1. Introduction
Besides risk zoning, defence structures such as dams or mounds are used to protect inhabited mountain areas. The interaction between avalanches and obstacles is not fully understood, especially the storing effects. In situ studies continue to be the most direct and certainly the best way to understand and quantify the mechanisms involved in avalanche dynamics and interaction with obstacles. Many instrumented sites including dams exist in various parts of the world. While the results of these studies are awaited, progress continues to be made in related fields, providing a unique source of information for approaching, understanding and modelling avalanche flows. The objective of our research is to determine the hydraulic effects of dams on dry and cohesion-less snow avalanches. Many conceptual behaviour laws have been proposed for snow, but until now none of them has been objectively validated. Recently, interesting scientific progress was made by Reference PouliquenPouliquen (1999) for the granular media. This was first achieved for an ideal granular medium (spherical glass beads and unique grain size). This theory was extended to more complex granular media such as polydispersed grains (Reference Chevoir, Prochnow, Jenkins and MillsChevoir and others, 2001). The dry and cohesion-less snow is usually regarded as granular material; we then considered the Pouliquen effective friction law valid for dry and cohesion-less snow. In this paper, we use a small-scale physical model and numerical model to quantify the effects of dams in terms of retained volumes.
The next section deals with the new deposition mechanism modelling using an empirical friction law. The simple laboratory experiments we performed are described in section 3. Experimental and numerical results are compared in section 4.
2. Deposition Mechanism: Modelling
The granular flow is here simulated using a depth-averaged model. The model uses the theory proposed by Reference Savage and HutterSavage and Hutter (1989) and the friction law proposed by Reference PouliquenPouliquen (1999). An entrainment and deposition model was developed and implemented, and was used to simulate avalanche flows and their interaction with obstacles (Reference Naaim, Faug and Naaim-BouvetNaaim and others, 2003). In the following one-dimensional equations, h is the flow depth and u is the depth-averaged velocity:
α is related to the velocity profile across the layer. Its effect is not significant in this model. We took α equal to 1. k is the ratio of the vertical normal stress to the horizontal normal stress. The isotropy is assumed here and k = 1. θ is the slope angle and μ(u; h) is the effective friction coefficient described in section 2.1. ϕe/d is the erosion–deposition flux detailed in Reference Naaim, Faug and Naaim-BouvetNaaim and others (2003).
The system takes into account the deposition process thanks to the mass-conservation equation. When a given infinitesimal layer (Δh) is deposited, its momentum is transferred to the ground. The new momentum of the moving mass is (h - Δh)u which corresponds to decreasing momentum. This is automatically taken into account by our conservative formulation of the shallow-water equations. Neither source nor sink of momentum is needed in this formulation.
2.1. Friction law
The friction law used is from Reference PouliquenPouliquen (1999). The effective friction coefficient is written as:
where B; μmin= tan(θmin) and μmax = tan(θmax) are parameters depending on the material properties and the bed roughness. B is linked to the parameters β and L defined in Reference PouliquenPouliquen (1999) by the following relation: B = β/Ld; where d is the mean diameter of the particles. We considered that the friction coefficient changes discontinuously at the Froude number 
0.136, below which steady granular flow has been found to be impossible (Reference PouliquenPouliquen, 1999).
2.2. Entrainment and deposition model
The entrainment and deposition model (Reference Naaim, Faug and Naaim-BouvetNaaim and others, 2003) assumes that the erosion/deposition rate is proportional to the excess or deficit shear stress at the bottom and uses the empirical functions hstop(θ) (defined in section 4.1) and hstart(θ). hstart(θ) is deduced from hstop(θ) by hstop
The deposition model is given by:
The erosion model is given by:
In our simulations, we assumed :
corresponding to a linear profile, and we used γ = 1. Through the extended previous works concerning the vertical velocity profile inside the granular flows (from Reference SavageSavage (1979) to Reference Andreotti and DouadyAndreotti and Douady (2001)), different profiles were exhibited. According to the flow and to material conditions, the velocity gradient at the bottom can be higher or smaller than the mean gradient. For simplicity, we approximated the bed velocity gradient by the mean velocity gradient.
The numerical solution of the full equations system is obtained using a finite-volume scheme. The topographic profile is recalculated after each time-step calculation, taking into account the deposition flux. The deposition condition corresponds to the deceleration condition. When it is reached, both deposition and deceleration start to operate. The deceleration decreases the velocity, and the deposition decreases the height. These two processes operate simultaneously and maintain approximately the same shearing rate (u/h) during the stopping process.
2.3. Upstream and downstream boundary conditions
The downstream condition represents the presence of the obstacle. This last is introduced in the model as follows. Three cases are considered:
When ht, the sum of the flow height h and the deposit height hd at the boundary condition, is lower than the obstacle height H, a total reflection of the flow is used.
When the obstacle height is between (ht) and the deposit height (hd), the output flow (Qo) is determined according to the incoming flow (Qi) by:
Finally, when hd exceeds H, the downstream boundary is considered free (Qo = Qi).
The first and last cases require classical treatment. The intermediate case is not common. The flow is clearly three-dimensional.We roughly simplified the problem by assuming the flow rate proportional to the free area. At the down-streamboundary condition, when the flow arrives, the total reflection induces a strong local increase in height. This generates a strong gradient height, which implies the start of deposition according to the deposition model. This process changes the topography till the dam is totally filled with the immobile material. This modification propagates upstream till attaining an equilibrium slope.
3. Experimental Set-Up
The experimental set-up we used consisted of an inclined channel (0.93m long and 0.2 m wide). A fence of height H was placed at the channel downstream end. A fixed volume of granular material was stored in a box (0.29m long and 0.2 mwide) situated at the top of the channel. The granular mass was released from the box, and the volume stored upstream of the fence was measured. The granular material comprised glass beads with a mean diameter of 1 mm.
Our study investigated the influence of two parameters on the stored volume Vs: (i) the channel inclination θ, and
(ii) the fence height H. A schematic view of the experimental set-up can be seen in Figure 1.
Fig. 1. Schematic view of the experimental-up.
4. Results Analysis
4.1. Empirical friction-law parameters
As seen in section 2, empirical friction parameters are needed for implementation in the numerical model. The function hstop(θ) corresponding to the thickness of the granular layer left by a steady uniform flow at the inclination θ was determined empirically by a specific experimental procedure detailed in Reference Pouliquen and RenautPouliquen and Renaut (1996), Daerr and Douady (1999), Reference PouliquenPouliquen (1999) and Reference Pouliquen and ForterrePouliquen and Forterre (2002).
The granular material used was glass beads flowing down a sandpaper roughness. The hstop(θ) empirical function corresponding to our experimental conditions is given in Figure 2. The value of θmax corresponding to hstop = 0 was determined accurately and was found to be equal to θmax = 28 ± 0.5˚. Several couples (θmin; B) can be used to fit the experimental data with the equations in section 2.1. The extreme values we obtained lead to:
Fig. 2. Variation of the stopping angle as a function of the thickness h. This curve was obtained for glass beads 1 mm in diameter and flowing down a rough plane with sandpaper roughness. Uncertainties were estimated around Δθ = 0.5˚ for the slope angle and θh = 1 mm for the thickness h of the granular layer.
The following parameters were used to describe the friction law implemented in the numerical model: μmax = 0.53 (θmax = 28˚), μmin = 0.38 (θmin = 21˚) and B = 34m-1.
4.2. Influence of the obstacle height
The empirical friction law implies three distinct types of deposition mechanism:
θ < θmin: the entire released volume is deposited before interacting with the fence. This case wasn’t therefore treated;
θmin < θ < θmax: a steady regime can be reached before the flow reaches the fence. The volume stored behind the dam comes from two contributions: the volume retained by the dam, and the volume stored along the channel. The latter is represented by the function hstop(θ). In our tests, we chose the slope corresponding to the average value: (θmax þ θmin)/2 = 24.5˚;
θ > θmax: the volume stored behind the obstacle results only from the obstacle effects (hstop(θ) = 0). We chose to study the case of θ = 29.5˚.
Experimental and numerical results are presented in Figure 3. The released mass was fixed to 7 kg, and the obstacle height was varied from H = 1cm to H = 8 cm. One can observe that the model correctly reproduces the experimental data. However, concavities of numerical and experimental curves are opposite for low obstacle heights, and the error increases when the obstacle height decreases, except for values close to zero where the curves converge: at zero for the case of θ > θmax and at Vs(hstop(θ))/V for the other case. The numerical model overestimates the volume stored for low obstacle heights and underestimates it, but less substantially, for higher obstacle heights.
Fig. 3. The retained volume and initial volume ratio according to obstacle height and initial height ratio: comparison between the computed and observed data for two slope angles, 24.5˚ and 29.5˚.
4.3. Influence of the channel inclination
The released mass was 7 kg, and the channel inclination was varied from θ = 22 to θ = 38°. The ratio of the fence height to the height of the initial volume released, H/hi, was fixed at 0.3 in order not to saturate the volume stored behind the obstacle (Vs = released volume) for low slope inclinations. As shown in Figure 4,
Fig. 4. Retained-volume/initial-volume ratio according to slope angle at fixed initial volume and fixed obstacle height (H/hi = 0.3).
the computed data and the experimental data are quite similar and the curves have the same concavities, the slope effect is well reproduced by the model
the computed data overestimate the stored volume
starting from the same value at low slope angle (22˚), when the slope angle increases, the computed data decrease more weakly than the observed ones.
However, we have to remember that the friction parameters (θmin; B) were determined empirically with a significant error for θmin (21˚ ± 1˚) and B (34m–1 ±7m–1), as explained in section 4.1. These errors partially explain the discrepancies in Figures 3 and 4. The model was built using quasi-steady uniform conditions, whereas the considered flows are far from these conditions. Furthermore, the impact effect when the front attains the obstacle is not considered in the model. Finally simple boundary-condition treatment was used.
5. Conclusion
The presented work, combining numerical and experimental approaches, allowed study of the deposition volume upstream of a fence. Granular material and granular behaviour laws were exploited. For different slope angles and different fence heights, we quantified the retained volume and showed relatively good agreement between the physical experiments and numerical simulations. In the case of dry granular behaviour flowing over a rough bed, we identified two contributions. The first is due to the fence and exists for all the slopes. The second is due to the friction along the rough channel and appears only for slopes where steady flow is possible. This latter modifies significantly the evolution of the retained volume according to the slope angle.
Acknowledgements
The authors are grateful to D. Bertrand, P. Lachamp, H. Bellot and F. Ousset who helped in performing the experiments.
