Collapse of a liquid-saturated granular column on a horizontal plane

Laboratory experiments on the collapse of a liquid-saturated granular column on a horizontal plane are reported. The trigger and the resulting dynamics of the collapse when occurring, as well as the shape of the final deposit, are characterized and analyzed in light of some dimensionless parameters, namely, the “column” Bond number Bo, the grain diameter to capillary length ratio $d/l_c$, the initial aspect ratio $a$, the Stokes number St, and the initial volume fraction $\varphi$, by varying the properties of the interstitial fluid and of the grains, the geometry, and the compaction of the initial granular column. The main contribution of this study is to: (i) provide a diagram of the different regimes of collapse shown to be mostly controlled by capillary effect, (ii) develop simple criteria that capture the transitions between each regime in terms of critical values of the Bond number and the ratio of the grain diameter to the capillary length, (iii) extend a predictive model of the runout for dry collapses to the more general case of liquid-saturated granular collapses, and (iv) quantify the influence of $a$, St, and $\varphi$ on the collapse dynamics and the shape of the final deposit in the capillary-free regime. A perspective description of the role of the interstitial fluid on the spreading of the granular medium, and more particularly of the driving role of the fluid, is discussed and argued on the basis of the present set of experiments.

Figure 1

Sketch of the experimental setup: A liquid-saturated granular column, defined by its initial length $L_i$ and its initial height $H_i$, initially on the left side of a sluice gate, is released into air and eventually forms a deposit characterized by the runout length $L_f$ and the final height $H_f$.

Figure 2

Snapshots of some water-saturated granular collapses for various grain diameters: (a) $d=120\mu \mathrm{m}$, (b) $d=500\mu \mathrm{m},$ and (c) $d=5\mathrm{mm}$. In Fig. 2, the failure angle ${\theta }_f$ with respect to the horizontal plane is indicated. In all cases, the initial aspect ratio is $a\approx 1.3$ and the initial volume fraction is $\varphi \approx 0.64$ (i.e., initially densely packed).

Figure 3

Diagram of the different regimes of collapse of an initially densely packed water-saturated granular column in the parameter space $(d,H_i)$ with $L_i=[3:15]\mathrm{cm}$.

Figure 4

Diagram of the different regimes of collapse of an initially densely packed water-saturated granular column in the parameter space $(d/l_c,\text{Bo})$: $\blacksquare$ static; $◆$ fluid-leaking; $▲$ block-avalanche; $⚫$ continuous-avalanche. Red dashed line: Eq. (1) with $\varepsilon =5.5\pm 0.5$; gray band: Eq. (2) with $\varepsilon =5.5$, $\varphi =0.64$ and ${22}^{\circ }\le \delta \le {28}^{\circ }$; blue area: region where $d_c/l_c\le d/l_c\le d_c^{\prime }/l_c$ with $d_c/l_c$ and $d_c^{\prime }/l_c$ given by Eqs. (3) and (4), respectively.

Figure 5

(a) Runout length $L_f/L_i$ and (b) final height $H_f/H_i$ as a function of the initial aspect ratio $a=H_i/L_i$. Symbols correspond to experiments performed in the block-avalanche regime ($▵$) and in the continuous-avalanche regime ($⚪$), while colors indicate the value of the Bond number $\text{Bo}=\rho g{H}_{i}d/\sigma$ (see color bar). Cross symbols correspond to the dry case with the same set of grains.

Figure 6

(a) Runout length $L_f/L_i$ in the parameter space $(a,\text{Bo})$. Experiments: ($▵$) block-avalanche regime, ($⚪$) continuous-avalanche regime. (b) Runout length $L_f/L_i$ as a function of Bo, with $a=2$. The Bond number is varied through the grain size $d$ ($d$ increasing from light to dark gray), the initial height of the granular column $H_i=[20,30]\mathrm{cm}$ (circles and diamonds, respectively) and the surface tension $\sigma \sim [0.03,0.07]\mathrm{N}{\mathrm{m}}^{-1}$ (closed and opened symbols, respectively). Predictions of Eq. (5): (- $·$ - $·$ -) balance of the driving pressure term and the friction term induced by capillary pressure [first and third terms on the right-hand side of Eq. (5)] with $\varepsilon \mu =2.2$, i.e., $\varepsilon =5.5$ and $\mu =0.4(\approx tan{\theta }_r)$; (- - -) inertialess solution with $\lambda \to \infty$, $\varepsilon \mu =2.2$ and $\mu =0.04$; (—) full solution with $\lambda =4.8$, $\varepsilon \mu =2.2,$ and $\mu =0.04$. The initial volume fraction is set to $\varphi =0.64$.

Figure 7

Sketch of the temporal evolution of the granular front position $x_f-L_i$ for a liquid-saturated granular collapse in the continuous-avalanche regime. The dynamics is decomposed into five stages: three stages where the front of the column is at rest ($I$, $III,$ and $V$) and two stages where the front is spreading ($II$ and $IV$). Indices $t$ and $r$ used in the definition of the transition times and distances denote “trigger” and “rest”, respectively.

Figure 8

(a) Runout length $L_f/L_i$ and (b) final height $H_f/H_i$ as a function of the initial aspect ratio $a=H_i/L_i$ for liquid-saturated granular collapses in the continuous-avalanche regime ($\bullet$). Note that here, the Stokes number St and the initial volume fraction $\varphi$ were varied. Inset: selected experiments where $\text{St}=42$ and $\varphi =0.63\pm 0.01$. For comparison, we also plot the results of dry collapses ($\times$) performed with the same experimental setup and the same grains ($d=3.15$ and $5.0\mathrm{mm}$). (—) $L_f/L_i={\lambda }_1{a}^{2/3}$, with ${\lambda }_1=4$ and $H_f/H_i={\lambda }_2{a}^{-2/3}$ with ${\lambda }_2=0.7$; (- - -) ${\lambda }_1=[2.5:6.5]$ and ${\lambda }_2=[0.5:0.9]$.

Figure 9

Trigger time $T_{t1}$ normalized by (a) the free-fall time ${\tau }_{\text{ff}}={[2{\rho }_{p}d/({\rho }_p-{\rho }_f)g]}^{1/2}$ and (b) the viscous time ${\tau }_v=18{\eta }_f/({\rho }_p-{\rho }_f)gd$ as a function of the Stokes number St ($a=2$ and $\varphi =0.64$). Insets: (a) Time evolution of the front position $x_f-L_i$ of the corresponding experiments. Colors indicate the range $\text{St}=[0.2:42]$ from light to dark gray. (b) Dimensionless trigger time $T_{t1}/{\tau }_{v}g\left(\varphi \right)$, with $g\left(\varphi \right)={(1-\varphi )}^{-2.8}$ and $\varphi =0.64$, as a function of St. (- - -) $T_{t1}/{\tau }_{\text{ff}}=T_{t1}/{\tau }_{v}g\left(\varphi \right)=1$; (- $·$ - $·$ -) $T_{t1}/{\tau }_v=20$. The gray area represents the range of St for which no second spreading phase was observed.

Figure 10

Characteristic times of the front dynamics as a function of St ($a=2$ and $\varphi =0.64$): ($▿$) $T_{r1}$–first stop, ($⚪$) $T_{t2}$–second start, ($▵$) $T_{r2}$–second stop. We substracted from these times the first trigger time $T_{t1}$ and scaled them by the column free-fall timescale $T_{\text{FF}}={(2H_i/g)}^{1/2}$. (- - -) $(T-T_{t1})/T_{\text{FF}}=\lambda =4.8$, (- $·$ - $·$ -) $(T-T_{t1})/T_{\text{FF}}\propto {\text{St}}^{-1}$. The gray area represents the range of St for which no second spreading phase was observed. The inset shows $(T_{r2}-T_{t2})/T_{\text{FF}}$ as a function of St.

Figure 11

Location of the first arrest $L_{r1}$ ($▿$) and the second arrest $L_{r2}\equiv L_f$ ($▵$) relative to that of the corresponding dry collapse $L_f^d$ as a function of St ($a=2$ and $\varphi =0.64$). ($+$) Results of the runout length $L_f^i$ for fully immersed dense granular collapses, extracted from Bougouin and Lacaze [28]. The gray area represents the range of St for which no second spreading phase was observed.

Figure 12

(a) Mean angle ${\theta }_m$ as a function of the Stokes number St ($a=2$ and $\varphi =0.64$). The gray area represents the range of St for which no second spreading phase was observed. (b) Height profile $h/H_i$ as a function of $(x-L_i)/L_i$ for $\text{St}\lesssim 1.5$ (top) and $\text{St}\gtrsim 7$ (bottom).

Figure 13

Temporal evolution of the height profile $h/H_i$ as a function of $(x-L_i)/L_i$ for a water-saturated granular collapse ($a=2$ and $\text{St}=42$) with two different volume fractions, i.e., $\varphi =0.616$ (top) and $\varphi =0.643$ (bottom). The time step between each profile is $0.02\mathrm{s}$.

Figure 14

Runout length $L_f$ relative to that of the corresponding dry collapse $L_f^d$ as a function of $\varphi$ ($\text{St}=42$): ($•$) $a=1$, ($⚪$) $a=2$. Inset: corresponding final height $H_f$ relative to that of the dry collapse $H_f^d$.

Figure 15

Hypothetical sketch of the evolution of the final runout length $L_f\equiv L_{r2}$ relative to that of the corresponding dry collapse $L_f^d$ as a function of $\text{St}/g\left(\varphi \right)$, with $g\left(\varphi \right)={(1-\varphi )}^{-2.8}$, for liquid-saturated granular collapses in air (${\rho }_p/{\rho }_f\approx 2.5$ and ${\rho }_f/{\rho }_a={10}^3)$. Triangles and circles correspond to the runout length $L_f\equiv L_{r2}$ of experiments shown in Figs. 11 and 14 with $a=2$, respectively. The gray area represents the range $\text{St}/g\left(\varphi \right)\lesssim 2\times {10}^{-2}$ for which no second spreading phase was observed.

Figure 16

Angle of avalanche ${\theta }_a$ ($⚪$) and angle of repose ${\theta }_r$ ($•$) as a function of the grain diameter $d$; (- - -) Mean values $\langle {\theta }_a\rangle ={28}^{\circ }$ and $\langle {\theta }_r\rangle ={22}^{\circ }$, for $d>D=550\mu \mathrm{m}$, used in Eq. (A1). Inset: Avalanche and repose angles scaled by the mean angle as a function of $d/D$; (—) Eq. (A1).

Figure 17

Sketch of the final deposit of a liquid-saturated granular collapse with different estimates of the runout $L_f$ ($\equiv L_{r2}$ defined in Fig. 7): $◆$ height front of two grain diameters, $★$ quasi-2D front and all grains in contact, $\blacksquare$ quasi-2D front, and $•$ farthest grain.

Figure 18

Runout length $L_f$ ($\equiv L_{r2}$ defined in Fig. 7) relative to that of the corresponding dry collapse $L_f^d$ as a function of (a) the Stokes number St and (b) the initial volume fraction $\varphi$ using different criteria defined in Fig. 17.

Figure 19

Sketch of the collapse of a liquid-saturated granular mass on a horizontal plane at an arbitrary time.

Figure 20

Collapse time $\lambda =T_{r1}/T_{\text{FF}}$ (defined in Fig. 7) as a function of the initial aspect ratio $a$ for ($⚪$) $d=3.15\mathrm{mm}$ and ($•$) $d=5.0\mathrm{mm}$. Here, time is normalized by the free-fall timescale $T_{\text{FF}}={(2H_i/g)}^{1/2}$. The dashed lines corresponds to the mean value of $\lambda =4.8$.