SNOW SLAB FAILURE DUE TO BIOT-TYPE ACOUSTIC WAVE PROPAGATION Rolf Sidler 2 Simon Fraser University, 8888 University Drive, Burnaby V5A1S6, Canada ABSTRACT: Even though seismic methods are among the most used geophysical methods today, their application in snow has been sparse. This might be related to the fact that commonly observed wave velocity, attenuation and reflection coefficients can not be well explained by the widely used elastic or visco-elastic models for wave propagation. Biot s well established model of wave propagation in porous media instead is much better suited to describe acoustic wave propagation in snow. This model predicts also a second kind of compressional wave that is commonly observed in snow, but not predicted by standard seismological models. The application of Biot s model is, however, notoriously difficult due to the large property space of the underlying porous material. Here we use the well defined properties of ice and air as well as empirical relationships to reduce the property space. This model predicts velocities of the shear and compressional waves of the first- and second-kind that compare well to measurements from the literature. Using this approach we construct realistic models of snow packs and evaluate snow-slab failure conditions due to wave induced stresses exceeding maximum snow strength by solving Biot s equations numerically. This has implications for the use of acoustic waves to evaluate snow properties and to predict snowpack failure due to the propagation of acoustic waves which is an important mechanism for explosive avalanche control and skier triggered avalanche release. KEYWORDS: avalanche control, modeling. 1. INTRODUCTION Acoustic wave propagation in snow has been investigated for many decades, but applications have been relatively sparse (Smith, 1969; Yamada et al., 1974; Gubler, 1977; Johnson, 1982). This might be due to to the porous nature of snow that can lead to seemingly inconsistant observations of acoustic velocities in a snowpack. Porous materials are inherent of an additional wave mode called the compressional wave of the second kind or simply the slow wave that is not predicted by the standard elastic or visco-elastic models of seismology (Biot, 1956; Smeulders, 2005). For most materials of seismological interest the slow wave mode is diffusive and mainly has the effect of attenuating the compressional wave. This behavior can be incorporated within a visco-elastic model. However, in fresh snow with low density the slow wave is a propagating wave mode and can be readily measured (Oura, 1952). Therefore a solution of Biot s (1962) differential equations is necessary to obtain an accurate description of acoustic wave propagation in snow. One of the main challenges for working with a Biottype porous material is the large property space that has to be determined. No less than ten properties of * Corresponding author address: Rolf Sidler, Simon Fraser University, 8888 University Drive, Burnaby V5A1S6, Canada email: rsidler@gmail.com the material have to be known. Here, we try to reduce this property space by considering the porous frame to consist of ice and the pore fluid to be air. This assumption reduces the unknowns to five. We further reduce the material properties to a single degree of freedom by using empirical relationships. Using this reduction model, we create a realistic model of a snowpack by assuming a exponential decrease of porosity from top to bottom and adding variability using a stochastic variable (Sidler and Holliger, 2005). We then solve Biot s (1962) equation of wave propagation with a pseudo-spectral approach to obtain the stress field as a function of time and evaluate snow failure by comparing the modeled stress field to the maximum stresses endured by the snowpack. 2. SNOW MODEL FOR POROUS MEDIA To reduce the property space of a porous snow model to one single degree of freedom we use the assumption that the porous frame of snow consists of ice and the pore space is filled entirely with air. In addition, to express the remaining four properties as a function 146
of porosity we use empirical relationships. The corresponding values and functions are shown in Table 1, where φ denotes the porosity. TABLE 1. Material properties for a Biot-type porous snow model as a function of porosity. Porous frame frame bulk modulus, K S 10 GPa matrix bulk modulus, K m K S (1 φ) 30.85 (7.76 φ) 3 shear modulus, µ S 2 density, ρ K m(1 2ν) 1+ν (1 φ) 916.7 kg permeability, κ 0.2 ((SSA) 2 (1 φ) ) 2 1 tortuosity, T 2 1 + 1 φ Pore fluid density, ρ a 1.29 kg/m 3 viscosity, η a 1.7 10 5 Pa s bulk modulus, K a 1.4 10 5 Pa φ 3 m 3 FIGURE 1. Frame bulk modulus of snow as a function of porosity. The solid line corresponds to the function given in Table 1. The signs indicate measurements. The equations for the matrix bulk modulus and the tortuosity are based on the Krief equation (Garat et al., 1990) and Berryman s (1980) considerations bases on geometry, respectively. To compute the shear modulus, the Poisson s ratio ν needs to be known. For snow the Poisson s ratio is by itself a function of porosity (Mellor, 1975). For simplicity we choose the linear relationship (1) ν = 0.38 0.36 φ, for this purpose. The permeability additionally depends on the specific surface area (SSA) of snow which is the pore surface for a specific volume (m 1 ). An empirical relationship for specific surface area as a function of density has been given by Domine et al. (2007) (2) SSA = 308.2 log(ρ) 206.0, but has to be applied with care as the values for SSA become negative for densities above ρ s = 512 kg/m 3. The frame bulk modulus and the shear modulus as a function of porosity are shown in Figures 1 and 2 respectively. In Figure 1 measurements from Smith (1969); Johnson (1982); Schneebeli (2004) and Reuter et al. (2013) for the frame bulk modulus are shown for comparison. In Figure 2 the empirical relationship for the shear modulus is compared to measurements of shear modulus by Smith (1969) and Johnson (1982). FIGURE 2. Shear modulus of snow as function of porosity as given in Table 1. The signs indicate measurements. To obtain the compressional wave velocities of the first and second kind in snow as a function of porosity, we solve Biot s (1956) differential equation of wave propagation with a plane wave approach (e.g., Pride, 2005; Carcione, 2007). The resulting velocities for the frequency of 1 khz are shown in Figure 3 and are compared to measurements from (Johnson, 1982). 147
FIGURE 3. Wave velocities for 1 khz for the compressional wave of the first and second kind are shown with solid and dashed lines, respectively. Signs indicate measurements presented by Johnson (1982). FIGURE 4. Maximum compressional and shear strength of snow. The black and red solid lines denote the strength to uniaxial and shear stresses, respectively. For comparison, maximum strength of compilations from Mellor (1975) and Shapiro et al. (1997) are shown with dotted lines for uniaxial stress and with dashed lines for the shear stress. 4. SNOWPACK FAILURE DUE TO ACOUSTIC WAVES 3. SNOW FAILURE In order to evaluate the conditions where the propagating wave field leads to a failure of the snowpack the maximum compressional and shear stresses have to be known. The nature of failure depends strongly on definition and correlates in some extent with the Young s modulus of the material. Mellor (1975) therefore suggests to use a fraction of the snow s Youngs modulus to define the maximum stress snow may withstand before it starts to fail. Here we use a fraction of 1 10 3 for the bulk modulus and a fraction of 0.5 10 3 for the shear modulus to define the stresses at which failure occurs. The corresponding shear and compressional strength as a function of porosity are shown in Figure 4 and compared to the compilation of snow strength measurements presented by Mellor (1975) and Shapiro et al. (1997). In the following a realistic model of a snowpack is presented by creating a realistic porosity distribution and using the relationships given in Table 1 to obtain a Biot-type porous model. We then simulate full waveform propagation and finally apply the snow strength criteria presented in Section 3 to obtain the locations where snow failure occurs. For the two dimensional porosity distribution of snow we assume an exponential decrease of porosity with depth with a porosity of 0.7 at the surface and a porosity of 0.5 at the bottom of the snowpack. To express the heterogeneity of the snowpack we add a two dimensional stochastic variable generated with the spectral method and based on a vonkármaán covariance function with a Hurst parameter of 0.8 and correlation lengths of 0.1 and 10 in vertical and horizontal direction, respectively (Sidler and Holliger, 2005). The resulting porosity distribution is shown in Figure 5. 148
FIGURE 5. Porosity distribution of a realistic snow pack based on an exponential decrease of porosity with depth and an additional stochastic component. The gray area indicates the air above the snowpack which is part of the model. FIGURE 6. Snapshot showing the deviatoric stress field after 6.8 milliseconds for the Biot-type poroelastic model based on the porosity distribution of Figure 5. The locations where the wave field has exceeded the maximum shear strength in the past are indicated with red in the background. 5. CONCLUSIONS By applying the relationships given in Table 1 and using the additional equations (1) and (2) the remaining nine properties of a Biot-type porous model can be obtained and full wave propagation can be simulated by solving Biot s (1962) differential equations with a numerical approach. For accurate heterogeneous poroelastic wave propagation an extremely dense grid spacing is necessary due to the slow wave which in general has a considerably shorter wave length than the fast compressional wave and the shear wave (Sidler et al., 2013). Here we use a grid with 385 x 185 nodes for a model with dimensions 19.2 x 3.8 m and solve the differential equations with a pseudo-spectral approach which is known to be efficient and accurate, but tends to be unstable if not properly implemented. To evaluate the locations where the maximal strength of the snowpack is exceeded during the time of the simulation snapshots of the tensional and deviatoric stress field are written out within short time intervals. All grid nodes of such snapshots are then evaluated for their specific maximum tensional and shear strength which are a function of the porosity at this location (see Figure 4). If a location exceeds its maximum strength it is marked as a point of failure. Figure 6 shows a snapshot of the deviatoric stress field 6.8 milliseconds after a horizontal shear source with the waveform of a Ricker wavlet and a central frequency of 1 khz has been exited. The locations where the maximum shear strength has been exceeded during the time of the simulations are indicated in red color behind the deviatoric stress field. We present a method to evaluate snow failure due to acoustic wave propagation in a poroelastic material. By assuming that the snow frame consist entirely of ice, the pore space of snow is filled entirely with air, and the use of empirical relationships for bulk matrix modulus, shear modulus, tortuosity, and permeability it is possible to obtain a Biot-type porous model of snow as a function of single variable, which we choose to be the porosity. By using this reduction model we obtain a realistic model of a snowpack by assuming a exponential decrease of porosity with depth and adding a stochastic variable to express the heterogeneity of the snowpack. Wave propagation in this snow model is then simulated by solving Biot s differential equations with a pseudo-spectral approach. To identify the locations of snow failure due to acoustic wave propagation the stress conditions in the snowpack are evaluated at small time intervals for the exceedance of the maximum axial and shear stresses. This methods allows to quantitatively explore acoustic wave propagation as a trigger for avalanches. A mechanism that is known to be of significant importance, but is rather enigmatic due to the lack of proper understanding. 6. AKNOWLEDGEMENTS This research has been founded by the Swiss National Sciences Foundation. References Berryman, J. G., 1980: Confirmation of Biot s theory. Applied Physics Letters, 37, 382 384. 149
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