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Co-seismic landslides represent a major hazard for populations
Landslides triggered by earthquakes are a major hazard in seismically active regions. They can destroy villages and cause hundred of injuries and fatalities (Marano et al., 2010, Petley, 2012, Catlos et al., 2016). For example, between 20.000 and 100.000 fatalities due to the 2008 Wenchuan earthquake in China have been attributed directly to landsliding (Huang and Fan, 2013). Coseismic landslides can also block roads and railways, which is especially problematic in isolated, narrow and inhabitated valleys with limited accessibility. They also deliver large volumes of sediments to rivers, modifying their dynamics, causing hydro-sedimentary hazards such as river aggradation or landslide dams (Collins and Jibson (2015), figure 1.2 a). Landslides can be the primary source of co-seismic damage to infrastructure (figure 1.2 b) and can prevent the functioning of transportation. Therefore, they induce economic losses that can be even more important than those caused by direct ground shaking (Bird and Bommer, 2004).
Large landslides in particular can be very destructive, such as shown by the Langtang landslide triggered by the Gorkha earthquake, that destroyed an entire village and killed more than 200 people. In natural hazard management, the size of such large and therefore destructive events is critical for the overall hazard anticipation (Strauss et al., 1989, Sornette, 2006). Unfortunately, because large landslides are rare compared to smaller ones, their occurrence is also the most difficult to predict (Geist and Parsons, 2014).
Earthquakes and coseismic erosion shape the landscape
The last decades have shown the fundamental importance of seismic rupture in mountain range building. Rocks move and uplift mainly during shallow earthquakes (Avouac, 2007), and the idea that seismic and interseismic deformation do not balance, but accumulate to create topography and form permanent structures (King et al., 1988), has been restored recently by observations (Le Béon et al., 2014) and numerical modelling (Simpson, 2015).
Moreover, earthquakes contribute significantly to the erosion of mountain belts. The total volume of coseismic landslides scales, at first order, with the earthquake magnitude (Keefer, 1994): during larger earthquakes, a larger volume of rock is converted to sediments that can be transported away the mountain belt by rivers. Landslides therefore need to be considered in the mass balance of large earthquakes (i.e., the mass that is uplifted by seismic rupture minus the mass that is removed by erosion). More and more accurate estimates of the erosion induced by co-seismic landsliding have been possible during the last decades with the development of high-resolution satellite imagery accessible for the scientific community. This has led to many complete or nearly complete co-seismic landslide inventories (Tanyaş et al., 2017). Recent studies have shown that the mass balance of large earthquakes can be variable. For example, the Chi-Chi earthquake generated more uplift compared to landsliding volume (Hovius et al., 2011); however the topographic budget of the Wenchuan earthquake was nearly neutral (Parker et al., 2011, Li et al., 2014). Recent numerical modelling suggest that intermediate size earthquakes (Mw 6-7.3) may cause more erosion than uplift (Marc et al., 2016a).
Moreover, the export time of landslide sediment is a debated question. If some studies suggest that rivers need more than several centuries to adjust to large landsliding events (Stolle et al., 2018), other studies propose an export time of several decades (Yanites et al., 2010, Howarth et al., 2012, Uchida et al., 2014, Wang et al., 2015, Xie et al., 2018). Other studies (Hovius et al., 2011, Croissant et al., 2017) propose an even shorter time scale (1-10 years) for the post-seismic export of landsliding sediments. These recent observations and numerical modelling are challenging the well accepted idea that large earthquakes always build topography.
Surface processes can trigger seismicity at human-life time scales
Surface processes, including sedimentation and erosion, involve the transfer of material (sediments, water) at the Earth’s surface. Sediment and water transfers induce stress changes at depth, with various periods (one day for tides, one year for the monsoon, to decades for co-seismic erosion). The amplitude of stress change decreases with depth, but appears in some cases to be sufficient enough to trigger shallow seismicity (e.g., Bollinger et al., 2007, Bettinelli et al., 2008). For example, figure 1.4 shows the annual variations of seismicity in Japan (Heki, 2003). Insets a and b show that large earthquakes (magnitude > 7) are more frequent (∼4 times) during summer than winter, with an annual periodicity. This is observed only in the area covered by snow during winter (fig. 1.4 c). The author of this study have explained this observation by the cycle of surface loading and unloading due to snow melt in spring and summer and snow cover in winter. Calculating the stress changes induced by this cycle on the crustal faults located below gives values of a few kPa. To give an order of idea, this is a hundred times smaller than the pressure of a bike tire.
Even though, the mechanisms of earthquake triggering by surface processes are not clearly understood yet, it appears that the rupture of shallow crustal faults can be activated by very small stress changes, as we also learned during the few last decades the triggered seismicity in the US (e.g., Chen and Talwani, 2001, Ellsworth, 2013). Therefore, at the time-scale of a seismic cycle, erosion is likely to influence the seismicity of active faults (Steer et al., 2014), and some feedbacks can be expected, for example between co-seismic erosion induced by landslides and shallow earthquakes. Nevertheless, those interactions have been poorly studied, and the conditions under which large erosional events can actually significantly interplay with seismicity remain unclear.
Rock strength and brittle failure
How much stress a material can hold and the critical stress that will induce failure is a very old problem. It is important for engineering, mining… The issue of interest to humans (for example landslide occurrence, or the failure of a bridge) implies macroscopic failure. However, macroscopic failure implies brittle failures at a wide range of scales, and involves two main modes of brittle deformation: crack propagation and sliding along pre-existing fractures. I hereinafter explain why rock strength and brittle failure is a scale-dependent problem; then I introduce rock friction, which is the macroscopic property controlling rock sliding; and finally I give the commonly used description of the macroscopic failure of rocks.
A scale-dependent problem
The most common method of studying rock strength is the compression of a cylinder of rock with a press (uniaxial compression, or tri-axial compression if the rock is confined). Strength measurements consist of measuring the non-dimensional strain ε under stress σ. The obtained stress-strain curve usually shows that the material first deforms elastically (equation 2.1) until a threshold is reached, inducing macroscopic failure of the material : σ=Eε (2.1).
where E is a constant called Young’s modulus, or elastic modulus.
In those experiments, the maximum stress the rock can hold is referred to as compressive rock strength.
Our modern understanding of rock strength arises from the discrepancy between early empirical estima-tions of rock strength, and analytical solution brought by the theory of matter. Orowan (1949) calculated the theoretical stress σt necessary to separate two atoms with an inter-atomic distance a, across a lattice plane : Eγ 1/2 σt = (2.2).
where γ is the specific surface energy (the energy per unit area necessary to break the bonds) and E is Young’s modulus. From this result, the typical stress necessary to break the atomic bonds in silicate rocks would be 1-10 GPa, which is several orders of magnitude greater than the actual rock strength, showing that the atomic scale is not appropriate to infer macroscopic rock strength.
So, what is a good scale to infer rock strength ? Experimental studies made on various rock types (e.g., Bieniawski, 1968, Jahns et al., 1966, Pratt et al., 1972, Hoek and Brown, 1997) have shown that rock sample strength decreases with increasing size. A theoretical explanation for this scale dependency was first proposed by Griffith and Eng (1921), Griffith (1924). They calculated that rocks yield with increasing scale because they contain defects (cracks) which concentrate stress at their tips, allowing failure at stresses much lower than the theoretical stress. However, the problem of macro failure can not be addressed simply by the modelling of an expanding crack, because it involves several processes at all scales (Scholz, 2002). For example, a fault could be at first order modelled by a linearly expanding crack. However, if we zoom on a portion of this fault, we will notice a lot of complex features resulting from meter to decameter-scale fractures (figure 2.2b). Those cracks create lower strength surfaces along which rock mass can slide under tectonic stresses. At even smaller scale, rocks also contain micro-cracks (Kranz (1983), figure 2.2c) whose propagation is involved in the micromechanics of what can be seen as frictional sliding at larger scales (Scholz, 2002). Therefore, modelling macro failure events requires to use empirical or semi-empirical laws determined from laboratory experiments (Hoek and Brown, 1980b), that integrate the variety of processes occurring at smaller scales (Carpinteri, 1994, Senent et al., 2013). The two next sections focus on those macroscopic properties and how they contribute to rock failure at the macroscopic scale.
Rock friction
Friction is the resistance to motion that occurs when a body slides tangentially to a surface on which it contacts another body. When a fracture already exists, sliding is controlled by the frictional resistance, which is a property of the interface more than a property of the material. Frictional strength acts in everyday life, and reducing friction is an important problem in engineering for machines with moving parts. Therefore, its first-order properties have been known since ancient times. The two main laws of friction were discovered by Leonardo da Vinci, and formulated 200 years later by Amonton (Amonton, 1699):
• Amonton’s first law : the frictional force is independent of the size of the surfaces in contact.
• Amonton’s second law : the frictional force Ff is proportional to the normal force Fn with a friction coefficient µ0 : Ff = µ0 Fn (2.3).
Byerlee (1978) compiled the shear stress necessary to slide a rock surface toward another from a variety of experiments, using either carbonate or silicate rocks (figure 2.3). He found that µ0 (∼ 0.6) was independent of lithology, and to first order, independent of velocity. Actually, at second order, friction slightly varies according to sliding velocity (Byerlee, 1970) and time (Rabinowicz, 1951). This is why, during frictional sliding, dynamic instability can occur (i.e, earthquakes). This arises from second-order frictional properties (section 2.1.2.1); the friction described by Byerlee (1978) is the static friction.
Macroscopic failure of rock
The principal tools that are commonly used in engineering as well as Earth sciences to describe rock macro-failure are the Coulomb criterion and Mohr circle analysis. The underlying idea is to define an empirical failure criterion for a potential rupture plane, based on macroscopic strength of the material. This failure plane can be a tectonic fault, or a potential landsliding plane characterized by a fracture, for example.
First, one can derive the relationship between normal and shear stresses acting on a plane submitted to bidirectional principal stresses σ1 and σ3 , and making an angle α with the direction normal to σ1 (figure 2.4 a). The shear stress τ and normal stress σn acting on this plane are (e.g., Jaeger et al., 2009):
σn = σn,1 + σn,3 = σ1 cos2 (α) + σ3 sin2 (α) − p (2.4).
τ = τ1 + τ3 = −σ1 sin(α)cos(α) + σ3 sin(α)cos(α) (2.5).
Linking model and landslide data
We consider a simplified and straight hillslope (figure 4.9 a) of length L and slope angle S. We aim at assessing the probability of unstable depth in this wedge, given simple geometric assumptions. At a coordinate x located in this wedge, we assume that any plane, defined by its local depth z and dipping angle α, is a potential rupture plane if 1) α is greater than the internal frictional angle Φ and 2) it intersects the surface topography upstream of the base of the hillslope. All planes meeting the two conditions in this wedge have the same rupture probability, i.e., we do not consider that rupture probability increases with increasing shear stress/shear strength ratio. Compared to Jeandet et al. (2019) who used a Mohr-Coulomb criterion, we neglect cohesion because that was found to mostly influence the stability of shallow and small landslides.
Therefore, the rupture factor at a depth z and rupture angle α is purely frictional: FR(z, α) = α Φ.
Presentation of the two numerical models
The first model I used, QDYN v1.1, (Luo et al., 2017) has been developed by Yingdi Luo et al. at the California Institute of Technology and released in 2017. The second model I used, FastCycle, is the outcome of the PhD work of Pierre Romanet (Romanet et al., 2018) at the Institut de Physique du Globe (Paris). Both models are based on the same physical approach. Their major difference lie in the acceleration algorithms implemented in FastCycle, which allow modelling of to model a population of faults with potentially complex geometries, instead of one planar fault.
Table of contents :
1 Preamble
1.1 From geological time scales to the seismic cycle time scale
1.2 Why do we care about short-term interactions between tectonics and erosion ?
1.2.1 Co-seismic landslides represent a major hazard for populations
1.2.2 Earthquakes and coseismic erosion shape the landscape
1.2.3 Surface processes can trigger seismicity at human-life time scales
2 Introduction
2.1 Earthquake and landslide mechanics
2.1.1 Rock strength and brittle failure
2.1.2 Earthquake mechanics
2.1.3 Landslide mechanics
2.2 Interactions between surface processes and earthquakes at short time scale (< 1000 years) . .
2.2.1 Landscape response to earthquakes
2.2.2 Seismic cycle response to surface processes
2.3 Earthquake and landslide sizes
2.3.1 Observations
2.3.2 Physical meaning of power-law distribution and b-value variations
2.3.3 Upper and lower limits to rupture size
3 Modelling landslide size distribution
3.1 Overview
3.2 Coulomb mechanics and relief constraints explains landslide size distribution
3.2.1 Introduction
3.2.2 Methods
3.2.3 Results
3.2.4 Discussion and concluding remarks
3.3 Supplementary material
3.4 General discussion
3.4.1 What is landscape strength ?
3.4.2 Consequences for landsliding volumes
4 How hillslope shape controls landslide size
4.1 Overview
4.2 Impact of finite hillslope geometry on large landslide probability
4.2.1 Introduction
4.2.2 Methods
4.2.3 Results
4.2.4 Discussion and concluding remarks
4.3 Supplementary material
4.4 General discussion
4.4.1 Linking model and landslide data
4.4.2 Implications for large landslide hazard in seismically active regions
5 Modelling the response of active faults to large erosional events
5.1 Overview
5.2 General presentation of the methods
5.2.1 Presentation of the two numerical models
5.2.2 How to model earthquakes of different sizes ?
5.3 Modelling fault response to large erosional events
5.3.1 Introduction
5.3.2 Methods
5.3.3 Discussion and concluding remarks
5.4 Supplementary material
5.5 Comparison of homogeneous and heterogeneous fault response
5.6 Perspective – modelling a fault network under normal stress change
6 Discussion
6.1 Main results
6.2 Rupture processes during the seismic cycle
6.2.1 Size of rupture events
6.2.2 Implications for interaction between landslides and earthquakes at short time-scales .
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