r}}(t) + {\mathbf{v}}(t)\varDelta t + \frac{{{\mathbf{F}}(t)}}{m}\varDelta t^{2} } \hfill \\ {{\mathbf{v}}(t + \varDelta t) = {\mathbf{v}}(t) + \frac{1}{2m}[{\mathbf{F}}(t + \varDelta t) + {\mathbf{F}}(t)]\varDelta t} \hfill \\ \end{array} } \right.$$
(17)
The force depends on the velocity by means of the viscosity term. We have checked that the results are not affected by the order of the algorithm.

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

In this section we show the simulation results and report some peculiarities that emerge from the analysis of the generated data. Concerning the dynamics, we observe the typical stick-and-slip dynamics of frictional systems, earthquake faults and landslides (Nielsen et al. 2010) that are also observed in other MD models as the seismic fault one (Ciamarra et al. 2010). The mean velocity (the average of the velocity of the particles in motion) and the mean kinetic energy increments of the simulated system over time (the difference of the kinetic energy between two subsequent time steps, averaged over all particles) are reported in Fig. 6. These observables are computed over the time interval (t, t + n ·Δt), where n is the number of discrete time steps between two subsequent events of particle detachment (Martelloni et al. 2012b). It is possible to note a first stick phase and a subsequent slip one (Heslot et al. 1994). The time behavior of the inverse of the mean velocity is plotted in Fig. 7, and there we can better identify the initial stick phase. The behavior of this simulation, in terms of the velocity, is similar to that of real landslides (Suwa et al. 2010). As mentioned above, we use the Fukuzono method of the inverse of velocity for the evaluation of the failure time of our simulated landslide (Fukuzono 1985). Simply, if the displacement velocity v at a slope surface increases over time, its inverse (1/v) decreases. When (1/v) approaches zero, failure occurs. Therefore, we evaluate the time of triggering by means of the calibration of the function adopted by Fukuzono,
$$\frac{1}{v} = [\beta \cdot (\alpha - 1)]^{{\frac{1}{\alpha - 1}}} \cdot (t_{r} - t)^{{\frac{1}{\alpha - 1}}}$$
(18)
where v is the mean velocity of the simulated landslide, t the time of simulation, tr the time of failure, and α and β are constant. We first apply this method to the initial part of the simulation, corresponding to the maximum variation in the inverse of velocity (green circle in Fig. 7). We then extend the calibration interval, including the subsequent five points, to consider all of them (red circle in Fig. 7).
Fig. 6

(Top) Mean velocity and (bottom) mean kinetic increment of simulated landslide (all particles in motion) versus simulation time

Fig. 7

(Top) Inverse of mean velocity versus simulation time and (bottom) application of the inverse velocity method (Fukuzono 1985) to our simulated system to estimate the triggering time

These evaluated triggering times vary from 150 to 220 simulation time steps. We note that according to Eq. (18), some curves are concave (1 < α < 2) and others are convex (α > 2). This is due to the increasing number of points considered for the calibration of Eq. (18), since, as we can see in Fig. 7, the shape of considered dataset varies (e.g., the ultimate calibration includes all circle points and the convexity increases). Then in Figs. 8 and 9 the landslide configuration is reported in the coordinate system of the slope (xz) for the extreme values of the evaluated range of time triggering. We observe an initial motion of the upper horizontal layer and an initial creep phase (in Figs. 8 and 9 the green particles are in motion, while the red ones are at rest). The infiltration states for each position of slope are reported for t = 150 and t = 220 in Figs. 8 and 9, respectively. Moreover, in Fig. 10 we report a system configuration of the same simulation at t = 600 and t = 630, where we note a slip phase with fractures, detachments and arching-like phenomena (compression zone).
Fig. 8

Simulated landslide in the coordinate system of the slope for t = 150 and simulated infiltration along the slope for t = 150

Fig. 9

Simulated landslide in the coordinate system of the slope for t = 220 and simulated infiltration along the slope for t = 220

Fig. 10

Simulated landslide in the coordinate system of the slope for t = 600 and t = 630

Other interesting results can be observed from a statistical point of view (Martelloni et al. 2012b): We perform some simulations varying the viscosity coefficient υ of Eq. (15). As reported in Figs. 11 and 12, we observe a transition in the distribution of the mean kinetic energy increments from Gaussian to power law, after decreasing the viscosity coefficient from a finite initial value up to zero. This behavior is compatible with the corresponding velocity increasing in the landslide after the decrease in the viscosity. In other words, this behavior is congruent with a stick-and-slip dynamics.
Fig. 11

Distribution of kinetic energy increments for υ = 0.01 (top) and for υ = 0.0025 (bottom)

Fig. 12

Distribution of kinetic energy increments for υ = 0 and distributions of time intervals relative to stick (red line) and slip (green line) phases for υ = 0

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).

We study the resulting distributions using three fit functions: Gaussian, lognormal and power law, respectively,
$$f(x) = a_{1} \cdot \exp \left( { - \left( {\frac{{x - b_{1} }}{{c_{1} }}} \right)^{2} } \right)$$
(19)
$$f(x) = \frac{{a_{1} }}{x} \cdot \exp \left( { - \left( {\frac{{\log x - b_{1} }}{{c_{1} }}} \right)^{2} } \right)$$
(20)
$$f(x) = a \cdot x^{b}$$
(21)
where x is the analyzed data. In the case of lognormal fitting, it is simpler to consider the logarithm of data, whose corresponding distribution is Gaussian. If f(x) is the density function of the variable x and y, the new variable such that x = log(y), the density function g(y) is equal to f[x(y)]·|dx(y)/dy|; therefore, if the original distribution f(x) is a Gaussian, we obtain the extra factor 1/|y| in the new distribution g(y).
The adopted estimators of the fitting accuracy are,
$$\begin{aligned} {\text{SSE}} & = \sum\limits_{i = 1}^{n} {\left( {y_{i} - \hat {y}_{i} } \right)^{2} } \\ R^{2} & = 1 - \frac{\text{SSE}}{\text{SST}};\quad {\text{SST}} = \sum\limits_{i = 1}^{n} {\left( {y_{i} - \bar{y}_{i} } \right)^{2} } \\ {\hat R}^{2} & = 1 - (1 - R^{2} )\frac{n - 1}{n - p - 1} \\ {\text{RMSE}} & = \sqrt {\frac{\text{SSE}}{n - m}} \\ \end{aligned}$$
(22)
i.e., SSE is the sum of squared residuals, R2 is the coefficient of determination, \({\hat {R}}^{2}\) is the Adjusted-R2, and RMSE is the root mean square error.
In Tables 1 and 2, we report the fit estimators, Eq. (22), and the optimal fit parameters (a1, b1, c1, a, b) of the resulting distributions according to Eqs. (19), (20) and (21).
Table 1

Fitting parameters concerning the mean kinetic energy increment distribution varying the coefficient of viscosity υ; parameters of fit goodness and parameters of obtained distribution

υ (Distribution)

0.01 (Gaussian)

0.0025 (Lognormal)

0 (Power law)

SSE

405.1

780.1

32.52

R2

0.972

0.9524

0.999

Adjusted-R2

0.9685

0.9418

0.9989

RMSE

5.032

9.31

1.078

a1

96.98

106.6

 

b1

9.713

6.361

 

c1

1.914

2.685

 

a

  

259.1

b

  

−2.11

Table 2

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

υ (Distribution)

0.01 (Power law)

0.0025 (Power law)

0 (Power law)

SSE

238.3

206.1

149.7 (1144)

R2

0.9973

0.9936

0.9992 (1)

Adjusted-R2

0.9972

0.9933

0.9991 (1)

RMSE

2.917

2.713

3.393 (13.81)

a

353.1

174.4

693.3 (5807)

b

−1.722

−1.469

−2.709 (−4.403)

Finally, we analyze how the number of particles in motion varies during a simulation, and also in this case, we observe a power law distribution with exponent equal to 0.8848 (Fig. 13). This represents a measure of the mobilized volume of the simulated landslide, although we should consider that this volume can expand.
Fig. 13

(Top) Number of particles in motion (green line) and particles at rest (red line). (Bottom) The corresponding power law distribution of particles in motion

4 Sensitivity analysis of the model

In this section we show the dependence of the model by its parameters. We perform several simulations to discriminate the parameters that influence the triggering of simulated landslide from the ones that influence the velocity in the propagation phase. The simulations are achieved considering both regular and random initial configurations of particles. In a random configuration, each particle is shifted from the regular grid (equilibrium state of interparticle potential) according to a Gaussian distribution of the displacements (an example is shown in Fig. 14). We remark that the range of the parameter values is only indicative since, for the moment, we consider only a theoretical model. In Figs. 15, 16, 17 and 18, we report the triggering time (considering the first particle in motion) versus the slope angle, the friction angle, the cohesion coefficient and the hydraulic conductivity, respectively. As expected, the triggering time is smaller in the case of a random initial configuration of grains than the case of a regular one. This is due to the fact that in the regular configuration the particle is in equilibrium state for that concerns the interparticle potential, while in the random configuration a nonzero force (which may add to the gravitational one) is present. In Fig. 19 the triggering time versus the hydraulic conductivity is shown, considering the initial nonzero pore pressure. Comparing this figure with Fig. 18, it is evident that in the former case the triggering is more rapid.
Fig. 14

Random configuration of the grains for the initial state of the system

Fig. 15

Triggering time, relative to the first particle in motion, varying the slope angle in case of regular and random configurations of grains

Fig. 16

Triggering time, relative to the first particle in motion, varying the friction angle in case of regular and random configurations of grains. The slope is equal to 45°

Fig. 17

Triggering time, relative to the first particle in motion, varying the cohesion coefficient in case of regular and random configurations of grains

Fig. 18

Triggering time, relative to the first particle in motion, varying the hydraulic conductivity in case of regular and random configurations of grains

Fig. 19

Triggering time, relative to the first particle in motion, varying the hydraulic conductivity in case of regular configurations of grains considering initial nonzero pore pressure

Let us investigate now the influence of the random variability. We perform a computation of the triggering time (based on the first particle in motion), varying the coefficient of random cohesion c′ for each particle i according to:
$$c_{i}^{{\prime }} = k_{c} + c_{r} \cdot r_{i} \cdot k_{c}$$
(23)
where kc is a constant, ri is a random uniform number, and cr is a tuning factor. In Fig. 20 the triggering time is reported as a function of cr, expressed as percentage in respect of the unit.
Fig. 20

Triggering time, relative to the first particle in motion, varying randomly the cohesion term according to Eq. (23)

However, a more sensible definition of the starting time is based on the motion of the center of mass of the system (Martelloni et al. 2012b). Therefore, we perform a set of onerous simulations in which the triggering time is measured when the displacement of the center of mass (respect to initial state) exceeds a certain value (mcdx, mcdy), varying also the thickness of simulated soil, i.e., the number of horizontal particle layers. Results are reported in Fig. 21. Clearly, in this case the triggering time is much higher than when considering the first particle displacement.
Fig. 21

Triggering time, relative to the center of mass displacement that exceeds a certain value variable, varying the thickness of simulated soil. The slope is equal to 45°

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.

We perform simulations by varying the coefficients μd and μdlow of Eq. (14) according to:
$$\left\{ {\begin{array}{*{20}c} {\mu_{d} = 0.8 + C_{\mu } \cdot k_{\mu } } \hfill \\ {\mu_{{d{\text{low}}}} = 0.2 + C_{\mu } \cdot k_{\mu } } \hfill \\ \end{array} } \right.$$
(24)
where kμ = 1, 2,…, 20, considering both regular and random initial configurations of the particles. In Fig. 22 we show the mean velocity versus the dynamic friction coefficients μd, with the multiplicative factor Cμ = 0.05. Again, the velocity of the initial random configuration is higher, and, as expected, the velocity decreases with the increasing in the coefficients, in an almost linear way.
Fig. 22

Mean velocity versus dynamic friction coefficient μd in case of regular and random configurations of grains

We also study the influence of the viscosity coefficient υ, as reported in Fig. 23. Here the dependence is manifestly nonlinear.
Fig. 23

Mean velocity versus viscosity coefficient in case of regular and random configurations of grains

Finally, we perform two simulations to characterize the influence of the randomness on the system. We consider the variations in the multiplicative factor Cμ of Eq. (24) on the dynamic coefficients μd and μdlow. In other words, being rnμ a uniform random number, we consider the equations:
$$\left\{ {\begin{array}{*{20}c} {\mu_{d} = 0.8 + C_{\mu } \cdot rn_{\mu } } \hfill \\ {\mu_{{d{\text{low}}}} = 0.2 + C_{\mu } \cdot rn_{\mu } } \hfill \\ \end{array} } \right.$$
(25)
In Fig. 24 we report the mean velocity versus Cμ, expressed as percentage in respect of the unit.
Fig. 24

Mean velocity of the simulated landslide varying randomly the dynamic friction coefficients of Eq. (14), μd and μdlow, according to Eq. (25)

We also analyze the influence, on the mean landslide velocity, of the size of random fluctuations kpos of particle positions in the initial configurations, defined as
$$\left\{ {\begin{array}{*{20}c} {x(i,j) = i + k_{\text{pos}} \cdot rn_{1} } \hfill \\ {z(i,j) = j + k_{\text{pos}} \cdot rn_{2} } \hfill \\ {k_{\text{pos}} = 0.025 \cdot k_{n} } \hfill \\ \end{array} } \right.$$
(26)
where kn = 1, 2, …, 10, i = 1, 2, …, nj (nj is the index of vertical layer), j = 1, 2, …, ni (ni is the index of horizontal layer), and rn1rn2 are a Gaussian-distributed random numbers. We perform ten Monte Carlo simulations (enough to detect the trend of the average value of the velocity) with the coefficient kpos that increases monotonically, as reported in Fig. 25. It is clear that the initial stress has a monotonic influence on the velocity and that the fluctuations, although important, are not dominating.
Fig. 25

A Monte Carlo simulation to analyze the size of the fluctuations on the initial state of the particle positions and how they influence the mean velocity of the simulated landslide; in the graph we report ten simulations and the average (black line)

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.

Acknowledgments

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.

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Physics and AstronomyCenter for the Study of Complex Dynamics (CSDC)Sesto Fiorentino (FI)Italy
  2. 2.INFN, Sez. FirenzeFlorenceItaly

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