### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Slider Block Model With State- and Velocity-Dependent Friction
- 3. Vaiont Landslide
- 4. La Clapière Landslide: Aborted 1982–1987 Acceleration
- 5. Discussion and Conclusion
- Appendix A:: Derivation of the Full Solution of the Frictional Problem
- Acknowledgments
- References
- Supporting Information

[1] Accelerating displacements preceding some catastrophic landslides have been found to display a finite time singularity of the velocity *v* ∼1/(*t*_{c} − *t*) [*Voight*, 1988a, 1988b]. Here we provide a physical basis for this phenomenological law based on a slider block model using a state- and velocity-dependent friction law established in the laboratory. This physical model accounts for and generalizes Voight's observation: Depending on the ratio *B*/*A* of two parameters of the rate and state friction law and on the initial frictional state of the sliding surfaces characterized by a reduced parameter *X*_{i}, four possible regimes are found. Two regimes can account for an acceleration of the displacement. For *B*/*A* > 1 (velocity weakening) and *X*_{i} < 1 the slider block exhibits an unstable acceleration leading to a finite time singularity of the displacement and of the velocity *v* ∼ 1/(*t*_{c} − *t*), thus rationalizing Voight's empirical law. An acceleration of the displacement can also be reproduced in the velocity-strengthening regime for *B*/*A* < 1 and *X*_{i} > 1. In this case, the acceleration of the displacement evolves toward a stable sliding with a constant sliding velocity. The two other cases (*B*/*A* < 1 and *X*_{i} < 1 and *B*/*A* > 1 and *X*_{i} > 1) give a deceleration of the displacement. We use the slider block friction model to analyze quantitatively the displacement and velocity data preceding two landslides, Vaiont and La Clapière. The Vaiont landslide was the catastrophic culmination of an accelerated slope velocity. La Clapière landslide was characterized by a peak of slope acceleration that followed decades of ongoing accelerating displacements succeeded by a restabilization. Our inversion of the slider block model in these data sets shows good fits and suggests a classification of the Vaiont landslide as belonging to the unstable velocity-weakening sliding regime and La Clapière landslide as belonging to the stable velocity-strengthening regime.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Slider Block Model With State- and Velocity-Dependent Friction
- 3. Vaiont Landslide
- 4. La Clapière Landslide: Aborted 1982–1987 Acceleration
- 5. Discussion and Conclusion
- Appendix A:: Derivation of the Full Solution of the Frictional Problem
- Acknowledgments
- References
- Supporting Information

[2] Landslides constitute a major geologic hazard in most parts of the world. The force of rocks, soil, or other debris moving down a slope can cause devastation. In the United States, landslides occur in most states, causing $1–2 billion in damages and more than 25 fatalities on average each year. Costs and casualty rates are similar in the European Union and often have even more catastrophic impacts in developing countries. Landslides occur in a wide variety of geomechanical contexts and geological settings and as a response to various loading and triggering processes. They are often associated with other major natural disasters such as earthquakes, floods, and volcanic eruptions.

[3] Landslides can occur without discernible warning. There are, however, well-documented cases of precursory signals, showing accelerating slip over timescales of weeks to decades (see *Voight* [1978] for a review). While only a few such landslides have been monitored in the past, modern monitoring techniques are allowing a wealth of new quantitative observations based on GPS and synthetic aperture radar technology to map the surface velocity field [*Mantovani et al.*, 1996; *Fruneau et al.*, 1996; *Parise*, 2001; *Malet et al.*, 2002] and seismic monitoring of slide quake activity [*Gomberg et al.*, 1995; *Xu et al.*, 1996; *Rousseau*, 1999; *Caplan-Auerbach et al.*, 2001]. Derived from civil engineering methods, the standard approach to mapping a landslide hazard is to identify the conditions under which a slope becomes unstable [e.g., *Hoek and Bray*, 1997]. In this approach, geomechanical data and properties are inserted in finite or discrete element numerical codes to predict the distance to a failure threshold. The results of such analyses are expressed using a safety factor *F*, defined as the ratio of the maximum retaining force to the driving forces. According to this approach a slope becomes unstable when *F* < 1. By its nature, standard stability analysis does not account for acceleration in slope movement [e.g., *Hoek and Brown*, 1980] but gives an all-or-nothing value. In this view, any specific landslide is essentially unpredictable, and the focus is on the recognition of landslide-prone areas.

[4] To account for a progressive slope failure (i.e., a time dependence in stability analysis), previous researchers have taken a quasi-static approach in which some parameters are taken to vary slowly to account for progressive changes of external conditions and/or external loading. For instance, accelerated motions have been linked to pore pressure changes [e.g., *Vangenuchten and Derijke*, 1989; *Van Asch et al.*, 1999]. According to this approach an instability occurs when the gravitational pull on a slope becomes larger than the resistance of a particular subsurface level. This resistance is controlled by the friction coefficient of the interacting surfaces. Since pore pressure acts at the level of submicroscopic to macroscopic discontinuities, which themselves control the global friction coefficient, circulating water can hasten chemical alteration of the interface roughness, and pore pressure itself can force adjacent surfaces apart [*Vangenuchten and Derijke*, 1989]. Both effects reduce the friction coefficient, leading, when constant loading is applied, to accelerating movement. However, this approach does not forecast slope movements. Other studies proposed that (1) rates of slope movements are controlled by microscopic slow cracking and (2) when a major failure plane is developed, the abrupt decrease in shear resistance may provide a sufficiently large force imbalance to trigger a catastrophic slope rupture [*Kilburn and Petley*, 2003]. Such a mechanism, with a proper law of input of new cracks, may reproduce the acceleration preceding the collapse that occurred at Vaiont, Monte Toc, Italy [*Kilburn and Petley*, 2003].

[5] An alternative modeling strategy consists of viewing the accelerating displacement of the slope prior to the collapse as the final stage of the tertiary creep [*Saito and Uezawa*, 1961; *Saito*, 1965, 1969; *Kennedy and Niermeyer*, 1971; *Kilburn and Petley*, 2003]. Controlled experiments on landslides driven by a monotonic load increase at laboratory scale have been quantified by a scaling law relating the surface acceleration *d*/*dt* to the surface velocity according to

where *A* and α are empirical constants [*Fukuzono*, 1985]. For α > 1 this relationship predicts a divergence of the sliding velocity in finite time at some critical time *t*_{c}. The divergence is, of course, not to be taken literally: It signals a bifurcation from accelerated creep to complete slope instability for which inertia is no longer negligible. Several cases have been described with this relationship, usually for α = 2, by plotting the time *t*_{c} − *t* to failure as a function of the inverse of the creep velocity (for a review, see *Bhandari* [1988]). Indeed, integrating equation (1) gives

These fits suggest that it might be possible to forecast impending landslides by recording accelerated precursory slope displacements. Indeed, for the Monte Toc, Vaiont landslide revisited here, *Voight* [1988b] mentioned that a prediction of the failure date could have been made more than 10 days before the actual failure by using a linear relation linking the inverse velocity and the time to failure, as found from equation (2) for α = 2. *Voight* [1988b, 1989] proposed that the relation (1), which generalizes damage mechanics laws [*Rabotnov*, 1969; *Gluzman and Sornette*, 2001], can be used with other variables (including strain and/or seismic energy release) for a large variety of materials and loading conditions. Expression (1) seems to apply as well to diverse types of landslides occurring in rock and soil, including first-time and reactivated slides [*Voight*, 1988b]. Recently, such time-to-failure laws have been interpreted as resulting from cooperative critical phenomena and have been applied to the prediction of failure of heterogeneous composite materials [*Anifrani et al.*, 1995] and to precursory increase of seismic activity prior to main shocks [*Sornette and Sammis*, 1995; *Jaume and Sykes*, 1999; *Sammis and Sornette*, 2002].

[6] In this work, we develop a simple model of sliding instability based on the rate- and state-dependent friction law, which can rationalize the empirical time-to-failure laws proposed for landslides by *Voight* [1988b, 1989]. This rate and state friction law has been shown to lead to an asymptotic time-to-failure power law with α = 2 in the late stage of frictional sliding motion between two solid surfaces preceding the elastodynamic rupture instability [*Dieterich*, 1992]. In addition, this model also describes the stable sliding regime, the situation where the time-to-failure behavior is absent. The rate and state fiction law has been established by numerous laboratory experiments (see, for instance, *Scholz* [1990, 1998], *Marone* [1998], and *Gomberg et al.* [2000] for reviews). The sliding velocities used in the laboratory to establish the rate and state friction laws are of the same order, 10^{−4}–10^{2} μm/s, as those observed for landslides before catastrophic collapse. State- and velocity-dependent friction laws have been developed and used extensively to model the preparatory and elastodynamical phases of earthquakes. In addition, analogies between landslide faults and tectonic faults have been noted [*Gomberg et al.*, 1995]. *Chau* [1995, 1999] first developed this analogy and used the rate and state friction law to model landslides and their precursory behavior. Chau transformed the problem into a formal nonlinear stability analysis of the type found in mathematical theory of dynamical systems, but no comparison with empirical data was presented. In addition and in contrast with the present work, Chau's analysis does not mention the existence of the finite time singular precursory behavior, which signals the time of the dynamical instability of the landslide.

[7] We should stress that it is extremely difficult to obtain all relevant geophysical parameters that may be germane to a given landslide instability. Furthermore, it is also a delicate exercise to scale up laboratory results to the scale of mountain slopes. Having said that, probably the simplest model of landslides considers the moving part of the landslide as a block sliding over a surface. We test how the friction law of a rigid block driven by a constant gravity force can be useful for understanding the apparent transition between slow stable sliding and fast unstable sliding leading to slope collapse. Within such a conceptual framework the complexity of the landsliding behavior emerges from (1) the dynamics of the block behavior, (2) the dynamics of interactions between the block and the substratum, and (3) the history of the external loading (e.g., rain, earthquake).

[8] Previous efforts at modeling landslides in terms of a rigid slider block have taken either a constant friction coefficient or a slip- or velocity-dependent friction coefficient between the rigid block and the surface. A constant solid friction coefficient (Mohr-Coulomb law) is often taken to simulate bed over bedrock sliding. *Heim* [1932] proposed this model to forecast extreme runout length of rock avalanches. In contrast, a slip-dependent friction coefficient model is taken to simulate the yield-plastic behavior of a brittle material beyond the maximum of its strain-stress characteristics. For rock avalanches, *Eisbacher* [1979] suggested that the evolution from a static to a dynamic friction coefficient is induced by the emergence of a basal gouge. Studies using a velocity-dependent friction coefficient have mostly focused on the establishment of empirical relationships between shear stress τ and block velocity *v*, such as *v* ∼ exp (*a*τ) [*Davis et al.*, 1990] or *v* ∼ τ^{1/2} [*Korner*, 1976] with, however, no definite understanding of the possible mechanism [see, e.g., *Durville*, 1992].

[9] In this work, we focus on two case studies, La Clapière sliding system in the French Alps and the Vaiont landslide in the Italian Alps. The latter landslide led to a catastrophic collapse after 70 days of recorded velocity increase. In the former case, decades of accelerating motion abated and gave way to a slowdown of the system. In section 2 we derive the different sliding regimes of this model, which depend on the ratio *B*/*A* of two parameters of the rate and state friction law and on the initial conditions of the reduced state variable. Sections 3 and 4 analyze the Vaiont and La Clapière landslides, respectively. We calibrate the slider block model to the two landslides and invert the key parameters. Our results suggest that the Vaiont landslide belongs to the velocity-weakening unstable regime, while La Clapière landslide is found to be in the stable strengthening sliding regime. Conclusions are presented in section 5. *Sornette et al.* [2004] investigate the potential of our present results for landslide prediction, using different methods to investigate the predictability of the failure times and prediction horizons.

### 5. Discussion and Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Slider Block Model With State- and Velocity-Dependent Friction
- 3. Vaiont Landslide
- 4. La Clapière Landslide: Aborted 1982–1987 Acceleration
- 5. Discussion and Conclusion
- Appendix A:: Derivation of the Full Solution of the Frictional Problem
- Acknowledgments
- References
- Supporting Information

[40] We have presented a quantitative analysis of the displacement history for two landslides, Vaiont and La Clapière, using a slider block friction model. An innovative concept proposed here was to apply to landslides the state- and velocity-dependent friction law established in the laboratory and used to model earthquake friction. Our inversion of this simple slider block friction model shows that the observed movements can be well reproduced with this simple model and suggests that the Vaiont landslide belongs to the velocity-weakening unstable regime, while La Clapière landslide should be in the strengthening stable regime. Our friction model assumes that the material properties embodied in the key parameters *B*/*A* and/or the initial value of the state variable of the friction law control the sliding regime. Even if the displacement is not homogeneous for the two landslides, the rigid block model provides a good fit to the observations and a first step toward a better understanding of the different sliding regimes and the potential for their prediction.

[41] For the cases studied here, we show that a power law increase with time of the slip velocity can be reproduced by a rigid slider block model. This first-order model rationalizes the previous empirical law suggested by *Voight* [1988b]. Following *Petley et al.* [2002], we suggest that the landslide power law acceleration emerges in the presence of a rigid block; that is, this corresponds to the slide of a relatively stiff material. *Petley et al.* [2002] report that for some other types of landslides in ductile material the slips do not follow a linear dependence with time of the inverse landslide velocities. They suggest that the latter cases are reminiscent of the signature of landsliding associated with a ductile failure in which crack growth does not occur. In contrast, they propose that the linear dependence of the inverse velocity of the landslide as a function of time is reminiscent of crack propagation, i.e., brittle deformation on the basal shear plane. Our contribution suggests that friction is another possible process that can reproduce the same accelerating pattern as crack growth on a basal shear plane [*Petley et al.*, 2002; *Kilburn and Petley*, 2003]. The friction model used in our study requires the existence of an interface. Whether this friction law should change for ductile material is not clear. The lack of direct observations of the shearing zone and its evolution through time makes the task of choosing between the two classes of models, crack growth versus state- and velocity-dependent friction, difficult.

[42] For La Clapière landslide the inversion of the displacement data for the accelerating phase 1982–1987 up to the maximum velocity gives *B*/*A* < 1, corresponding to the stable regime. The deceleration observed after 1988 implies that not only is La Clapière landslide in the stable regime but, in addition, some parameters of the friction law have changed, resulting in a change of sliding regime from a stable regime to one characterized by a smaller velocity, as if some stabilizing process or reduction in stress was occurring. The major innovation of the frictional slider block model, which is explored further by *Sornette et al.* [2004], is to embody the two regimes (stable versus unstable) in the same physically based framework and to offer a way of distinguishing empirically between the two regimes, as shown by our analysis of the two cases provided by the Vaiont and La Clapière landslides.