Simplified simulation of rock avalanches and subsequent debris flows with a single thin-layer model: Application to the Prêcheur river (Martinique, Lesser Antilles)
Introduction
The remobilization by water of old or recent volcanic materials, during or even long after an eruption, generates sediment-laden flows called lahars that travel in ravines and rivers tens to hundreds of kilometers away from the volcano (Vallance and Iverson, 2015, Thouret et al., 2020). Thus, they can be major threats to populations and infrastructures. Non-eruptive lahars can be correlated to landslides that create loose debris reservoirs. Numerical simulations considering both the landslide that creates the reservoir and its remobilization as lahars can improve hazard assessment. However, the modeling process is not straight-forward because the initial landslide and the subsequent lahar are two different phenomena.
The initial landslide can take various forms, as water-laden debris avalanches or dry rock avalanches (Hungr et al., 2014). In a first approximation, the physical and rheological properties of materials (such as density or basal friction coefficient) can be considered homogeneous both in space and time, which simplifies the quantification of the propagation (McDougall and Hungr, 2005). In comparison, the subsequent lahars are more complex: they can propagate as hyperconcentrated flows (HFs) or debris flows (DFs). In the following, we will thus talk about lahars to refer to DFs and HFs alike. Following (Coussot and Meunier, 1996, Vallance and Iverson, 2015, Thouret et al., 2020), we define DFs as homogeneous mixtures of water and granular rock material with volumetric solid fraction higher than 60%, similar velocities for the solid and fluid phases and densities above 1800 kg m−3. HFs feature solid fractions between 20% and 60%, a vertical separation of the two phases and densities below 1800 kg m−3. We may expect that the remobilization of a small amount of solid materials will produce HFs, while fast remobilization by liquefaction of a large debris reservoir will turn into a DF (Vallance and Iverson, 2015). However, a DF initiated in the upper section of a river may well turn into HF at its tail because of dilution and settling, while its front increases its solid content due to bed erosion. Further dilution downstream can then transform completely the DF into a HF (for a conceptual view of such a process, see Fig. 2 in Thouret et al., 2020).
The combined effects of particle collision and friction, lubrication, advection and suspension in presence of an interstitial fluid, are difficult to model in a single framework (Andreotti et al., 2013, Delannay et al., 2017). Thus, current solutions where the dynamics of elementary volumes of fluid and/or of each solid particle are considered (in 2 or 3 dimensions) often focus on reproducing some of the physical processes, but never all of them. Discrete element modeling (DEM) is now widely used to model dry and wet granular flows at the laboratory scale (e.g. Durán et al., 2012, Lefebvre-Lepot et al., 2015, Windows-Yule et al., 2016). Applications to field scale simulations are given for instance by Zhao and Shan (2013) and Leonardi et al. (2014) for DFs, and by Yan et al. (2020) and Wu and Hsieh (2021) for rock avalanches. Another approach is to consider a single-phase flow and solve the Navier-Stokes equations (e.g. Hu et al., 2015). However, both DEM and continuous models often require huge computing resources and/or depend on too many user-defined parameters, which is incompatible with the limited knowledge of the flowing material we have in practice.
Over the past decades, thin-layer models have been increasingly used to study debris and rock avalanches, as well as lahars (see McDougall (2017) for a general review, and Thouret et al. (2020) for lahar modeling). Their main assumption is that the landslide thickness is negligible in comparison to its length. In turn, flow description is reduced to flow thickness and flow thickness-averaged velocity, which simplifies greatly the governing equations in comparison to 3D models. In their simplest form, thin-layer models describe an homogeneous flow and dissipate energy solely by considering a stress applied at the base of the flow. For instance, with the Coulomb rheology the only rheological parameter is the friction coefficient μS = tan(δ), with δ the friction angle. If the topographic slope θ is higher than δ the flow accelerates, and decelerates and stops otherwise (inertial effects and spatial variations in flow thickness may change temporarily this first-order behavior). Such models proved to reproduce well rock and debris avalanches as well as debris flows (Hungr et al., 2007, Pirulli and Mangeney, 2008, Favreau et al., 2010, Lucas et al., 2014, Pastor et al., 2018a). More elaborate numerical codes also model, for instance, two-phase flows (Iverson and George, 2014, Bouchut et al., 2015, Bouchut et al., 2016, Mergili et al., 2017, Pastor et al., 2018b), three-phase flows (fluid, coarse solid fraction, fine solid fraction, Pudasaini and Mergili, 2019), and erosion along flow path (Iverson, 2012, Pirulli and Pastor, 2012). However, these developments often rely on empirical relations (e.g. for erosion laws McDougall, 2017). Besides, thin-layer equations with complex rheologies are mostly derived on simple topographies (e.g. Pastor et al., 2009, Baker et al., 2016), and the lack of analytical solutions makes it difficult to test the robustness of associated numerical tools. Furthermore, although complex rheologies may model more realistic dynamics, they come at the cost of an increased number of parameters, such as erosion rates, erodible thickness, viscosity, drag coefficient or densities of each phase (e.g. George and Iverson, 2014, Mergili et al., 2017). These parameters can be difficult to calibrate if not enough data are available. Besides, when they are not known, the high number of degrees of freedom may artificially improve back-analysis studies.
In practice, experts conducting hazard assessment studies may neither have the time nor the financial resources to carry out a thorough analysis with detailed but complex numerical models. The question is: to what extent can we expect realistic results from simple physically based thin-layer models for rock avalanche and DF simulations? The answer strongly depends on the available field data. In this work, we present a modeling approach with empirical but simple rheologies involving no more than two parameters. To enhance the quality of simulation results, we make an extensive use of field data to define realistic simulation scenarios and characterize past events for model calibration. We will use the thin-layer model SHALTOP (Bouchut et al., 2003, Bouchut and Westdickenberg, 2004, Mangeney-Castelnau et al., 2005, Mangeney et al., 2007b), that proved to reproduce accurately analytical solutions for the dam-break problem (Mangeney et al., 2000, Lucas et al., 2007), and was used successfully to model gravitational flows at the field scale with a simple Coulomb friction law (e.g. Favreau et al., 2010, Lucas et al., 2014, Moretti et al., 2012, Moretti et al., 2015, Moretti et al., 2020, Peruzzetto et al., 2019). In comparison to other thin-layer models, SHALTOP also takes into account precisely topography curvature effects that can be significant for fast gravity driven flows (Peruzzetto et al., 2021).
Because they have the highest potential impact on infrastructures and populations, we focus on extreme events (avalanches of volumes >1 × 106 m3, and high discharge DFs). We choose the Prêcheur river in Martinique island (Lesser Antilles, French Caribbean) as study site (Fig. 1), where such events are documented and where stakes are high, as large DFs threaten the Prêcheur village at the mouth of the river (Fig. 2). In a first calibration step, we will use topographic surveys and aerial photographs to construct the initial conditions of (i) a rock avalanche that occurred in 2018 and (ii) a major debris flow that occurred in 2010. Granulometric data help choosing the rheological law, and a range of possible rheological parameters is identified in the literature (see Table 1). By reproducing the travel distance and main dynamic characteristics of the rock avalanche, and the flooded area and travel time of the DF (deduced from aerial photographs and seismic recordings in both cases), we calibrate more precisely rheological parameters. With these fine-tuned parameters, we can then consider the forward prediction of a rock avalanche simulation, whose initial conditions are deduced from geomorphological and geological observations. The resulting deposits are then remobilized instantaneously in another simulation to model the propagation of a high discharge DF. Because in the Prêcheur river rock avalanches do not, in general, transform directly into DFs (Aubaud et al., 2013), we do not consider such a continuous transition in this work.
In Section 2 we present in more details our study site, along with the data used to construct simulation scenarios and calibrate our model. Simulation scenarios used for model calibration and forward prediction are presented in Section 3, and the numerical model SHALTOP is detailed in Section 4. Simulation results are then given in Section 5. In Section 6, we investigate the influence of initiation mechanism on simulation results. The latter are discussed in Section 7.
Section snippets
Data
In this section, we present the geological and geomorphological context of our study site, along with the data used to define simulation scenarios. Topographic surveys will be used to define the bed topography and initial volumes. To calibrate the numerical model, we use aerial photographs that give the travel distances and flooded areas of past events. Seismic recordings are used to estimate flow velocity and duration. The granulometry of deposits is also used to choose the rheology in DF
Simulation scenarios for calibration and forward prediction
We focus on the modeling of extreme events: rock avalanches with volumes above 1 × 106 m3 and high discharge DFs. In the following we present two such events used for model calibration, and explain how we construct the topography and initial volumes for model calibration. A forward-prediction scenario is then described. This is summarized in Table 2.
Numerical model
The SHALTOP thin-layer numerical code simulates the dynamics and emplacement of flows on general topographies (Bouchut et al., 2003, Bouchut and Westdickenberg, 2004, Mangeney-Castelnau et al., 2005, Mangeney et al., 2007a). It has been successfully tested to reproduce both real landslide (e.g. Brunet et al., 2017, Moretti et al., 2015, Peruzzetto et al., 2018) and laboratory experiments (Mangeney-Castelnau et al., 2005, Mangeney et al., 2007a). In SHALTOP, the material layer moving on the
Rock avalanche back-analysis
The travel distance of the RA_2018 rock avalanche scenario with various friction coefficients is displayed in Fig. 8. The extent of the Jan. 4, 2018 deposits (dashed green line in Fig. 8) is best reproduced with μS = tan(14°) =0.25. This is less than μS = tan(18.4°) =0.33, that is derived from the empirical law of Lucas et al. (2014) (see Section 7.1.1 for further discussion). With μS = tan(14°), the flow dissipated energy rate reproduces correctly the main seismic energy increase phase (Fig. 6b, at
Influence of successive destabilizations on rock avalanches simulations
To investigate the influence of retrogressive destabilizations on runout prediction, we release the 1.5 × 106 m3 of the RA_2018 in two successive steps, instead of one. In the resulting RA_2018_2 scenario, 0.8 × 106 m3 are first released at the cliff bottom (A in Supplementary Figure 6a), and the rest (B in Supplementary Figure 6a) collapses 13 s later. The two volumes are constructed arbitrarily by separating the extent of the initial mass of the RA_2018 scenario approximately at the middle of the
Choice of rheological parameters
In this study, the friction coefficient μS used in the rock avalanche forward prediction simulation is chosen after a calibration step, as often done in the literature (e.g. Sosio et al., 2012, Pastor et al., 2018a). To our knowledge, it is difficult to estimate μS directly from physical characteristics of the materials. Indeed, simulations of laboratory experiments involve high friction coefficient (for instance, μS = tan(30°) in Gray et al., 1999) that fail to reproduce deposits and dynamics
Conclusion
In this work, we have modeled a rock avalanche, and the subsequent remobilization of the deposits as a high discharge debris flow, with a single thin-layer numerical code, SHALTOP. SHALTOP is used empirically, with a maximum of two rheological parameters (Coulomb or Voellmy rheology). We focus on extreme events, and in particular high discharge DFs, in a risk conservative approach. The simplicity of the modeling solution is compensated by an extensive use of field data to define realistic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank Janusz Wasowski and two anonymous reviewers for their comments. We gratefully thank the French Ministère de la Transition Ecologique et Solidaire (MTES), the BRGM for funding this work for 2017–2020 as well as the ERC contract ERC-CG-2013-PE10-617472 SLIDEQUAKES and the DEAL Martinique for their contribution. Numerical computations were performed on the S-CAPAD platform, IPGP, France. We also thank the staff of OVSM-IPGP, BRGM Guadeloupe and BRGM Martinique for their contribution to
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