Shaking conditions required for flame structure formation in a water-immersed granular medium
© Yasuda and Sumita; licensee Springer. 2014
Received: 25 November 2013
Accepted: 28 May 2014
Published: 20 June 2014
Flame structures found in sedimentary rocks may have formed from liquefaction and gravitational instability when the sediments were still unconsolidated and were subject to shaking caused by earthquakes. However, the details of the process that leads to the formation of the flame structure, and the conditions required for the instability to initiate and grow remain unclear. Here, we conduct a series of small-scale laboratory experiments by vertically shaking a case containing a water-immersed layered granular medium. The upper granular layer consists of finer particles and forms a permeability barrier against the interstitial water which percolates upwards. We shake the case sinusoidally at different combinations of acceleration and frequency. We find that there is a critical acceleration above which the instability develops at the two-layer interface. This is because the upward percolating water temporarily accumulates beneath the permeability barrier. For larger acceleration, the instability grows faster and the plumes grow to form a flame structure, which however do not completely penetrate through the upper layer. We classify the experimental results according to the final amplitude of the instability and construct a regime diagram in the parameter space of acceleration and frequency. We find that above a critical acceleration, the instability grows and its amplitude increases. Moreover, we find that the critical acceleration is frequency dependent and is smallest at approximately 100 Hz. The frequency dependence of the critical acceleration can be interpreted from the combined conditions of energy and jerk (i.e., the time derivative of acceleration) of shaking, exceeding their respective critical values. These results suggest that flame structures observed in sedimentary rocks may be used to constrain the shaking conditions of past earthquakes.
There is considerable evidence which indicate that there is a critical condition for liquefaction and subsequent instability to occur. Compilation of liquefaction-related phenomena shows that there is a minimum seismic magnitude, which increases with epicentral distance, required for these phenomena to occur (Manga and Brodsky 2006; Manga and Wang 2007). The threshold at which liquefaction occurs has also been investigated using laboratory experiments. For example, oscillatory deformation experiments of water-saturated granular matter and simultaneous pore-water pressure measurements have been conducted to constrain the strain needed for liquefaction (e.g., Vucetic 1994). Experiments for the situation in which a permeability barrier exists have also been conducted to study the formation process of water film and sand boils. In these studies, a one-dimensional tube test, to which an instant shock is applied (Kokusho 1999; Kokusho and Kojima 2002), or horizontal shake tables (Kokusho 1999; Yamaguchi et al. 2008) were used. Experiments more closely simulating natural conditions have also shown that a variety of structures can form as a consequence of shaking (Moretti et al. 1999).
On the other hand, the response of a dry granular matter under vertical shaking has intrigued physicists for a long time. Experiments have revealed instabilities such as surface waves, the Brazil nut effect, and granular convection (Duran 2000). Detailed parameter studies have shown that a variety of instabilities can occur, depending on the combination of acceleration and frequency of shaking (Burtally et al. 2002). Recently, shaking conditions required for granular convection have been investigated (Hejmandy et al. 2012). Experiments for liquid-saturated cases have also been conducted and have revealed that a number of novel phenomena can occur (e.g., Schleier-Smith and Stone 2001; Voth et al. 2002; Leaper et al. 2005; Clement et al. 2009, 2010).Here, we consider the situation observed in Figure 1 and attempt to explain how the observed structures may have formed, and consider constraining the required shaking conditions. In order to answer these questions, we conduct a series of experiments in which a water-immersed granular medium with a permeability barrier is shaken vertically under a range of accelerations and frequencies.
where Δρ is the particle-water density difference, g is the gravitational acceleration, and η is the viscosity of water. An efficient size grading occurs for a mixture of 0.05 mm and 0.22 mm particles, which can be understood as follows. The approximate fraction F of the small particles that can be included in the lower layer (consisting primarily of large particles) can be estimated using the time scale for a particle to settle ∼H/V s. Assuming that the particles are initially homogeneously distributed, the fraction F of the small particles that have settled during the time needed for all of the large particles to settle is F ∼ (H/V s(la))/(H/V s(sm)) ≃ 0.06, where the subscripts sm and la indicate small and large particles, respectively. A small F(≪1) value indicates an efficient size grading. We also note that, since permeability scales as ∝d2 (e.g., Andreotti et al. 2013), this particle size ratio results in a permeability ratio of 19, and the upper layer becomes a low permeability layer.
where A is the amplitude and f is the frequency. An accelerometer (352A24, PCB Piezotronics, New York, USA) is attached to the case, and the signal is amplified (482C, PCB Piezotronics, New York, USA) and recorded by a digital oscilloscope (ZR-RX70, Omron, Kyoto, Japan). We use a high-speed camera (EX-F1, Casio, Tokyo, Japan) at 300 fps to record the motion within the cell. An LED lamp is used to synchronize the acceleration and image data.
We calculate the acceleration from the voltage output of the accelerometer. Here, we calculate the root-mean-square of the acceleration data to obtain the average peak acceleration. This gives the exact peak acceleration for a sinusoidal shaking.
We conduct experiments under a peak acceleration range of 1.4 to 78.3 m/s2 and a frequency range of 10 to 5,000 Hz. For comparison, this frequency range extends beyond the typical high frequency response limit (approximately 10 Hz) of seismometers (Shearer 1999). Digital images are analyzed using MATLAB. The spatial resolution of the images is 0.118 ± 0.002 mm per pixel. We binarized the images for the time interval of t = 0 to 9.9 s to obtain the heights of the granular layer surface and the two-layer interface, and analyzed how these heights change with time.
Growth of an instability
Acceleration and frequency dependence
Additional file 2: Movie 2 in QuickTime. Acceleration dependence at a fixed frequency of 50 Hz replayed at ×0.4 speed. These are four selected examples from those shown in Figure 5b. Accelerations in m/s2 (and the resulting regime) are 2.1 (Ib: percolation), 7.9 (II: transition), 19.3 (II: transition), and 40.5 (III: flame), respectively. (MOV 7 MB)
Additional file 3: Movie 3 in QuickTime. Frequency dependence at a fixed acceleration of 5.0 ± 0.4 m/s2 replayed at ×0.4 speed. These are four selected examples from those shown in Figure 6b. Frequencies in Hz (and resulting regime) are 10 (Ib: percolation), 100 (II: transition), 150 (III: flame), and 3,000 (Ib: percolation), respectively. Note that the flame structure forms at the intermediate frequency of 150 Hz. (MOV 7 MB)
We conducted a total of 73 experiments with different combinations of shaking acceleration and frequency, and classified the results into the following four regimes using the relative amplitude δ z′ and the total compaction of the whole granular layer δ h at t = 9.9 s (i.e., 4.9 s after the shaking stops): regime Ia (no change), when δ z′ < 0.1 mm and δ h < 0.1 mm; regime Ib (percolation), when δ z′ < 0.1 mm and δ h ≥ 0.1 mm; regime II (transition), when 0.1 ≤ δ z′ < 0.6 mm; and regime III (flame), when δ z′ ≥ 0.6 mm. Here, regimes I to III are defined using the same threshold values of δ z′ used to define stages I to III, but at t = 9.9 s. δ h is used to subdivide regime I into regimes Ia and Ib. For all experiments in regimes II to III, δ h ≥ 0.1 mm, and therefore, we do not use δ h to classify these regimes. For the amplitude data shown in Figures 5b and 6, the different resulting regimes are indicated by the line thicknesses.
Similar non-dimensionalizations have been used previously (e.g., Pak and Behringer 1993; Melo et al. 1995; Duran 2000; Schleier-Smith and Stone 2001; Burtally et al. 2002; Voth et al. 2002; Leaper et al. 2005; Eshuis et al. 2005, 2007). Here, we define the onset of instability when the relative amplitude is δ z′ ≥ 0.1 mm (i.e., onset of the transitional regime). Figure 7b indicates that the minimum acceleration for the instability to occur is Γ = 0.58 at a frequency of f′ = 2.5.
Effects of other parameters
In addition to the shaking conditions, there are many other changeable parameters in our experiments. Here, we briefly describe the effects of other parameters which help to better understand the dynamics occurring in our experiments. Here, we define the experiments for the parameters shown in Figure 2 as the reference case.
First, we describe the results of two dry experiments (i.e., without water) and otherwise the same as the reference case. We find that the results are strikingly different. Most importantly, flame structure does not form in the dry experiments. At m/s2 and f = 50 Hz, the small particles comprising the upper layer settle into the pore space of the large particles comprising the lower layer and formed a mixture layer whose thickness increased with time. Furthermore, the granular layer as a whole gradually tilts. At m/s2 and f = 50 Hz, in addition to tilting, a large-scale whole layer granular convection with a length scale comparable to the case width occurs.
Second, we describe the results of experiments using a wider case (width 194 mm, height 104 mm, thickness 26 mm) and otherwise the same as the reference case. From a series of experiments at fixed accelerations of and 8.8 m/s2 and different frequencies, we find that the final amplitude reached a maximum at a frequency of approximately 100 Hz, the same as the reference case. We also find a similar frequency dependence of the critical acceleration and a similar wavelength of the instability of λ ∼ 10 mm, indicating that the effects of width dimension are not evident.
Third, we describe the results of different layer thicknesses. When there is no permeability barrier (i.e., one layer consisting of a particle size d = 0.22 mm with a thickness 21.9 mm), neither sand boil nor flame structure formation occurs. We also compare three experiments with the same thickness 33.8 ± 2.0 mm of the granular layer but with different thickness fractions of the upper layer: 0.19, 0.28, and 0.78. For m/s2 and f = 50 Hz, for all three cases, a similar flame structure forms. However, we find that the relative amplitude at t = 9.9 s decreases by 35% as the upper layer becomes thicker.
The origin of the flame structure
We consider that the flame structure in our experiments forms as a consequence of Rayleigh-Taylor type instability and not from surface instability, resonance, or granular convection. There seem to be at least three requirements for Rayleigh-Taylor instability to occur: the presence of a permeability barrier, a large fluid viscosity, and granular matter being dense. We explain each of these in detail below.
First, whenever a flame structure developed, it always formed at the permeability barrier. This indicates that the accumulation of water at the permeability barrier is necessary for the flame structure to form, from which we infer that the instability is of Rayleigh-Taylor type.
Second, granular convection did not occur in the water-immersed experiments, whereas it did occur in the dry experiments. On the other hand, in the dry experiments, the small particles comprising the upper layer settled downwards and the flame structure did not form. Here, we evaluate the importance of viscosity by comparing the terminal velocity given by the Stokes velocity V s (Equation 1) and the free fall velocity where the particle diameter scale d is taken as the height scale. The ratio of these two velocity scales is the Stokes number St = V s/V f (e.g., Andreotti et al. 2013). If S t ≪ 1, viscous drag is large such that the terminal velocity is attained within a particle diameter scale (viscous regime), and vice versa for S t ≫ 1 (free-fall regime). If we use the particle size d of the upper layer (d = 0.05 mm), for the water-immersed case, we obtain S t ∼ 0.08 < 1. In contrast for the dry case, we obtain S t ∼ 6 > 1. This suggests that the suppression of particle motions by viscosity played a role for the accumulation of water at the permeability barrier.
Third, in our experiments, the particles were initially closely packed, and even when they were shaken, we did not observe any standing surface waves indicating that the particles remained close to a close packed state. Even for the dry experiments, granular matter did not detach from the bottom of the case or bounce (e.g., Eshuis et al. 2007). The thickness of the whole layer scaled by the particle size is N ∼ 270. A large value of N results in a larger frictional dissipation (Eshuis et al. 2007), which causes the granular matter to remain dense even when shaken.
An interpretation of a regime diagram
Here, we consider the origin of the critical acceleration and its frequency dependence. Our experiments indicate that the instability occurs when Γ > Γc = 0.58, which we indicate by vertical lines in Figure 7a,b. The critical value Γc = 0.58 can be understood in terms of the force balance. The inertial stress of a granular layer is where h is the thickness scale. Frictional stress is ∼μ Δρ gh, where μ is the friction coefficient. μ estimated from the angle repose θ r of a pile of glass beads is μ = tanθ r ∼ 0.4 to 0.5 (Samadani and Kudrolli 2001; Higashi and Sumita 2009). It follows that the ratio of the inertial stress to the frictional stress is ∼ Γ /μ, where we used the definition for Γ (Equation 3). Inertial stress must exceed the frictional stress to displace a particle from which we expect Γ > μ is required for instability to occur. Our experimental result of Γ > 0.58 for instability to occur is consistent with Γ > μ ∼ 0.4 to 0.5.
We use Equation 6 and draw a critical line in Figure 7a,b for S c = 0.01, which separates regimes Ib and II fairly well. We note however that the high frequency limit may alternatively be interpreted in terms of critical shaking velocity v c (Hejmandy et al. 2012), since the critical velocity line also scales as .
In Figure 7a,b, we draw a line for J c = 290, which separates regimes Ib and II fairly well. The conditions under which these three dimensionless parameters are supercritical, i.e., Γ > Γc, S > S c, and J > J c, correspond to the colored domain bounded by the three lines in Figure 7a,b.
Finally, we derive the frequency of approximately 100 Hz at which the critical acceleration is minimum. From combining the conditions S c ∼ 0.01, J c ∼ 290, and Γ ∼ 0.22, which is the value of Γ at the intersection of the critical S, J lines (see Figure 7b), we obtain f′ = f · d/V s ∼ 3, which corresponds to approximately 100 Hz. We also obtain A ∼ 0.05d which is the amplitude of shaking relative to the particle size comprising the upper layer under this condition.
Comparison with the sedimentary rocks
Our experimental parameters (particle size pair, densities, layer thickness ratios) were not chosen to closely match or scale those of the sedimentary rocks in Figure 1. Therefore, our regime diagram (Figure 7) cannot be directly applied to these outcrops. If one wishes to constrain the exact shaking condition required for the flame structure in these outcrops to form, a similar set of experiments using the same particles from these outcrops need to be conducted. Nevertheless, we consider that the basic features of the regime diagram will remain unchanged.
Here, we point out several features which are common with the flame structures in Figure 1 and in Figure 3. First is that they both have upward pointing cusps. Similar cusps have been previously observed in Rayleigh-Taylor instability of a liquid-immersed granular medium (e.g., Michioka and Sumita 2005; Shibano et al. 2012). In addition, for both cases, the preserved flame structures do not fully penetrate through the upper layer. Second is that in the experiments, even when liquefaction occurs, only the uppermost part of the lower layer seems to be fluidized. We may approximate the situation in which water accumulates beneath the permeability barrier as a thin low viscosity layer with a thickness δ underlying a thick high viscosity layer. Here, we use the results of linear stability analysis (Whitehead and Luther 1975), from which we can constrain δ from the wavelength of instability λ as δ < λ/π. It follows that for λ ∼ 21 mm, δ < 7 mm, implying that only the uppermost part of the coarse lower layer is fluidized. Similarly, for the outcrops, using the wavelength of λ ∼ 40 mm, we obtain δ < 13 mm, suggesting that only the uppermost one fifth or less of the tuff layer (layer 2 in Figure 1) became fluidized. Field observations indicate that the thickness of the tuff layer varies in the range of 20 to 80 mm, whereas the wavelength is found to be around λ ∼ 40 mm. The constancy of the wavelength suggests that the thickness of the fluidized layer is also approximately constant, regardless of the thickness of the tuff layer.
We conducted a series of experiments in which a two-layered water-immersed granular medium, where the upper layer forms a permeability barrier, is shaken vertically at different combinations of accelerations and frequencies. We find that above a critical acceleration, the instability develops at the two-layer interface and grows. For a sufficiently large acceleration, a flame structure which is similar to those observed in sedimentary rocks forms. We also find that the critical acceleration is frequency dependent and is minimum at approximately 100 Hz. In other words, there is an optimum frequency band in which the flame structure most easily develops. These results were interpreted by combined conditions of inertial stress, energy, and jerk of shaking exceeding their critical values. Although further work is needed to clarify how the optimum frequency band depends on the parameters such as fluid viscosity and particle size, our experiments suggest that the occurrence of a flame structure may be used to constrain the shaking conditions when these structures formed.
We thank Y. Shibano and A. Namiki for discussions and technical advice, and the two anonymous reviewers for helpful comments on the manuscript. This work was supported by KAKENHI (22109505, 24244073, 24510246).
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