Hydromechanical triggering of landslides: From progressive local failures to mass release

Authors


Abstract

[1] Water infiltrating during intense rainfall on steep slopes gradually weakens the wet soil mass, inducing localized failures that may initiate a cascade of load redistributions and successive failures propagating across a hillslope. The challenge of linking the progressive nature of local events culminating in an abrupt landslide is addressed by a new hydromechanical triggering model that links key hydrologic processes with threshold-based mechanical interactions. A hillslope is represented as assembly of soil columns interconnected by frictional and tensile mechanical bonds represented as virtual bundles of fibers. Increasing water load exerted on mechanical bonds causes gradual failure of fibers until restraining forces are exceeded. Following failure at the soil-bedrock interface, the load on a column is redistributed to its neighbors via intact mechanical (primarily tensile) bonds which, in turn, may also fail and transmit the load downslope as compressive stresses. When soil internal compressive strength is exceeded, a load-bearing column may liquefy and initiate a landslide release that could propagate downslope or retrogressively upslope. The model reproduces observed power law frequency magnitude relationships of landslides with exponents ranging between −1.0 and −2.2 in agreement with landslide inventory data. We applied a criticality measure defined by Ramos (2011) to evaluate the specific influences of slope angle, soil texture, and root reinforcement on attainment of hillslope criticality in which a small local failure may trigger release of a large soil mass. The model provides new insights on the conditions giving rise to an abrupt transition from a seemingly stable hillslope to a catastrophic landslide.

1. Introduction

[2] Landslides are natural geomorphic processes shaping mountainous regions and redistributing mass on steep slopes. Expansion of human activity and infrastructure to steep mountainous regions increases exposure to this ubiquitous natural hazard. Sidle and Ochiai [2006] estimated the direct costs associated with rebuilding or replacing infrastructure at several billion dollars per year, not considering indirect costs related to construction of engineered barriers and temporary loss of site functionality. Among the different types of mass movements (e.g., falls, topples, slides, and flows) and cover materials involved (rock, debris, and soil) this study focuses on rapid shallow translational landslides in soils with relatively low cohesive strength [Varnes, 1978], most frequently induced by intense and/or prolonged rainfall [Sidle and Ochiai, 2006; Schuster and Wieczorek, 2002].

1.1. Hydrologic Triggering of Shallow Landslides: Mechanisms and Modeling

[3] Physically based models attempt to link dynamics of hydrologic status and pathways with the mechanical state of a hillslope and the onset of failure. Soil mass on steep slopes is stabilized by frictional and capillary forces and by reinforcing elements and cohesion. Water infiltration during rainfall may increase the total weight of the soil mass and associated downslope forces [Chigira and Yokoyama, 2005] as well as reduce internal soil strength [Mitchell and Soga, 2005; Lu et al., 2010]. Moreover, based on local hydrologic conditions (e.g., infiltration, initial water content, and hydraulic conductivity), a perched water table or increased groundwater table may occur resulting in positive water pressure and reduced effective stress [Iverson, 2000]. In extreme cases, rapid increase in soil water content may result in deformation of soil volumes until they become liquefied and are released in an abrupt motion [Iverson et al., 1997]. During deformation, soil porosity and shear strength may attain values characteristic of the so-called critical state [Gabet and Mudd, 2006] attainable from compaction of loose soil or dilation of initially densely packed soil. In either case, when shearing rate exceeds soil drainage capacity, water pressure may peak and cause liquefaction of the deforming soil mass [Iverson et al., 2000].

[4] For most applications, details of soil failure and liquefaction are seldom considered and mass release is attributed to a simple factor of safety (FOS) calculation, i.e., the ratio between driving and resistive forces often computed based on an infinite slope approach [O'Loughlin and Pearce, 1976; Wu et al., 1979; Casadei et al., 2003]. Some models express FOS as a function of seepage flow (transmissivity) and rainfall rate [Montgomery and Dietrich, 1994; Fernandes et al., 2004]. Others have considered effects of pore pressure on landslide initiation based on numerical solution of the Richards equation [Iverson, 2000; Tsai and Yang, 2006], including attempts to resolve unsaturated flow processes over an entire catchment [Simoni et al., 2008]. To consider effects of soil volumes on stability in up- and downslope directions, FOS can be formulated for slopes consisting of soil slices with force equilibrium for each slide and between neighbored slices. While Bishop [1955] and Janbu [1973] considered normal forces between adjacent slices but ignored interslice-shear force, both force components were accounted for by Morgenstern and Price [1965]. To describe impact of local deformations at smaller scale that are not considered in FOS analysis, finite element methods have been studied [Griffith and Lane, 1999; Zheng et al., 2005; Eichenberger et al., 2010]. Many of the numerical methods rely on continuum representation of key mechanical processes where landslides slowly and continuously deform. Such a continuous deformation may be in agreement with processes in deep-seated slow landslides but often fail to capture the abruptness characterizing hazardous shallow landslides. As abruptness we define in this study the release of large soil masses at once with no apparent indicators for the imminence of mass release.

1.2. Landslide Scaling Laws and Criticality

[5] Occurrence of shallow landslides is not always preceded by detectable surface deformation, but may be associated with internal changes such as build-up of positive pressure related to subsurface flows [Huang et al., 2008] that may relax close to failure [Germer and Braun, 2011], or marked by increased acoustic emissions (release of elastic waves) [Cadman and Goodman, 1967] or gradual internal shearing of soil material [Iverson et al., 2000]. Petley et al. [2005] attributed the resulting abrupt failure (characterized by very short time from first indicators to mass release) to a gradual process of progressive failure occurring at smaller scales where small failures events (e.g., cracks) gradually form and abruptly coalesce to a continuous failure plane. While Petley et al. [2005] discussed the concept of progressive failure in the context of microcracks in deforming cohesive soil, progressive failure may occur also in noncohesive soils by reduced internal strength and liquefaction We associate local failure of mechanical elements as indicators of “progressive failure” that may ultimately propagate across the entire system marking attainment of criticality [Jensen, 1998; Ramos, 2010]. Note that this definition of criticality or critical state where a local perturbation can lead to release of large masses is different than the term used in critical state mechanics marking transition of a sheared soil to “frictional fluid” [Schofield and Wroth, 1968].

[6] Critical systems exhibit no characteristic length scale and often yield a power law of failure size distribution [Sornette, 2004]. Criticality associated with landslides is indeed manifested at larger scales in the form of power law scaling exhibited in the number, areas, or volumes of landslides [Stark and Hovius, 2001]. For landslides larger than volume math formula or area math formula the frequency of a landslide as a function of landslide volume math formula or area math formula can be expressed as probability density math formula or math formula [Stark and Guzzetti, 2009]:

display math
display math

with power law exponents math formula and math formula. The entire range of landslide size (including small landslides that may be less frequent) is represented by a unimodal distribution [Malamud et al., 2004; Guzzetti et al., 2002; Stark and Hovius, 2001]. For the power law tail of such unimodal probability density with the standard constraints that (1) the area under the PDF is 1 and (2) maximum landslide size is not limited (infinite), the probability density can be expressed as Pareto distribution [Stark and Hovius, 2001] with exponent math formula and a proportionality factor math formula (corresponding expressions for landslide area). For a Pareto distribution the exponent math formula corresponds to the power law exponent of the complementary cumulative probability distribution function, integrating the probability density math formula from a landslide size math formula to infinity. Note that a finite value of such integration is only fulfilled for math formula and math formula, respectively. Smaller exponent values require definition of an upper cutoff (defining largest landslide size) for validity of power law.

[7] For measured or modeled landslide distribution with total number math formula of landslides, we estimated the probability density as [Malamud et al., 2004]

display math

with the number of landslides math formula within a range of volumes math formula that is increasing with increasing landslide volume. The frequent use of aerial images to estimate landslide properties favors representation of landslide area of the scar (volume is more difficult to obtain) [Guzzetti et al., 2002; Piegari et al., 2009; Van Den Eeckhaut et al., 2007; Turcotte, 1999] than analysis of landslide volumes [Dai and Lee, 2001; Imaizumi and Sidle, 2007]. The exponents of power law characterizing relationship between landslide area and frequency range between 1.4 and 3.5 for math formula [Van Den Eeckhaut et al., 2007], and between 1.0 and 1.9 for landslide volumes math formula [Brunetti et al., 2009]. The average values deduced from the studies above are math formula = 2.4 and math formula = 1.4, indicating that exponent values are smaller for volume statistics. The smaller absolute exponent value can be justified by a power law relationship between landslide volume and area with exponent values determined as ranging from 1.1 to 1.3 [Larsen et al., 2010] or as 1.5 [Guzzetti et al., 2009] using inventory data, and 1.3 to 1.4 based on limit equilibrium principles [Klar et al., 2011]. Note that these reported values may involve different types of landslides and triggering mechanisms not limited to shallow rapid translational slides that are in the focus of this study. In addition, in most cases the landslides included in the statistics are not triggered by the same rainfall event. To isolate potential effects of rainfall characteristics on the power law, we present in Figure 1 data from landslide inventories obtained from the Prealps region of Switzerland [Rickli and Graf, 2009; Rickli, 2001; Rickli and Bucher, 2003; Rickli et al., 2008] where the triggered landslides were attributed to a single rainfall event.

Figure 1.

Relationships between landslide frequency and volume deduced from event-based landslide inventories from Switzerland (see references in text). The results are characterized by power law distributions with exponent values |ζV| decreasing from (a) to (d). The power law tail of probability density is shown for landslide volumes larger than a minimum volume between 30 and 50 m3. The landslides were triggered by heavy rainfall events (year of rainfall event, name, and size of the catchment are specified in each figure).

1.3. Model Concepts of Abrupt Failure and Criticality

[8] Abruptness and criticality of landslides considered in this study arise from mechanical interactions among many soil columns in which mechanical thresholds resembling concepts of self-organized criticality (SOC) were embedded within a hydromechanical framework. We emphasize that complete representation of landslides as SOC systems would require geological time scales for the full spectrum of SOC events to be manifested; hence we limit application of SOC principles to two primary aspects, namely (1) progressive failure preceding a landslide and (2) emergence of power law size-frequency scaling relationships resulting from interactions of many elements with threshold mechanics. A key ingredient in delineating progressive failures is the application of formalism based on the fiber bundle model (FBM), representing all soil strength-imparting components (capillarity, cements, friction, and roots). To clarify the links between the hydromechanical triggering model and the concepts of SOC and FBM, we introduce next the key elements of these two concepts.

1.3.1. Self-Organized Criticality (SOC)

[9] A system driven to a critical state by internal self-organization of many interacting elements is denoted as a SOC system [Bak, 1996; Jensen, 1998; Turcotte, 1999; Hergarten, 2002; Christensen and Moloney, 2005]. The spontaneous organization of system components may result in a so-called marginally stable state corresponding to a dynamic critical state. Sornette [2004] and Koslowski et al. [2004] characterize SOC systems by the following properties: (1) power law frequency/magnitude relations result without any apparent external tuning, (2) external driving rates are significantly slower than internal relaxation processes, (3) the presence of a marginally stable state, and (4) a large number of interacting entities that show a highly nonlinear (threshold) response. Among the many SOC models we introduce two types that were previously used to simulate mass movements.

1.3.1.1. Sandpile Cellular Automaton

[10] To represent dynamics of avalanches of grains in growing dry sandpile, Bak et al. [1987] have introduced a cellular automaton consisting of a two-dimensional regular grid (denoted as the BTW model). The grid cells in the model may store a prescribed number of “mass units” (sand grains). At each time step, a mass unit is added to the system at a random position until at some cell the number of sand grains reaches its threshold capacity (often set equal to the number of nearest neighbors). Subsequently, that content of cell is redistributed instantaneously by sending one grain to each of the neighbors which in turn may exceed their own capacity and redistribute their mass further. The number of cells that redistribute their grains in response to addition of a single sand grain to the system is denoted as an event or an avalanche. It can be shown that avalanche size and frequency follow a power law with exponent of math formula close to −1 [Hergarten, 2002]. The notion that addition of a single grain may induce avalanches with a wide range of size distributions have been confirmed in laboratory experiments using sand or rice grains [Yoshioka, 2003; Held et al., 1990; Lörincz and Wijngaarden, 2007], however, the resulting values of power laws math formula were typically higher (between 1.1 and 2.5) than predicted by the theoretical BTW-sandpile model. This difference in power law values indicates that the frequency of large avalanches is lower in real systems and the redistribution mechanism in the sandpile model is too simplified, mainly due to frictional forces reducing stress release in real sandpiles [Hergarten, 2003].

[11] Nevertheless, the BTW model has been proposed as an analog for landslide triggering and dynamics [Turcotte, 1999; Hergarten, 2002]. The model was adapted to model gravity-driven failure in a slope by Piegari et al. [2006] considering slope dependent stress distribution when a force threshold (expressed as factor of safety) was exceeded for a random loading process. A similar model with randomly distributed stress thresholds was previously proposed by Katz [1986] for modeling brittle material failure.

1.3.1.2. Spring-Block Models

[12] In the original BTW model no mechanical properties between grid cells were defined and mass redistribution was not restricted. This limitation was rectified by the spring-block model proposed by Olami et al. [1992] (denoted as the OFC model) that considers blocks of mass interconnected by springs. Block motion is controlled by friction at the block base and mechanical interactions between neighboring blocks. When forces acting on a block overcome friction at its base, the block slips and forces are instantaneously redistributed along mechanical bonds which may trigger additional displacement (the number of displaced blocks determines avalanche size). The resulting power law exponent for the initial OFC model math formula was −1.23 for conditions where the force of a displaced block was entirely transferred to neighboring elements [Hergarten, 2003]. The OFC model was recently modified by Faillettaz et al. [2010] considering gravity-driven motion of blocks and introducing reduced friction and progressive damage of mechanical bonds between neighboring blocks.

1.3.2. Fiber Bundle Model (FBM)

[13] The third ingredient for critical failure in soil covered hillsopes is provided by the fiber bundle model (FBM) originally introduced by Peirce [1926] and Daniels [1945]. The model considers a large number of equal-length fibers whose strengths obey a prescribed distribution connected to stiff plates at both ends. Incremental loading of the bundle of fibers results in distribution of the load uniformly among all intact fibers, until the weakest fiber breaks and its load is redistributed to remaining intact fibers. The load redistribution may trigger breakage of additional fibers due to the added load increment (released from the broken fiber). The process may cascade and result in an avalanche-like failure of a certain number of fibers. Early models have used fibers representing cotton yarns, however, the formalism is broader and may be applied to various load bearing elements composed of many discrete units such as ice crystals in weak snow layers [Reiweger et al., 2009], roots in hillslopes [Schwarz et al., 2010a, Cohen et al., 2011], and soil material at various scales [Cohen et al., 2009]. In addition to obvious “fiber like” elements such as cementing agents, roots, and capillary water in soil (grains interconnected by bond of water retained by capillary forces), Cohen et al. [2009] discussed frictional and compressive force chains [Majmudar and Behringer, 2005] that may also be represented as fiber-like elements. Because fibers can break long before mass is released, FBMs are a tool to quantify so-called precursor events. Precursors are defined here as any failure events at small scale that may indicate imminent mass release.

[14] As for the sandpile cellular automaton and spring-block models, the statistics of avalanches (the number of fibers failing during a load increment) follow a power law with theoretically established exponent math formula with value of −5/2 [Hemmer and Hansen, 1992]. However, as a bundle approaches its global failure (i.e., the entire bundle breaks) the power law exponent converges to a value of −3/2 [Pradhan et al., 2006] reflecting transition in which large events become more frequent. In contrast with the BTW and OFC models presented in previous subsections, a marginally stable critical state (fulfilling conditions of SOC systems) is not possible for FBM unless fibers can “heal” after failure [Moreno et al., 1999]. The FBM formalism provides unique modeling capabilities that not only accommodate gradual loss of mechanical strength prior to abrupt failure, but also capture discrete mechanical events that may be measured independently by acoustic or electrical sensors [Amitrano et al., 2005] and thus provide a means for noninvasive monitoring of system mechanical state.

1.4. Objectives

[15] The primary objective of this study was to develop a new hydromechanical modeling framework for simulating rainfall-induced rapid and shallow landslides that explicitly considers the following processes and characteristics: (1) describe hydrologic pathways and resulting spatial and temporal water content distributions, (2) quantify evolution of soil mechanical properties affected by hydrology, (3) represent and track local failure events occurring prior to landslide release, (4) capture abruptness (large soil masses released at once) and criticality (local perturbations cascade and lead to mobilization of soil material), and (5) reproduce the resulting power law distribution of landslide sizes at a catchment scale. The proposed modeling framework (denoted as landslide hydromechanical triggering model, LHT) differs from other physically based landslide models in its capability to seamlessly link physically based hydromechanical hillslope behavior with processes leading to abruptness and criticality associated with hydrologic triggering of a landslide. These traits emerge from implementation of SOC and FBM ingredients within the hydromechanical model. In short, we will describe a hillslope as a system of interconnected soil columns taking into account that mechanical soil properties are defined by small scale phenomena like root reinforcement, frictional grain contacts, and bonds of water bound by capillary forces. Table 1 summarizes key traits of the various models maintaining a separation between the operational hillslope scale of the model with many interacting soil columns, and local failure and precursor events at smaller scale of an FBM mechanical bond.

Table 1. Model Concepts to Reproduce Abruptness and Criticality in Material Failure and Mass Releasea
Model TypeScale/SizeSystem CompositesAbruptnessCriticalityMarginal Stable State
Local Failure ConditionRelaxationRecoveryAvalanche Definition|ζ|
  • a

    While the sandpile cellular automaton [Bak et al., 1987] and the spring-block model [Olami et al., 1992] are examples for self-organized criticality (SOC) systems, the fiber bundle model (FBM) [Hemmer and Hansen, 1992] and the landslide hydromechanical triggering model LHT introduced in this study (combining elements of SOC and FBM) do not evolve towards a sustainable critical state.

Sandpile cellular automatonSandpile 10−1 – 100 mStack of sand particlesHeight of stack ≥ thresholdAll particles redistributed to neighborsHeight threshold recoveredNo. of stacks of sand with redistribution∼1Yes
Spring-block modelHillslope 101 – 103 mRigid soil blocksForce at base > frictionForce redistributed along bondsNew random strengthNo. of blocks with changed position1.23Yes
Mechanical bonds (soil strength)Load on bond > strengthLoad transfer to other bondsNo
Fiber bundle modelSoil sample 10−4 – 10−1 mRoots, water bonds, force chainLoad per fiber > fiber strengthLoad redistributed to neighborsNoNo. of breaking fibers5/2No
Hydromechanical triggering model LHTHillslope 101 – 102 mRigid soil blockLoad at base > frictionLoad redistributed along bondsNoVolume of released mass1.0–2.2No
Load > compress. strengthMass removedNo
Mechanical bonds (soil strength)Load on bond > tensile strengthLoad redistributed to other bondsNo
Roots, water bonds, force chainLoad per fiber > fiber strengthLoad redistributed to neighborsNo

[16] Moreover, by considering results from different realizations of similar hillslopes, we may draw general conclusions regarding larger catchment scale relationships. In section 2 we introduce aspects of progressive failure, followed by details of the hydromechanical hillslope model in section 3 (and the Appendix). In sections 4 and 5 simulation results of the triggering model are presented and discussed, and the main findings of the study are summarized in section 6.

2. Rainfall-Induced Progressive Failure in Triggering Model with Interacting Soil Columns

[17] The process of hydrologic “loading” of a hillslope involves rainfall infiltration and runoff and associated mechanical alteration of soil strength and soil water content distribution. Figure 2 illustrates schematically the application of SOC concepts to a landslide triggering model highlighting gradual failure of mechanical bonds (represented as FBM) and propagation of a perturbation (local failure) across the system. A hillslope is discretized into hexagonal soil columns with cross section math formula, local surface elevation math formula, and distance math formula between column centers. The soil depth math formula for each column was deduced from (spatial) random distribution defined by an average soil depth, a coefficient of variation, and a spatial correlation length. The surface elevation and soil depth define bedrock elevation math formula. Based on random distribution of soil depth, 100 realizations of a hillslope are generated to analyze statistics of size and shape of modeled landslides.

Figure 2.

Analogy between landslide triggering model and concept of self-organized criticality (SOC) illustrated with a modified version of the sandpile cellular automaton (introduced by Bak et al. [1987]) shown in the inset (numbers denote sand grains in a cell, arrows indicate particle redistributions when threshold of four particles is reached). The hillslope consists of soil columns with hexagonal cross section. Water flowing through subsurface modifies mechanical load (represented by red spheres) and strength. A local perturbation (indicated by introducing a black sphere) initiates a chain reaction, moving loads in the downslope direction. A column's failure depends on its stabilizing forces and its mechanical interactions with neighbored columns (here represented by fiber bundles between adjacent columns). By assuming that a soil column can bear four “load units,” the blue line marks the perimeter of columns in which load redistributions initiated by black sphere occurred.

2.1. Soil Mechanical Interactions

2.1.1. Basal Failure at Soil-Bedrock Interface

[18] The downslope driving force for a soil column follows closely the classical slope stability analysis [Selby, 1993; Mitchell and Soga, 2005] with the infinite slope model that can be applied for shallow translational slides [Skempton and DeLory, 1957]. To compute stability based on factor of safety [Casadei et al., 2003; Salciarini et al., 2006] we consider maximum downslope difference of the height math formula between adjacent soil columns to determine local slope angle. The total mass math formula of a soil column and its weight math formula (acting along gravity) are given as

display math

with acceleration due to gravity math formula, hexagonal cross section of a soil column math formula, soil depth math formula, volumetric water content math formula, porosity math formula, and densities of water math formula and soil minerals math formula. The weight math formula can be subdivided into downslope directed driving force math formula and counteracting normal force math formula, with slope angle math formula defined by difference of math formula between adjacent columns and grid spacing math formula. By relating the forces to the cross section at the base along the slope math formula the normal stress math formula and “driving” component math formula equal

display math
display math

[19] The main components of force acting on a soil column are shown in Figure 3. Resisting forces are summarized as shear strength math formula and take into account effective normal stress including weakening or reinforcing effects of soil water and cohesion math formula (root reinforcement is only considered in a lateral direction but not at the base). Soil internal strength is modified by water content due to increased pore pressure [Iverson, 2000] and reduced effective stress [Vanapalli et al., 1996]. For free water table of depth math formula with seepage flow parallel to slope, resisting forces are reduced by a term attributing positive pore pressure math formula:

display math

with friction angle math formula. For unsaturated conditions at the interface between soil and bedrock the water related forces math formula are reinforced according to the effective stress formulation introduced by Bishop [1960] with soil strength enhanced proportionally to capillary pressure or head math formula:

display math

with a proportionality factor math formula relating shear strength math formula and capillary pressure head math formula < 0. We ignore in this study other rheological functional relationships between bulk soil tensile strength and water content as described by Mitchell and Soga [2005] and by Ghezzehei and Or [2001].

Figure 3.

Basic units of triggering model with soil columns of hexagonal cross section, soil depth Hsd, grid spacing LG, and surface and bedrock elevations zsu and zbr. Slope angle β activates column weight W as driving force opposed by shear stress τS linking soil-bedrock friction computed as product of normal stress σN and tangent of friction angle. Rainfall and infiltration rates r and i, matrix flux Jh, bedrock loss Jbr, and fast flow along soil-bedrock interface Jsat, affect average water content θ and capillary head h < 0. When driving force overcomes frictional resisting forces, load is redistributed to tensile bonds of strength τT. When all tensile bonds are broken, compressive stresses (τC) are exerted in the downslope direction. The progressive failure of soil columns and load redistribution to adjacent cells is characteristic of self-organized critical (SOC) systems. Here we combine SOC and fiber bundle models (FBM) by expressing soil strength by capillary forces with FBM (visualized here as deformable bonds).

2.1.2. Precursor Events and the Fiber Bundle Model

[20] When math formula the column base fails and it must be stabilized by neighboring elements. But prior to complete failure at the soil-bedrock interface (i.e., math formula), certain conditions math formula may induce local failures. These are associated with driving forces that are smaller than soil shear strength yet are larger than friction math formula. The excess stress math formula is supported by cohesion math formula and water content dependent soil strength math formula that is related to capillary frictional forces but also includes grain to grain interlocking (“jamming”) and cementing agents. The soil strength associated with water menisci (and soil cohesion) is represented by a fiber bundle model (FBM) consisting of many capillary and frictional bonds. Even for small loads, the weakest bonds (fibers) may fail and their load is redistributed to neighboring bonds. Because FBMs provide a natural framework to describe progressive failure, we model mechanical properties related to math formula (and cohesion in a few cases) with this approach. A group of failing fibers at a time step could be interpreted as indicator for incipient failure (precursor event).

2.2. Load Redistribution Along Mechanical Bonds

[21] When driving forces (per area) exceed resisting forces ( math formula) within a soil column, neighboring columns attempt to stabilize the failing column by sharing the load. The FBM formulation of soil strength math formula (related to capillary forces) and cohesion math formula (e.g., by clay minerals or cementing agents) postulates that when all fibers are broken for math formula, both math formula and math formula attain a residual value math formula according to math formula with residual strength factor math formula. Hence, not only the stress difference math formula must be redistributed, but a larger value math formula must be sustained by neighboring soil columns. The lower the residual strength math formula, the higher the resulting load (stress multiplied by area of soil-bedrock interface) transferred to intact bonds. Each soil column with hexagonal cross section has six neighbors. We define a mechanical connection between two adjacent columns (consisting of many fibers) as “mechanical bond.” As “tensile bonds” we define mechanical connections to neighboring columns in upslope or horizontal directions and connections in downslope direction as “compressive bonds.” We assume that load is first redistributed along tensile bonds. The load (per interfacial area) redistributed after basal failure to lateral (tensile) bond is given by stress math formula:

display math

with cross-sectional area of a soil column with hexagonal cross-section math formula, local slope angle math formula of the failing soil column, intersecting area math formula between two adjacent columns, and number of tensile bonds (fiber bundles) math formula in the soil column that failed at the base. As for soil strength at the base of a column, the lateral capillary bonds are represented by virtual fiber bundles. When stress carried by a mechanical bond (the fiber bundle) exceeds its tensile strength, the bundle fails and load must be redistributed to other intact tensile bonds.

2.3. From Small Local Failures to Onset of a Landslide

[22] Even after all tensile bonds failed, a soil column may remain locally stable by “leaning on” or exerting a compressive load on intact downslope soil columns. However, when the compressive load exerted on a soil column exceeds its intrinsic compressive strength, the soil matrix fails (becomes “fluidized”) and the column can no longer support a load. As summarized in section 1.1, landslides are mobilized when the stress induced strain causes destabilizing positive pressures and particles are rearranged into a critical liquefied state. Because the modeling of strain and related mechanical property would make the model much more complex, we chose a simplified formal condition for soil fluidization defined by a water-dependent compressive strength threshold [Mullins and Panayiotopoulos, 1984]. A soil column fails when the average stress on compressive bonds math formula exceeds threshold defined as

display math

with number of bonds (interfaces) math formula under compression, a compressive load math formula acting on the column intersectional area math formula of a compressive bond, soil cohesion math formula, friction angle math formula, and soil strength provided by capillary forces math formula.

[23] When a soil column is fluidized, upslope columns supported by the now failed column may destabilize and fail or redistribute their weight to other downslope columns that in turn may also be fluidized causing a chain reaction. The landslide is modeled as an instantaneous event and dynamics of landslide soil volume motion are not described in this model. The model is simplified by removing fluidized soil columns without tracking their mass motion beyond instantaneous effects on subsequent load distribution.

3. Hydromechanical Modeling of Hillslope Processes

[24] An important ingredient in the modeling framework is a detailed account of the spatial and temporal distribution of the rain water impacting load and the soil mechanical properties. For parsimony and simplicity we link Brooks and Corey [1964] soil hydrologic parameterization with the Lu et al. [2010] formulation for unsaturated soil strength. Consequently, the number of free parameters is reduced considerably leading to characterization of soil specific hydromechanical properties based on a single parameter math formula deduced from soil water characteristic curve according to the Brooks and Corey model. As highlighted by Rawls et al. [1982] each soil textural class is characterized by a typical value of the parameter math formula also known as the “pore size distribution index” [Lenhard et al. 1989].

[25] The hydrological model considers infiltration capacity, surface water flow, interflow including fast water flow along soil-bedrock interface, and lateral unsaturated flow within the soil matrix. While fast water flow along the soil-bedrock interface can be interpreted as macropore or preferential flow, we do not consider other fast flow processes along large pores or soil pipes in the present form of the model. The hydrologic model is then used to deduce evolution of water content distribution on the hillslope (details of the hydrological model are presented in Appendix A).

3.1. Soil Strength As a Function of Water Content and Root Reinforcement

[26] As soil water content increases during a rainfall event, the weight of the wet soil column and hence downslope driving forces increase, while concurrently the intrinsic soil mechanical strength decreases. Dating back to the formulation of effective stress by Bishop [1960], soil strength is enhanced proportionally to capillary pressure math formula. Lu et al. [2010] used water saturation math formula (defined in Appendix A, equation (A1)) as proportionality factor linking capillary pressure with soil strength math formula. By choosing the Brooks and Corey [1964] model parameterization for the relationship between water saturation math formula and capillary head math formula, the water content dependent soil strength math formula is given as

display math

with capillary pressure head math formula, air-entry value math formula, and pore size distribution parameter math formula [Brooks and Corey, 1964]. The quantification of soil strength and its relation to water saturation is presented in Appendix B.

[27] For simplicity we follow in this study Montgomery and Dietrich [1994] in setting cohesion value math formula to zero (note that cohesion will be included in a single simulation to estimate its effect) as many shallow rapid landslides occur in soils with low cohesion [Sidle and Ochiai, 2006]. Because cohesion increases with clay content, we apply the proposed model only to soil textural classes with low clay content (soil textural classes sand, loamy sand, sandy loam, loam, silt loam, sandy clay loam). Because soil textural class and pore size distribution parameter math formula are related, the analyzed soil textural classes are restricted to values of math formula larger than 0.2 [Rawls et al., 1982].

[28] In contrast with bare soil surfaces, vegetated hillslopes are reinforced by plant roots. Root reinforcement has been studied extensively and was often introduced directly into a factor of safety calculations [Gabet and Dunne, 2002; Schwarz et al., 2010b]. Here we have taken a slightly different approach by assigning additional strength imparted by roots math formula to tensile mechanical bonds between adjacent cells. Root reinforcement is dependent (among other factors) on the type of vegetation, depth, and distance from tree trunk [Schwarz et al., 2010c]. Schmidt et al. [2001] reported maximum values up to close to 100 kPa for tree roots but values on root reinforcement may depend on method of quantification (measurement of strength of individual roots, shear tests with vegetated columns, in situ tests with shear box, or back calculations). In a recent work based on back calculations Schwarz et al. [2010b] deduced lateral root reinforcement values between 5 and 25 kPa along the scarp, while Sidle and Ochiai [2006] reported values between 1 and 4.3 kPa for back calculations, highlighting that values based on back calculations are in general smaller than for other methods. To capture a typical reinforcement range, we varied the value of tensile root reinforcement between 0 and 30 kPa. A simple approach to account for root reinforcement is additive [Schmidt et al., 2001] and the combined effects of soil strength and vegetation is expressed as tensile strength math formula.

[29] While soil strength math formula depends on water content and soil texture, there is an additional mechanical property determining soil strength, the internal friction angle, which was set constant and independent of water content or soil texture. As shown by Cho et al. [2006] the friction angle is determined primarily by particle shape and not by their size. While Cho et al. [2006] reported values for sandy soils with average friction angle math formula of 32°, Mitchell and Soga [2005] proposed a value of 26° for wet and dry quartz. Here we used an average of values listed by Lu et al. [2010] with internal friction angle of 30°.

3.2. Fiber Bundle Model Statistics

[30] The FBM representation of mechanical bonds provides a means for capturing details of progressive failure events developing within stressed hillslope elements. The FBM consists of a prescribed number of parallel fibers math formula drawn from a distribution of strengths and connecting adjacent (stiff) surfaces. The bundle may be gradually loaded in a stress controlled mode, or stretched by displacement of the surfaces in a strain controlled mode, both modes lead to failure of the weakest fibers (assuming identical stiffness for each fiber). The bundle's strength distribution and residual strength after failure enable representation of a wide range of mechanical behaviors. In Figure 4 two strain controlled stress-strain experimental results of Fannin et al. [2005] are represented using the FBM. It is interesting to observe that the stress-strain response of soil reinforced by roots is represented by two bundles (for soil and roots) that are activated and peak at different strains, defined by activation or stiffness of roots that increases with decreasing fraction of intact soil fibers. The bundles used in Figure 4 were drawn from lognormal distribution of fiber strengths. For simplicity of the landslide triggering model developed in this study, we employ a single FBM with uniform fiber strength distribution [Pradhan et al., 2006] to represent tensile (mechanical bonds) and basal soil strength of partially wet soil.

Figure 4.

Strain-controlled experiments [Fannin et al., 2005] simulated with fiber bundle model (FBM). (a) Direct shear test represented by a FBM with 1000 fibers with strength values deduced from lognormal distribution with residual strength of broken fibers equal to 38% of intact fiber. The inset shows a random distribution of fibers at the base of a soil column constituting the modeled hillslope with hexagonal cross section. Broken and intact fibers are shown in white and black, respectively. (b) Direct shear test of a soil sample with roots represented by combination of two FBMs. Fibers of the FBM representing roots are loaded depending on the number of intact fibers in the soil FBM. In addition to the parameters of the lognormal distribution of the two FBMs, the ratio between number of fibers representing soil and roots (10:3) and the loading of the root fibers as a function of intact soil fibers were optimized.

[31] Next, we link water content modified soil strength (equation (9)) with a bundle composed of fibers with strengths (dimension of force) ranging uniformly between 0 and math formula. We chose bundle size composed of math formula = 10,000 fibers for a hexagonal cross section of math formula, and fiber numbers proportional to the cross-sectional area for other planes. An applied load math formula is sustained by the intact fibers math formula and the residual strength of broken fibers. For uniform distribution, the number of intact fibers math formula and applied load per fiber math formula are related according to math formula. A balance between applied load math formula and strength of intact and residual fibers is formulated as follows:

display math

with the coefficient math formula for residual strength defined as the ratio of intact to broken fiber strength. By solving equation (10) for math formula an expression for the number of intact fibers is obtained:

display math

[32] To determine the bundle peak strength, the derivative of equation (11) with respect to math formula was computed. The stress carried by each intact fiber at maximum strength equals math formula. The peak strength math formula must be equal to the soil strength defined in equation (9) and thus determines the maximum strength of a fiber math formula:

display math

with the area math formula that is represented by the fiber bundle. Inserting equation (12) into (11) defines the number of intact fibers for each load math formula and can be used to estimate the number of broken fibers for changing loads. According to Cohen et al. [2009] and measurements of Win [2006] we set math formula = 0 for tensile stress between soil columns. Cohen et al. [2009] reported values of math formula in the range of 0.26 to 0.50 for shear stress and we fitted values of 0.20 and 0.38 for the series shown in Figure 4. For the simulation of landslide triggering in this study we have chosen math formula = 0.3.

3.3. Link Between LHT and Other Models of Progressive Failure

[33] The landslide hydromechanical triggering (LHT) model described in sections 2 and 3 is related to the models of self-organized criticality (sandpile cellular automaton BTW and spring-block model OFC) in various aspects: The basic elements of LHT are soil blocks with individual strength thresholds as in case of BTW and OFC. While the load (BTW model) or strength values (OFC) were assigned randomly in original SOC models, they are determined by mechanical laws for LHT. For the BTW model the entire load is redistributed at failure to the neighbored elements without limitations by mechanical interconnections and initial strength value is reestablished immediately. This is different in LHT with (1) only a fraction of load redistributed to neighbors, (2) strength reduced to residual value that does not recover for the time scales addressed in this study (single rainfall event), and (3) load redistributions limited by the mechanical strength of the bonds. In that respect the LHT model is similar to OFC where bonds have well defined mechanical properties and it is possible to redistribute only a fraction of force to neighbored blocks. In difference to both SOC models, the LHT model is sort of a two-scale model with mechanical properties of bonds represented by fiber bundle models (FBM). While the SOC-like load redistribution rules are expressed for soil columns with scales defined by soil depth (meter scale), FBM are used to represent much smaller processes at scale of roots or particle contacts (mm scale). The link between the two scales and model concepts is based on equations (11) and (12), linking applied load or stress at column scale to the individual strength of the fibers at grain scale. Another difference between the proposed LHT and “real” SOC model lies in the method used to deduce failure statistics: For SOC models the system evolves into a sustainable critical state (marginally stable) where landslide (or avalanche) statistics can be determined by collecting failures at different time steps within the same slope; for the LHT model no sustainable critical state exists at hydrological time scales and many realizations of similar systems are required, recording the size of the first landslide in each simulation to deduce failure statistics (or to consider large spatial scales, e.g., watersheds).

4. Simulation Results of Landslide Triggering Model

[34] The landslide hydromechanical triggering (LHT) model was used to systematically analyze effects of soil textural class (Brooks and Corey parameter math formula), rainfall intensity math formula, slope angle math formula, and root reinforcement math formula on landslide characteristics (released volume and elapsed time to the onset of mass release). For each combination of parameters characterizing a hillslope (soil texture, slope angle, and root reinforcement), we generated 100 realizations of random soil depth distributions defined by average depth 1.5 m, standard deviation 0.25 m, and spatial correlation length of 5 m. Each simulated hillslope consisted of nearly 3000 soil columns with hexagonal cross section representing 50 by 50 m slope, with hexagonal grid spacing math formula = 1.0 m (lateral distance between centers of touching soil columns). The initial water content was set to field capacity (corresponding to capillary head of math formula = −3.4 m). For each realization a constant rainfall rate math formula was applied, and the resulting spatial distribution of soil water content as well as soil strength were computed at a temporal resolution of 1 min until a landslide occurred. For each realization, the volume of released soil, the shape of the failure plane (length and width), and time of first and subsequent (second) mass release were recorded. Additionally, we monitored the timing and number of breaking fibers in each soil column during load redistribution.

4.1. Precursor Events and Evolution of Failure Patterns

[35] The increase in water content math formula during a rainfall event with intensity math formula is accompanied with increasing weight-dependent downslope driving force math formula and normal (frictional) resisting forces math formula, whereas soil strength math formula decreases. Eventually, the net driving force math formula exceeds math formula at the column base leading to failure of the basal fiber bundle. The occurrence of such basal failure zones may be highly localized as shown in Figure 5a. The potential of such local failure to trigger a landslide depends on strengths of lateral mechanical bonds (enabling load redistribution) and on mechanical resilience of neighboring soil columns. For the example shown in Figure 5a with root reinforcement math formula = 5 kPa, the load is redistributed until a wide region of the slope fails at the base, lateral bonds break, and the ensuing compressive stresses fluidize certain soil columns.

Figure 5.

Evolution of failure patterns (left) and precursor events (right). (a) After 14 h of rainfall the first bundles break at base of soil columns in the left lower corner of the system (dark gray). Additional spots with broken basal bundles appear after 17 h (intermediate gray) and connect at 18 h (bright gray), initiating load redistributions and failure of lateral bonds until compressive stress overcomes material strength at positions indicated by orange color. The release of these columns triggers further relaxations with more cells becoming fluidized (darker red colors for columns released later in the relaxation process). (b) Hourly statistics of breaking fibers for the same time steps shown in (a) compared to the theoretically deduced statistics of a FBM close to failure of the entire bundle (dashed red line, FBMf) with frequency-magnitude exponent ζ = −3/2. An event is the number of breaking fibers in each column in a load redistribution step. All these events are counted for the corresponding hour and statistical properties are deduced for this hourly interval. The small inset shows that with increasing time the absolute value of power law exponent (determined for statistics of small events) becomes smaller and converges to the theoretical value of ζ = −3/2.

[36] During load redistribution, weak fibers in lateral and basal fiber bundles gradually fail, thereby producing a rich statistical picture of precursor events. We present an example of FBM precursor event statistics for three time steps in Figure 5b. Small events (an event is defined as number of fibers breaking during one load redistribution step in a soil column) are more frequent and the frequency/magnitude relationship can be expressed as power law for a certain range of event sizes (Figure 5b). The power law exponent math formula (determined for statistics of small events) was constant during early stages and converged with time to the theoretically predicted value of 3/2 for bundles close to complete failure [Pradhan et al., 2006]. A similar evolution in the exponent value math formula close to imminent failure was also reported by Pradhan et al. [2006] and Cohen et al. [2009]. The results shown in Figure 5b are scaled by normalized distributions with area of 1.0 below the frequency/magnitude curve. The absolute frequencies of the events varied with time, with larger and more frequent events occurring close to mass release. For the three time periods shown in Figure 5b, the total number of events (fibers breaking in a column during load change) was 25% and 35% for the 14th and 17th hour after onset of rainfall, respectively, by relating the number of breaking fibers to the value obtained for the last hour when failure occurred (18th hour, 100%).

[37] In Figure 5 the landslide was initiated close to the top and subsequently propagated downslope. We found that both upslope and downslope propagation of landslides could occur after landslide initiation (i.e., occurrence of a first fluidized soil column). While for systems with no root reinforcement landslide progression was equally likely in both directions, the fraction of downslope propagating landslides increased with increasing root reinforcement with 85% of all landslides propagating in downslope direction for math formula = 15 kPa.

4.2. Elapsed Time to Landslide Triggering

[38] Here we analyze the time needed to trigger a landslide for different rainfall intensities math formula, slope angles math formula, and soil textural classes determined by parameter math formula. We also calculate time intervals between first and subsequent landslide occurrence.

4.2.1. Effect of Rainfall Rate r on Time to Failure

[39] The assumed constant rainfall intensity math formula renders elapsed time proportional to cumulative rainfall math formula (defined as product of time and rainfall intensity). For a hillslope with soil textural class defined by math formula = 0.3 (corresponding to sandy loam with math formula = 0.32 according to Rawls et al. [1982]), slope angle math formula = 38°, and with rainfall intensities ranging from 2.5 to 100 mm h−1, the time of first landslide occurrence in each of 100 hillslope realizations (random soil depth distribution) was computed. For intensities larger than 10 mm h−1, runoff occurred and not the entire rainfall water contributed to the loading and weakening of soil columns. By increasing rainfall intensity math formula from 2.5 to 10 mm h−1 (no surface runoff) the rainfall duration required for triggering a first landslide was reduced from 32.5 to 8.1 h (average of 100 realizations) in soils without vegetation root reinforcement ( math formula = 0), and from 40.4 to 20.2 h with root reinforcement of math formula = 15 kPa. The corresponding cumulative rainfall amounts ranged from 101 to 81 mm for slopes with and without root reinforcement, respectively. The distribution of elapsed times to landslide initiation is completely scalable by cumulative rainfall which leads us to conclude that rainfall intensity did not play an important role in this range of intensities and soil textural classes. We note that for higher rainfall intensities surface runoff occurred and this water is removed from the system and does not contribute to increase load or reduced soil strength. For a rainfall intensity of math formula = 20 mm h−1 the cumulated amounts of rainfall were 125 mm for root reinforcement ( math formula = 15 kPa) and 95 mm in bare soils ( math formula = 0 kPa) and were higher than for conditions without runoff (101 and 81 mm). Irrespective of rainfall intensity (in the range from 2.5 to 100 mm h−1) the landslide occurred for average water contents of 0.27 for soils without vegetation and at 0.28 for root reinforcement of math formula = 15 kPa (for soil texture math formula = 0.3). Note that the conclusion of small sensitivity to rainfall rate math formula has probably to be revised when taking into account different flow mechanisms like fast flow paths along macropores from surface to failure plane at the soil-bedrock interface [Sidle et al., 2000] that could accelerate weakening and failure process.

4.2.2. Effect of Slope Angle β on Time to Failure

[40] Elapsed time to first failure depends on the balance of resisting and driving forces and hence on a local factor of safety (LFOS) defined for each soil column. The dependency of forces on slope angle math formula necessitated scaling of elapsed time. For the scaling we analyzed conditions corresponding to a factor of safety of 1.0 for an infinite slope analysis (i.e., driving force math formula equals to math formula) for average slope angle math formula. Similar to the scaling approach of Keefer et al., [1987] we thus related cumulative rainfall math formula to a critical value math formula that corresponds to cumulative rainfall needed to obtain FOS = 1.0 for the infinite slope. This calculation was based on assumed average soil depth of 1.5 m, soil textural class math formula = 0.3, average surficial slope angle math formula, and for simplicity, assuming all rainfall infiltrates for the infinite slope FOS.

[41] Landslides for nonvegetated hillslopes ( math formula = 0 kPa) were triggered before attaining (infinite slope) mean FOS = 1.0 (Figure 6a). This is attributed to the localized definition of LFOS (with local slope angle determined by the height difference between centers of adjacent soil columns) which may drop below unity while the average FOS for the hillslope remains above 1.0. Considering different slope angles math formula for the same soil textural class math formula = 0.3, differences after scaling with math formula are small (Figure 6a) with failure occurring between 0.7 and 0.8 of scaled time. However, the volumes of triggered landslides increase with steeper slope angles, spanning a range of several orders of magnitude. We relate larger landslide volumes to narrower distributions of local LFOS for steep slopes (data not shown). When a soil column fails, the forces on adjacent columns are similar hence failure cascades throughout the hillslope. For more gentle steepness with wider variation of FOS, it is more likely that a perturbation is contained by adjacent more stable soil columns. Note that these conclusions are affected by the assumption of soil depth independent of slope angle values. As proposed by Gabet and Dunne [2002] landslide volumes may be smaller in steep slopes due to reduced soil depth.

Figure 6.

Scaling of time to trigger landslides with rainfall intensity r of 5 mm per hour in a slope of size 50 × 50 m. The arrows show the medians of landslide size of 100 realizations. The cumulative rainfall P was scaled by the cumulative rainfall P* that corresponds to a factor of safety of 1.0 for an infinite slope analysis. (a) The change of slope angle β (soil texture λ = 0.3) from 35° (blue triangles) to 38° (green squares) and 40° (red disks) has minor effect on time of landslide occurrence but affects volume of landslides (size increasing with slope angle β). Due to variation of local factor of safety, landslides occur before average FOS is equal to 1. (b) For different soil textural classes (slope angle β = 38°) with Brooks and Corey parameter λ ranging from 0.2 (blue triangles), to 0.3 (green squares) and 0.4 (red disks) the landslides are triggered later for less coarse soils due surface runoff delaying weakening of soil strength. With root reinforcement τroot = 15 kPa of lateral bonds, landslides are generally triggered later than in nonvegetated hillslopes.

4.2.3. Effect of Soil Textural Class Parameter λ and Root Reinforcement on Time to Failure

[42] Soil textural class determined by parameter math formula affects FOS due to capillary-dependent soil strength [Bishop 1960; Lu et al., 2010] and potential alteration of other soil hydraulic properties and pathways (i.e., onset of runoff). For initial conditions with pressure head math formula = −3.4 m, the initial soil strength is higher in soils with small value of math formula (less coarse textured), and more cumulative rainfall math formula is needed to reach critical LFOS (Figure 6b). The scaling of cumulative rainfall math formula by a critical value math formula that corresponds to cumulative rainfall needed to attain a factor of safety of 1.0 (for slope angle math formula = 38°) enables removal of differences to initial conditions. However, we found that for soil texture characterized by math formula = 0.2 (corresponding to loamy soil) landslides are triggered at later times relative to coarser soil textures, for a mean (infinite slope) FOS close to 1.0. This “delay” was attributed to increased surface runoff component that resulted in a lower rate of hydrologic loading and slowed the onset of mechanical weakening. In all cases landslides are triggered later for reinforced bonds ( math formula = 15 kPa) simply because basal failure is more readily stabilized by stronger mechanical bonds with neighboring columns. With the exception of math formula = 0.2 (delayed triggering due to runoff), landslides for reinforced bonds were triggered when standard FOS approached 1.

4.2.4. Time Interval (Waiting Time) Between First and Second Landslide

[43] Another important temporal aspect of landslide triggering is the time interval between release of the first and second landslide in each realization of simulated hillslope (Figure 7a). Following the first failure (with released mass removed from the modeled system), the hillslope is mechanically primed and just a rather small amount of additional rain water is needed to trigger a second landslide. In many cases, the second landslide was triggered within a few minutes after the first release, however, in some realizations it took 2 h until the second landslide was triggered. For hillslopes with no root reinforcement, the time interval between the first and second landslide was longer, indicating higher localization in the absence of lateral reinforcement and thus reduced spatial dependency with the first release event. For root reinforced slopes, however, the load is redistributed over a wider footprint area and failure zones may become interconnected (see Figure 5a). This result is in agreement with findings of Dietrich et al. [2007], Schwarz et al. [2010b], and Gabet and Dunne [2002] that showed that larger areas up to 1000 m2 can be stabilized and interconnected due to tensile strength of roots. Mechanical loads in these large footprint areas are similar due to load redistribution along bonds and it can be expected that an additional perturbation in the same interconnected region will be triggered for conditions very similar to those during the first landslide.

Figure 7.

Power law relationships for waiting time/frequency and landslide volume/frequency indicating criticality of the modeled system. Results are shown for 100 realizations of a hillslope with size 50 × 50 m, soil texture defined by Brooks and Corey parameter λ = 0.3, slope angle β = 38° and enduring rainfall intensity of r = 5 mm per hour. (a) The time interval between first and second landslide in a realization can be characterized by power law with larger absolute values of exponent |ζT| for mechanical bonds reinforced by τroot, indicating that a long time between two failures is unlikely in reinforced systems. (b) For the volume of triggered landslides the absolute value of power law exponent |ζV| decreases with reinforced mechanical bond strengths, indicating that large landslides are more frequent in reinforced systems.

[44] The results shown in Figure 7a are expressed as a power law. Because power laws are valid for time intervals in critical states [Sánchez et al., 2002; Baiesi, 2009] these findings could be interpreted as an indicator for criticality of the system. However, due to the model assumption to remove the mass of the first landslide from the system, the results for these time intervals may be oversimplified and we may not be in a position to draw general conclusions based on these limited findings. Instead, we explore aspects of criticality based on the frequency/magnitude distribution of modeled landslide volumes presented in section 4.3.

4.3. Frequency-Magnitude Statistics of Modeled Landslides

[45] We obtained results from 100 realizations for each combination of hillslope soil textural class math formula, slope angle math formula, rainfall intensity math formula, and vegetation root reinforcement math formula, in which released soil volume for the first triggered landslide in each realization was recorded. As indicated in Figure 6, the sizes (volumes) of these landslides may vary over several orders of magnitude and their frequency/volume statistics can be characterized by a power law with exponent math formula (Figure 7b). Landslide probability density was determined from the derivative of their cumulated size distribution (equation (2)). The resulting landslide volume distributions for hillslopes with and without vegetation root reinforcement math formula are depicted in Figure 7b. For the root reinforced lateral bonds, larger landslides are more frequent corresponding to smaller math formula for root reinforced bonds. This result corresponds to the findings of Schwarz et al. [2010b] and Dietrich et al. [2007] that larger volumes can be stabilized by lateral roots enabling release of larger soil mass during failure, while primarily small volumes are released for weak lateral bonds (no root reinforcement) due to limited load redistribution.

[46] To analyze trends in power law exponents math formula, we varied rainfall rate math formula between 2.5 and 10 mm h−1 (no runoff), root reinforcement math formula between 0 and 30 kPa, soil textural parameter math formula from 0.2 (silt loam) to 0.4 (loamy sand), and slope angle math formula from 35 to 40 deg. The resulting power law exponents (absolute values) varied between 1.0 and 2.2. Small values of math formula close to unity were found for root reinforced bonds and steep slopes. High values of math formula close to 2.0 were typical for hillslopes without root reinforcement. While rainfall intensity had virtually no effect on power law exponent (note that we did not include vertical preferential flow paths), increasing root reinforcement math formula from 0 to 6 kPa resulted in a monotonic decrease in math formula from 1.9 to 1.3 ( math formula = 5 mm h−1, math formula = 38°, and math formula = 0.3). The corresponding power law frequency/magnitude statistics are shown in Figure 8.

Figure 8.

Frequency/magnitude statistics of modeled landslides with power law exponents ζV for root reinforcements τroot varying from 0 to 6 kPa. The first landslide triggered in each of 100 realizations of a hillslope with size 50 × 50 m, rainfall rate r = 5 mm h−1, slope angle β = 38° and soil textural class λ = 0.3 was taken into account to determine probability density. By increasing lateral bond strength τroot from (a) to (d), the power law exponent |ζV| decreases from 1.9 to 1.3. Only power law tail of probability density is shown for minimum landslide volumes between 1 and 4 m3.

[47] For stronger mechanical tensile bonds, soil columns with broken fiber bundle at the soil-bedrock interface can form large clusters (see Figure 5a). We hypothesize that when finally tensile bonds fail and downslope soil columns are fluidized, the released stress is too high to be carried by neighboring intact soil-bedrock bases or other bonds and neighboring columns instantly exceed their strength threshold as well. Such conditions result in large landslides and thus small values of power law exponents math formula. Note that the range of exponents given in Figure 8 is similar to values deduced from real landslide inventories shown in Figure 1. However, this should not be interpreted as an explanatory result for the real inventory because the model does not consider heterogeneities included in real hillslopes and is based on simplification of cohesion-less soil. Because this may be an oversimplification and Stark and Guzzetti [2009] proposed explicitly that enhanced cohesion increases the absolute value of power law exponent, we carried out a few simulations with nonzero but small cohesion value of 3 kPa for root reinforcement of 6 kPa. The results are shown in Figure 9a and indicate an increase of exponent math formula from 1.3 to 2.0. Our model findings suggest that basal reinforcement (by cohesion or root anchorage) leads to higher absolute power law values (in agreement with findings of Stark and Guzzetti [2009]), while strong lateral bonds without reinforcement at the base result in smaller values (confirming study of Faillettaz et al. [2004]).

Figure 9.

Effect of cohesion csoil and slope angle β on frequency/magnitude statistics of modeled landslides. The first landslide triggered in each of 100 realizations of a hillslope with size 50 × 50 m, rainfall rate r = 5 mm h−1, and soil textural class λ = 0.3 was taken into account. (a) By introducing moderate cohesion value csoil of 3 kPa, the power law exponent |ζV| increases from 1.3 to 2.0 (τroot = 6 kPa; β = 38°). (b) By increasing slope angle β from 36° to 40°, the exponent |ζV| drops from 1.7 to 1.0 (τroot = 15 kPa).

[48] The following analysis illustrates that power law exponents are dependent on various factors. Figure 9b depicts effects of slope angle with exponent values math formula decreasing from 1.7 to 1.0 with increasing slope angle math formula from 36° to 40° ( math formula = 5 mm h−1, math formula = 0.3, and math formula = 15 kPa). A similar trend was confirmed for other soil textural classes ( math formula of 0.2 and 0.4). This could be attributed to larger destabilization of failed soil columns by redistributing larger loads to adjacent columns (see section 5.1). Additionally, as discussed in context of Figure 6, the FOS distribution is narrower for a slope of 40° and containment of failure propagation by stronger soil columns becomes less likely. Finally, by changing soil textural class parameter math formula from 0.2 to 0.4, the exponent value math formula decreased only slightly from 1.7 to 1.5 ( math formula = 5 mm h−1, math formula = 38°, and math formula = 15 kPa). We relate the smaller frequency of large events (corresponding to high values of math formula) to stronger compressive strengths for soils with small math formula (less coarse textured soils). However, in comparison with slope angle and root reinforcement, the soil textural class exerted a smaller effect on the power law exponent. The power law exponent values and their relevance to quantify criticality of the system will be discussed in more detail in section 5.

4.4. Resulting Landslide Shapes

[49] As shown in Figures 1 and 8, the frequency/magnitude distribution of modeled landslides is in agreement with statistics deduced from landslide inventories. In the following we address the question how realistic simulated landslide shapes and volumes are relative to measured landslides. An aerial image depicted from Alpnach in central Switzerland (Figure 10a) illustrates landslide distribution triggered by the 2005 rainfall event (5000 landslides triggered in entire country). The insets with realizations of modeled landslides exhibit similar variability of shapes and sizes reproduced by the triggering model. In one model case, two landslides are triggered within the same time step (1 min) during load redistribution after release of the first soil column. While the size of the landslides can only be roughly estimated from airborne imagery in Figure 10a, a more quantitative comparison is possible for images of the landslide inventories in which individual landslides were measured in the field by Rickli [2001], Rickli and Bucher [2003], and Rickli et al. [2008]. In Figure 10b a landslide with reported width and length is shown together with the modeled landslide of the same dimension.

Figure 10.

Comparison of modeled and observed landslide shapes and sizes. (a) The left photo shows landslides triggered by an intense rainfall during 2005 in Alpnach, Switzerland (photo from Cantonal Office for Forest and Spatial Development, Canton Nidwalden). The insets show modeled landslides in a hillslope of 50 × 50 m with similar shapes and sizes. Note (1) that the modeled and measured landslides in the left upper corner are concave, while the others are convex in shape and that (2) two landslides were released at same time step in simulation shown in upper right corner. (b) The photo at the right was taken by the Swiss Federal Research Institute WSL in course of recording landslide inventories (some data shown in Figure 1). The landslide size was measured and the inset shows a model realization with similar size and shape.

[50] In addition, we determined the aspect ratio (length along the slope divided by width of the landslide) and compared them to literature data (Figure 11). The average aspect ratio of the simulations was 2.7 and is comparable to published values of 3.3 [Marchesini et al. 2008], 3.4 [Gabet and Dunne, 2002], and 2.6 [Parise and Jibson, 2000] but deviates from the aforementioned inventory data of Rickli and co-workers with an average aspect ratio of 1.3. The landslide shape results are preliminary, and we expect presence of strong dependency on local heterogeneities and other subsurface features as well as on details of load redistribution rules.

Figure 11.

Shape of modeled landslides expressed by aspect ratio (length divided by width of landslide). For 100 realizations of a hillslope with size 50 × 50 m, soil texture defined by Brooks and Corey parameter λ = 0.3, slope angle λ = 38°, bond reinforcement of τroot = 15 kPa, and enduring rainfall intensity r = 5 mm per hour, the aspect ratio and landslide volume are presented. In insets rather large landslides with aspect ratios of 2.1 and 5.1 are shown. The arrows pointing at the abscissa denote average aspect ratio deduced from simulations in this study and from literature data.

5. Criticality and Scale Invariance

[51] The frequency/magnitude statistics for simulated hillslopes in this study are characterized by a wide range of landslide volumes expressed as a power law with exponent math formula. The scale invariance criterion is fulfilled irrespective of the value of the exponent math formula, however, it may not necessarily represent criticality. The question whether hydrologically-induced landslide triggering is governed by concepts of self-organized criticality needs to be addressed separately [Hergarten, 2002; Turcotte 1999]. As pointed out by Sornette [1994] a power law is not a sufficient criterion for SOC which could also result from underlying scale invariant spatial water distribution [Pelletier et al., 1997]. Sornette [1994] showed that for various phenomena power law relationships are obtained by driving the system to a “global failure” (similar to the FBM when eventually the entire bundle breaks) and not by avalanches in a marginally stable state as postulated by SOC formalism. Clearly the hydrologic time scales investigated in this study are too short to warrant definitive statements regarding whether or not a hillslope attains a marginally stable state over time. Such determination could only be made over much longer (geological) time scales. Instead, we adopt Ramos [2011] analyses that provide a framework to quantify system criticality based on value of power law exponent and ratio of average and maximum avalanche size.

5.1. Scale Invariance and Power Law Exponent Values

[52] A first difference between theoretical SOC systems and real (or simulated) landslides is manifested in the values of the power law exponent math formula. While theoretical values lie in the range of 1.0 for the sandpile model (BTW model, Bak et al. [1987]) and 1.23 for the spring-block model (OFC model, Olami et al. [1992]), most exponents deduced from landslide inventories and simulated in our study exhibit larger exponent values. Figure 12a summarizes power law exponent values for different model classes and field and laboratory data. For the field data we rely on Rickli [2001], Rickli and Bucher [2003], and Rickli et al. [2008], and the reviews of Brunetti et al. [2009] for frequency/volume ratios ( math formula) and Van Den Eeckhaut et al. [2007] for frequency/area ( math formula) relationships. Note that we have used data from the aforementioned reviews that are related to rainfall induced landslides only. Laboratory data with piles of sand, rice, and beads were reported by Aegerter et al., [2003], Altshuler et al., [2001], Costello et al., [2003], Frette et al., [1996], Held et al., [1990], Lörincz and Wijngaarden, [2007], Nerone et al., [2003], and Ramos et al., [2009].

Figure 12.

Criticality and scale invariance of simulated hillslope scenarios. (a) Exponent of frequency/magnitude distribution for field and laboratory experiments, landslide hydromechanical triggering model (LHT, this study), sandpile cellular automaton (BTW), and spring-block model (OFC). Values of BTW and OFC correspond to systems governed by self-organized criticality (SOC). The absolute exponent values of the presented triggering model and those deduced from field and laboratory measurements are in general higher than the values for SOC systems. This is related to load redistribution dampened by frictional losses at soil-bedrock interface. (b) Criticality factor CF defined as ratio between modeled average size of a landslide and its value for an ideal critical system. This ratio was introduced by Ramos [2011] to quantify criticality with CF close to 1 for systems governed by SOC. The modeled system becomes more critical for steep slopes, high reinforcement, and coarse textured soils. The black line is drawn to highlight the emerging trend and is not deduced from theoretical considerations. The dashed lines and shaded area denote the range of frequency/magnitude exponents reported for SOC systems (BTW and OFC model).

[53] For the BTW and OFC model with values of math formula close to unity, a perturbation may propagate across a large region, however, such large avalanches are not observed in typical shallow landslides. For the LHT model small values close to 1.0 were obtained only for very steep slopes. Theoretically, such small exponent values are only possible for well-defined maximum landslide size (e.g., corresponding to size of the system); without an upper bound of landslide volume the integration of probability density function with power law exponents math formula would not converge to 1.

[54] As pointed out by Olami et al. [1992], Hergarten [2003], and Ramos [2011], the exponent math formula depends on the fraction of redistributed load, with large math formula > 1 when only a fraction of the released load is redistributed to neighbors. While for the BTW model the entire mass (load) is redistributed to neighbors, the fraction of redistributed load was changed systematically in the OFC model [Olami et al., 1992], increasing the power law exponent math formula from 1.23 to 3.5 by decreasing the fraction of load transmitted to neighbored blocks from 1.0 to 0.2. Consequently, one would expect exponent values math formula close to unity (hence behavior similar to classical SOC systems) for hillslopes with large ratio between released and applied load. To assess this ratio for soil columns, we first compute the total downslope directed force as sum of math formula (with slope angle math formula and soil column weight math formula), and the load exerted by neighboring soil columns math formula. For presentation purposes we express the latter term as math formula with larger dimensionless coefficient math formula for high external loads. Setting the post failure residual strength to zero, the ratio between released and applied load denoted as stress-release-ratio math formula is

display math

[55] For systems with math formula values close to 1, the failure is similar to theoretical SOC systems (exponent math formula close to 1) with large amount of released load and higher potential to trigger a chain reaction. According to equation (13) this behavior is expected for large slope angles math formula and large amount of redistributed load (high math formula). Because large math formula requires strong bonds for load redistribution, hillslopes are similar to theoretical SOC for large root reinforcement and large slope angle. In section 5.2 we present a method to quantify hillslope criticality in the SOC sense.

5.2. Criticality

[56] Ramos [2011] defined a criterion for criticality based on statistical properties of SOC systems in the context of phenomena exhibiting burst-like redistributions and avalanches. Ramos [2011] invoked properties of systems attaining idealized critical state (no frictional losses or energy dissipation) in which the correlation length diverges and the average avalanche size math formula increases linearly with system size. Ramos [2011] established an analogy between mean avalanche size for a power law size distribution with a sharp cut-off value at maximum “event” size math formula (that is proportional to math formula) and an idealized critical system where the average size math formula equals

display math

with a power law exponent math formula representative for idealized SOC systems. Based on a study by Chessa et al. [1999] demonstrating that various SOC systems can be characterized by an exponent of math formula = 1.27, Ramos [2011] expressed the ratio of the observed average avalanche size math formula to the average size of a SOC system math formula as the criticality factor math formula:

display math

with the observed average math formula and maximum size (volume) math formula for a theoretical SOC system. In agreement with the definition by Ramos [2011] we considered as a maximum landslide size math formula the entire simulated slope volume. For systems in a critical state math formula is close to 1. The calculated criticality factors ( math formula) for landslides simulated in this study are depicted in Figure 12b. Criticality factor values math formula are smaller than 1 and increase with strength of tensile mechanical bonds math formula, with slope angle math formula, and with soil textural class parameter math formula. Apparently, the criticality value is high for conditions where a local perturbation may propagate across the entire hillslope. This condition occurs when lateral bonds are strong with large values of math formula, increasing the likelihood of adjacent bond failure when such disproportionally high stress is abruptly released. Hence slopes with strong root reinforcement attain “more critical” states manifested by smaller exponent values in agreement with a conceptual model of Faillettaz et al. [2004] considering two-threshold cellular automaton for failure at the base and between adjacent elements. They found that the larger the ratio between thresholds at the “lateral bond” and at the base, the smaller the absolute value of the resulting power law exponent. The increase of criticality factor and decrease of power law exponent value math formula with increasing root reinforcement is also in agreement with findings presented in section 5.1 in which stress release ratio math formula is larger for large root reinforcements. The enhanced load release can also explain the increase of criticality factor and decrease of math formula with increasing slope angle math formula.

[57] For coarse textured soils (large math formula) with low soil strength, a small perturbation may overcome resisting forces and destabilize a soil column. The criticality factor increases slightly with soil texture parameter math formula but the variations are small compared to effects of slope angle and root strength. As shown in Figure 12b the criticality factor math formula is inversely related to the power law exponent value math formula. Criticality is characterized by both values ( math formula and math formula) close to unity. Based on the analysis in this section and the formalism of Ramos [2011], we conclude that during extended rainfall events a hillslope may converge to a temporally critical state due to local failures and load redistribution among soil columns. Power law frequency/magnitude relationships (exponent) are shaped by frictional losses at the soil-bedrock interface and the stress release to neighbored soil columns. The factors contributing to attainment of criticality are summarized in the trends depicted in Figure 12b.

6. Summary and Conclusions

[58] The abruptness of rainfall-induced shallow landslides was simulated by employing concepts of self-organized-criticality (SOC) within a hydromechanical physically based hillslope model. Rainfall was applied at a constant rate to model hillslopes composed of mechanically interacting discrete soil columns until internal mechanical failure in some of the columns resulted in a chain reaction of local failures leading to a landslide. When driving forces exceed shear strength at a soil column base, failure occurs and the column's load is redistributed to neighboring columns via tensile or compressive mechanical “bonds.” These local failure and load redistribution events may result in local compressive stresses exceeding soil strength of a load bearing column causing its liquefaction (inability to carry load). The failure may propagate and destabilize other columns in downslope (redirection of compressive stress) and upslope directions (released by removed supporting mass). Mechanical strength related to capillary forces at the soil-bedrock interface and between neighboring columns was represented by fiber bundles. The findings of this study are summarized as follows:

[59] 1. The size and frequency statistics of simulated landslides obey a power law with exponents similar to values inferred from real landslide inventories.

[60] 2. The absolute values of power law exponents for the hydrologic triggering model (and for landslide inventories and sandpile experiments) were larger (1.0 to 2.2) than values obtained for theoretical SOC systems (1.0 to 1.3).

[61] 3. The higher exponent values for modeled and observed landslides compared to theoretical SOC systems were attributed to dissipative processes (e.g., frictional losses at soil-bedrock interface) that reduced the redistributed load (hence reduced ratio between released and applied load).

[62] 4. By increasing lateral bond strength (effect of roots) from 0 to 6 kPa, the exponent dropped from 1.9 to 1.3 (increasing cohesion would lead to increase of exponent value).

[63] 5. A criticality factor introduced by Ramos [2011] was computed to determine conditions and evolution to criticality of a hillslope.

[64] 6. For steep hillslopes (slope angle 40°) with strong bond reinforcement the highest criticality values resulted for the triggering model.

[65] 7. Time to failure (landslide release) could be scaled by cumulative rainfall and factor of safety (this result may need modifications when different preferential flow paths would be accounted for).

[66] 8. By representing mechanical bonds with fiber bundles, the dynamics close to mass release could be analyzed.

[67] 9. Frequency/magnitude statistics of breaking fibers (local failures) followed a power law with exponents changing close to failure.

[68] Although simulation results of landslide volumes and shapes were in reasonable agreement with field measurements, the results may not be generalized as these are affected by several model simplifications. Most prominently, we simplified the description of liquefaction and mobilization of soil volumes by defining compressive strength threshold and did not take into account strain dependent transition to a mechanical critical state with pressure release. Also by removing failed columns from the system, the effect of failed columns on load redistribution is suppressed. Additionally, the failure plane was always chosen at the soil-bedrock interface while in nature it is possible that plane evolves closer to surface. Furthermore, lattice size and resolution effects were not discussed here (however in a few tests we obtained similar power law exponents math formula and proportionality factors for different lattice sizes). Finally, different hydromechanical functions (for example taking into account other preferential flow mechanisms and models for water content dependent soil strength) and more complex load redistribution rules (so far it was only distinguished between compressive and tensile bonds), or more complete hydrologic models including Richards equation could be used. However, the expected modifications in saturation and material strength compared to the model choices presented here are not expected to significantly affect the abruptness and criticality reproduced with the triggering model and observed in natural landslides. The model presents a first step into describing system criticality in terms of standard physical hydromechanical variables. Some of the FBM rich statistics and associated power law evolution could be combined with monitoring systems (such as acoustic emission sensors to measure emission of elastic waves during loading) to enable short term prediction of eminent failure.

Appendix A:: The Hydrological Model

A1. Parameterization of Soil Hydrologic Properties

[69] To parameterize hydraulic properties we used the model of Brooks and Corey [1964]:

display math
display math
display math

with water content math formula, effective water saturation math formula, capillary pressure math formula, air-entry value math formula, pore size distribution parameter math formula, and residual and maximum water content math formula and math formula. Because soils during intense and enduring rainfall may exceed maximum water contents reached in laboratory experiments, we set math formula with porosity math formula. Based on the data set from Rawls et al. [1982] for average parameter values for nine soil classes, we expressed the hydraulic properties math formula, math formula, math formula, and water saturated hydraulic conductivity math formula as a function of parameter math formula:

display math
display math
display math
display math

with length in meters and time in hours. Note that math formula defines the width of soil water characteristics and hence the soil pore size distribution. For large values a small change in capillary head corresponds to a large decrease of water saturation.

A2. Infiltration Capacity and Overland Flow

[70] For each time step math formula a constant rainfall volume math formula with rainfall rate math formula and hexagonal cross section math formula is added to each soil column. If this volume is larger than free pore space of a column or the infiltration rate math formula, water is ponding at a height math formula and is routed along maximum downslope gradient. Based on infiltration theory of Philip [1957] and estimate of sorptivity (capacity to adsorb water based on capillarity) according to Parlange and Smith [1976], the infiltration rate equals

display math
display math

with infiltration rate math formula as a function of time math formula, sorptivity math formula expressed as a function of initial water content math formula and soil water diffusivity math formula with soil water capacity math formula as a derivative of a water retention curve, and water content dependent hydraulic conductivity math formula. The time math formula relates to an infiltration rate equal to rainfall intensity and math formula to time compression approximation taking into account free storage capacity for period math formula:

display math
display math

A3. Subsurface Flows

[71] Infiltration during time step math formula increases the water content within the soil element. Assuming equilibrium water content distribution with capillary head math formula = 0 at soil-bedrock interface and math formula at soil surface with soil depth math formula, the maximum water volume math formula without formation of a free water table is given by

display math

[72] If water volume per soil column math formula is above this value, a perched free water table of height math formula forms that was determined by solving numerically the following equation:

display math

[73] When the level of this free water table math formula with bedrock level math formula is higher than in an adjacent cell, it can flow along height difference math formula (positive if math formula is higher than in adjacent cell) according to Darcy's law.

[74] The water flux math formula along the soil-bedrock interface equals the water volume math formula flowing along the driving pressure head difference through the water-saturated part of interfacial cross section math formula and can be expressed as

display math

with saturated hydraulic conductivity math formula at the interface between soil and bedrock that is assumed to be higher than saturated conductivity of the soil matrix due to fractures and gaps. We chose math formula for simulations. This value is a rough estimate motivated by the observations of Tromp-van Meerveld and McDonnell [2006] with water flow velocities between 4 and 8 m per hour along the soil-bedrock interface for vertical matrix saturated conductivity of 0.64 m per hour, assigning the high conductivities to soil pipes and other large pores occurring particularly at the soil-bedrock interface.

[75] In addition to saturated water flow along the soil-bedrock interface, subsurface water flow occurs through the soil matrix. To assess soil water flow driven by differences in gravitational height and capillary head, we assign a capillary head math formula to each soil column based on an average water content of each soil column and water retention curve as described by equation (A1b). Water flows between two cells along a difference in total water pressure math formula with gravitational height math formula computed as an average of bedrock elevation math formula and surface height math formula. Water is redistributed to all adjacent neighbors with positive difference math formula between the cell and its adjacent neighbor according to the law of Buckingham-Darcy. The water flux through the soil matrix along the pressure head difference equals

display math

with hydraulic conductivity math formula as a function of water content and grid distance math formula.

Appendix B:: The Mechanical Model

B1. Parameterization of Soil Mechanical Properties

[76] For shear and compressive strengths for unsaturated soils we have used a formulation based on capillary head math formula (equations (6), (8), and (9)). Bishop [1960] extended the effective stress for unsaturated condition where math formula is expressed as linear function of head math formula, with proportionality factor math formula. While Lu et al. [2010] proposed using water saturation math formula to estimate math formula, Khalili and Khabbaz [1998] defined a relationship between math formula and math formula based on review of shear strength experimental data given as

display math

with capillary pressure math formula and air-entry value math formula.

[77] Khalili and Khabbaz [1998] reported small (absolute) exponent values for fine textured soils and higher values for coarse soils, nevertheless, they postulated that equation (B1) would provide reasonable estimates for a wide range of soils. In Figure B1 we compare the relationship proposed by Khalili and Khabbaz [1998] with the interpretation of math formula based on effective saturation by Lu et al. [2010] (that was used in our study). In Figure B1 we show math formula for the values of math formula reported by Rawls et al. [1982] ranging from 0.13 to 0.59. While the estimates of math formula for coarse textured soils are in the range reported by Khalili and Khabbaz [1998], the values for fine textured soils are far above values defined by equation (B1). To examine the validity of saturation math formula as estimator of math formula for rather fine textured media, we used data from Vanapalli et al. [1996] that were also analyzed in the study of Lu et al. [2010]. We first fitted the Brooks and Corey model to measured water retention curve data (Figure 11 in Vanapalli et al. [1996]). In one case we fitted four model parameters ( math formula, air-entry value, porosity, and residual water content) and in the second case we fitted math formula only (expressing other model parameters as functions of math formula as shown in equation (A2)). Then the resulting Brooks and Corey parameters were used to compute soil strength as math formula with density of water math formula, gravity acceleration math formula, and effective saturation math formula expressed by the Brooks and Corey model (equation (A1a)). We compared the predicted soil strength with values shown in Figure 14 in Vanapalli et al. [1996]. In the comparison we corrected measured values by subtracting cohesive strength obtained for complete water saturation (10 kPa), and expressed measured shear strength as soil strength math formula considering reported friction angle of 23°. As shown in Figure B1 the predictions do not overestimate soil strength for the range of absolute capillary pressure values relevant for this study ( math formula < 10 m). The predictions based on a single fitted parameter ( math formula) remained accurate for the entire range of measured capillary pressure values. The analysis lends credence to soil strength estimates used in this study also for less coarse textured soils.

Figure B1.

Comparison of the soil strength model used in this study with other references. (a) Khalili and Khabbaz [1998] proposed that the proportionality factor χ linking soil strength τh and capillary head h is not very sensitive to soil texture and can be expressed by a rather narrow range as indicated by black lines with solid line for best fit of experimental data and dashed lines for envelope of measured values. The values for χ obtained in this study for the range of soil textures (defined by parameter λ) used by Rawls et al. [1982] seem to be too high for fine textured material. (b) The validity of our model for more fine textured material was tested for data reported by Vanapalli et al. [1996]. The measured water retention function was fitted according to the Brooks and Corey model and soil strength was predicted with χ expressed as effective water saturation. While the prediction based on a fit of all four Brooks and Corey parameters was better for small values of capillary head, the prediction based on fitted λ (expressing other parameters as a function of λ according to equations (A2)) described the measured values fairly well for the entire range of capillary heads. For comparison soil strength of the three soil materials used in the landslide triggering study are shown as well (λ = 0.2, 0.3, and 0.4).

Notation
math formula

area assigned to fiber bundle, m2.

math formula

intersectional area between two cells, m2.

math formula

cross section of hexagonal cell, m2.

math formula

area of a landslide, m2.

math formula

minimum landslide area for validity of power law distribution, m2.

BTW

Bak, Tang, and Wiesenfeld sandpile model

math formula

soil cohesion, kg/(m s2).

math formula

criticality factor.

math formula

soil water capacity, m1.

math formula

water diffusivity, m2 s−1.

FBM

fiber bundle model.

math formula

force factor proportional to force applied by adjacent soil columns.

math formula

strength ratio of broken and intact fiber (residual strength).

math formula

load (force) applied to fiber bundle, kg m s−2.

math formula

compressive force on downslope soil column, kg m s−2.

math formula

load applied by neighbored soil columns, kg m s−2.

math formula

weight component along slope, kg m s−2.

math formula

weight component normal to slope direction, kg m s−2.

FOS

factor of safety, ratio of resisting to driving forces.

math formula

weight (force) along gravity, kg m s−2.

math formula

gravity acceleration, m s−2.

math formula

capillary head, m.

math formula

air-entry value of soil material, m.

math formula

soil depth, m.

math formula

water ponding height at surface, m.

math formula

height of free water table in soil column, m.

math formula

infiltration rate, m s−1.

math formula

water flow into bedrock, m s−1.

math formula

water flow in soil matrix, m s−1.

math formula

water flow along soil-bedrock interface, m s−1.

math formula

hydraulic conductivity for water unsaturated condition, m s−1.

math formula

hydraulic conductivity at soil-bedrock interface, m s−1.

math formula

hydraulic conductivity under water saturated condition, m s−1.

math formula

horizontal distance between centers of hexagonal cells, m.

LFOS

local factor of safety, defined by local soil depth and slope angle.

LHT

landslide hydromechanical triggering model (this study).

math formula

mass of soil column, kg.

math formula

number of compressive bonds in a soil column.

math formula

number of fibers in a bundle.

math formula

number of intact fibers in a bundle.

math formula

total number of landslides.

math formula

number of tensile bonds (fiber bundles) in a soil column.

OFC

Olami, Feder, and Christensen spring-block model.

math formula

landslide area probability density, m2.

math formula

landslide volume probability density, m3.

math formula

cumulated rainfall, m.

math formula

cumulated rainfall for factor of safety equal to 1, m.

math formula

rainfall rate, m s−1.

math formula

water sorptivity, m s[1/2].

math formula

average size of a landslide, m3.

math formula

average landslide size for a theoretical SOC system, m3.

math formula

maximum size of a landslide, m3.

SOC

self-organized criticality.

math formula

ratio of released and applied load.

math formula

time, s.

math formula

time until water is ponding at surface, s.

math formula

time when infiltration rate equals rainfall rate, s.

math formula

water volume in a soil column, m3.

math formula

maximum water volume in a column without free water, m3.

math formula

volume of a landslide, m3.

math formula

minimum landslide volume for validity of power law distribution, m3.

math formula

weight per area acting on soil-bedrock interface, kg/(m s2).

math formula

bedrock elevation, m.

math formula

elevation of center of soil column, m.

math formula

elevation of free water table, m.

math formula

surface elevation, m.

math formula

power law exponent of Pareto distribution.

math formula

slope angle, deg.

math formula

friction angle, deg.

math formula

number of landslides in certain range of landslide size.

math formula

range of landslide volumes, m3.

math formula

general exponent of frequency/magnitude relationship.

math formula

exponent for landslide frequency/area relationship.

math formula

power law exponent for systems in critical state.

math formula

exponent for time interval (waiting time) between two landslides.

math formula

exponent for landslide frequency/volume relationship.

math formula

volumetric water content.

math formula

initial volumetric water content at field capacity math formula = −3.4 m.

math formula

residual volumetric water content.

math formula

maximum volumetric water content.

math formula

effective water saturation of soil column.

math formula

Brooks and Corey parameter of water retention curve.

math formula

density of soil minerals, kg m−3.

math formula

density of water, kg m−3.

math formula

load applied to each intact fiber in a bundle of fibers, kg m s−2.

math formula

normal stress related to weight of soil column, kg/(m s2).

math formula

maximum strength of a fiber in a fiber bundle model, kg m s−2.

math formula

compressive stress, kg/(m s2).

math formula

soil strength provided by capillary forces, kg/(m s2).

math formula

tensile shear strength provided by vegetation, kg/(m s2).

math formula

residual soil strength after failure at base, kg/(m s2).

math formula

shear strength, kg/(m s2).

math formula

basal shear strength reduction in presence of free water, kg/(m s2).

math formula

total tensile strength related to capillary and root effects, kg/(m s2).

math formula

soil porosity.

math formula

proportionality factor relating shear strength and capillary pressure.

Acknowledgments

[78] The work on landslide triggering models was supported by Swiss National Science Foundation SNSF (project 200021-122299, ‘Local and regional hydrologic and geomorphic factors determining landslide patterns’) and the Competence Centre Environment and Sustainability CCES of the ETH Domain (‘Triggering of Rapid Mass Movements in Steep Terrain TRAMM'). We are grateful to Christian Rickli (Swiss Federal Institute for Forest, Snow and Landscape Research, WSL) for providing access to landslide inventory data and Fabian Rüdy for illustrations used in Figures 2 and 3. We greatly appreciated the insightful comments of Roy Sidle, Colin Stark, and an anonymous reviewer on previous forms of this manuscript.

Ancillary