As a simple model of submarine landslides, laboratory experiments were performed to determine the effects of Darcy number on the collapse of granular columns in water with loose packing. The Darcy number describes the tendency for pore fluid pressure developed by the contraction process to reduce the friction for loosely packed columns. The experimental results reveal that the collapse depends on the Darcy number, *Da*, the bed slope, and the initial aspect ratio of the column. Predictive models are provided for the spreading duration, spreading velocity, runout distance, and finial height of the deposit. Comparison of the present results with those of the previous studies reveals that the effects of *Da* on collapse vary with the packing conditions. Specifically, a positive runout-*Da* relationship exists in cases of loose packing, while a negative relationship is revealed for dense packing. A simple dynamic analysis is performed to explore observation, which is a consequence of different pore pressures occurring inside columns owing to different packing.

## I. INTRODUCTION

Submarine granular flow, a flow of the mixture of solid particles and fluid driven by gravity or shear, usually occurs in continental margins. This flow is an important transport mechanism of sediments into the deep ocean^{1} and it can destroy underwater facilities.^{2} Besides this, the flow may cause tsunami,^{3} threatening coastal zones. Due to its importance in science and engineering, submarine granular flow has drawn increasing attention from researchers. The collapse of submerged granular column (hereinafter referred to as the submarine granular collapse) is widely used to study the submarine granular flows because of the simple configuration of the columns.^{4–8}

When the particle density is much larger than the fluid density, the fluid has an insignificant effect on the collapse. In this case, the most important factor affecting the submarine granular collapse is the aspect ratio of the granular column *A* = *H*_{i}/*L*_{i}, where *H*_{i} = the initial height and *L*_{i} = the initial length.^{9,10} The aspect ratio determines the runout distance and the deposit morphology. The duration of the collapse process is proportional to $Hi/g$ where *g* is the acceleration due to gravity. When the bed is inclined, the incline angle, *θ*, is the other important parameter,^{11} which positively affect the collapses. When the particle density is comparable with the fluid density (e.g., sand and water), the surrounding fluid plays an important role in the submarine granular collapse. In addition to *A*, the initial packing is another important parameter due to the pore pressure feedback.^{6} In the early stage of submarine granular collapse, the contraction/dilation occurs for the initial loose/dense column, inducing high/low pore pressure inside the column. The high/low pore pressure favors/disfavors the collapse. The Stokes number, $St=\rho sg\u2032d3/182\rho f\nu f$, where *g*′ = *g*(1 − *ρ*_{f}/*ρ*_{s}) and *ρ*_{s/f} = the solid/fluid density,^{8} and the Archimedes number (the square of the Stokes number)^{5} could affect the collapse. The Stokes number represents the ratio of the sediment inertial force to the fluid viscous force. For the initially densely packed columns, previous experimental studies^{5,8} suggest positive Δ*L* − *St* relationships, where Δ*L* is the runout distance. For the initially loosely packed columns, the experimental and the numerical results of Lee, Huang, and Yu^{7} indicate that smaller particles have longer runout distances, indirectly suggesting a negative Δ*L* − *St* relationship. At present, it is not understood why *St* has different effects on Δ*L* for the different packing columns.

This study speculates that the possible reason for the discrepancy in the Δ*L* − *St* relationships may be the different pore pressure occurring in differently packed columns. Since the physical meaning of *St* does not reflect the pore pressure, this study defines a new dimensionless number, which is called the Darcy number, Da. Although *Da* is similar to 1/*St* mathematically, *Da* reflects the pore pressure forces physically. Effects of *Da* and *θ* on the collapse duration, spreading velocity, runout distance, and final height of deposit are studied herein by conducting experiments of submarine granular collapse in cases of loose packing. Predictive models are provided for the collapse duration, spreading velocity, runout distance, and final height of deposit. A simple dynamic analysis is performed to discuss the effect of *Da* and *θ* on the collapse and explain why *Da* (or *St*) has a different effect on the collapse for different packing.

## II. EXPERIMENTAL SETUP

Fig. 1 schematically presents the experimental set up, where Δ*L* is the runout distance and *H*_{f} is the finial height. A Perspex water tank, 200-cm long, 15-cm wide and 60-cm high, is used. A false bottom made of a Perspex plate is posited in the tank, and is made rough by gluing a layer of sands with diameter *d* = 1.24 mm on it. Two vertical Perspex walls are installed in the slots made on the side walls to create a reservoir for granular materials. The right wall is designed as a gate which can be removed in an upward direction. Medium sands (M), coarse sands (C), and very coarse sands (VC) are used as the solid particles. The density of sand is *ρ*_{s} =2580 kg/m^{3}, and the angle of repose is *θ*_{r} ≈ 31°. The liquid is water with *ρ*_{f} = 1000 kg/m^{3} and *ν*_{f} = 10^{−6} m^{2}/s. The other particle properties and experimental conditions are listed in Table I. This study newly defines $Da=\rho f\nu f/\rho sg\u2032d3$ and uses it to discuss the collapses. The Darcy number describes the tendency of pore fluid pressure developed by dilation/contraction to change the friction; it reflects the importance of the pore pressure feedback on the collapse (discussed in Section IV).

Particle . | ||||||
---|---|---|---|---|---|---|

type . | d (mm)
. | Da(×10^{−3})
. | L_{i} (cm)
. | H_{i} (cm)
. | A = H_{i}/L_{i}
. | θ(°)
. |

M | 0.45 | 16.6 | 4, 10, 14 | 3-23 | 0.2-5.6 | 0 |

C | 0.64 | 9.77 | 4, 10, 14 | 3-23 | 0.2-5.6 | 0, 5, 10 |

VC | 1.24 | 3.62 | 4, 10, 14 | 3-23 | 0.2-5.6 | 0, 5, 10 |

Particle . | ||||||
---|---|---|---|---|---|---|

type . | d (mm)
. | Da(×10^{−3})
. | L_{i} (cm)
. | H_{i} (cm)
. | A = H_{i}/L_{i}
. | θ(°)
. |

M | 0.45 | 16.6 | 4, 10, 14 | 3-23 | 0.2-5.6 | 0 |

C | 0.64 | 9.77 | 4, 10, 14 | 3-23 | 0.2-5.6 | 0, 5, 10 |

VC | 1.24 | 3.62 | 4, 10, 14 | 3-23 | 0.2-5.6 | 0, 5, 10 |

In preparing the loosely packed columns, the particles are poured into the reservoir. A whisk driven by a drill motor is used to stir the particles until all the particles have been suspended. A loosely packed column is obtained after the particles settle. The stir process to create loosely packed columns was also adopted in the previous studies of Rondon, Pouliquen, and Aussillous^{6} and Lee, Huang, and Yu.^{7} The initial concentration of the loosely packed column is approximately 0.55; such value is measured by using the same procedure and a two-liter beaker instead of the Perspex water tank. The collapse is initiated by opening the gate after removing the whisk for approximate 1 minute. A high-speed camera (Fastec-IL5) is used to record the collapse processes at the frame rate of 120 frames per second and a resolution of 2560 x 1000 pixels.

## III. RESULTS

### A. Observed collapsing processes

A loosely packed submarine granular collapse is very similar to a dry granular collapse.^{7} The collapses are of three types on a flat bed. The first type occurs when *A* < *A*_{H} (where *A*_{H} ≈ 0.7, suggested by a two-phase simulation^{7}). The failure surface intersects with the top surface of the column in the beginning. The granular mass above the failure surface slides down, and that below the failure surface remains static. The resulting deposit is trapezoidal in shape. The second type occurs when *A* > *A*_{L} (where *A*_{L} ≈ 3, suggested by a two-phase simulation^{7}). The failure surface is buried deep inside the column. The granular mass above the failure surface falls, forms a pile, and then spreads forward. The finial deposit has either a triangular shape or a “Mexican hat” shape. The third type occurs when *A*_{H} ≤ *A* ≤ *A*_{L} and is a transition between the other two types. The deposit of this type of collapse is triangular in shape. The present experimental results also reveal these three collapse types for *θ* = 0°, 5°, and 10°. *A*_{H} = 0.7 − 1.3 and *A*_{L} = 2.3 − 3.5 are obtained in the present experiments.

Fig. 2 shows the photographs of the observed underwater granular collapse for Particle VC with *L*_{i} = 4 cm and *H*_{i} = 15 cm (*A* = 3.5) over the three different slopes (*θ* = 0°, 5°, and 15°). The flow-front motion for *θ* = 0° is displayed in the inset of Fig. 2, where *x*_{f} is the location of the flow front projected to the *x* axis. The instant of time when the gate is fully open is defined as *t* = 0 s. The submarine granular collapses on different slopes have similar collapse processes. They fall down, form piles, spread forward, and finally deposit. The deposits are triangular. The bed slope does not seem to affect the collapse type significantly; however, it affects the kinematic properties such as spreading velocity and duration, and deposit morphology such as runout distance and finial height.

As shown in the inset of Fig. 2, the three stages of the flow-front motion can be identified: (I) an acceleration stage (*t* ¡ 0.2 s), (II) a nearly constant velocity stage (0.2 s ¡ *t* ¡ 0.53 s), and (III) a deceleration stage (*t* ¿ 0.53 s). The velocity at stage (II) is defined as the spreading velocity, *u*_{f}, for discussion, see Section III B. Concerning the collapse duration, I follow Rondon, Pouliquen, and Aussillous^{6} to represent the collapse duration by *t*_{95}, which is the time it takes for the sliding material to travel a distance equal to 95% of the runout distance Δ*L*, as shown in the inset of Fig. 2. It is more precise to determine *t*_{95} than to determine the time for the entire process.

### B. Collapse duration and spreading velocity

For the dry granular collapses, previous experimental results^{9} suggest that the collapse duration is proportional to $(Hi/g)1/2$. For the submarine granular collapse, the two-phase simulation of Lee, Huang, and Yu^{7} reveals that $t95\u223c(Hi/g\u2032)1/2$ when *A* < *A*_{H} and *L*_{i} have an effect on *t*_{95} when *A* > *A*_{H}. Figs. 3 (a)–(c) display the measured collapse durations for various *A*, *θ*, and *Da*. Two scaling parameters are selected to define the dimensionless collapsing duration: $(Hi/g\u2032)1/2$ and $(Li/g)1/2$. It can be seen in the insets of Figs. 3 (a)–(c) that $t95/(Li/g\u2032)1/2\u223cA1/2$, suggesting that *t*_{95} depends linearly on $(Hi/g\u2032)1/2$. However, the computed results^{7} suggest that *t*_{95} depends on *L*_{i} for *A* > *A*_{H}. The inconsistency may be resulted from the different particle types, affecting friction coefficients and parameters related to the drag force. This study used natural sands but the previous studies used glass beads. Referring to Figs. 3 (a)–(c), both *Da* and *θ* have positive effects on $t95/(Hi/g\u2032)1/2$. Changing *Da* from 3.62 × 10^{−3} to 16.6 × 10^{−3} causes 52.2% difference in $t95/(Hi/g\u2032)1/2$. Increasing *θ* from 0° to 10° leads to approximately 30% difference in $t95/(Hi/g\u2032)1/2$.

Regarding the spreading velocity, the two-phase simulation of Lee, Huang, and Yu^{7} suggests that $uf\u223c(g\u2032Hi)1/2$ when *A* < 3 but $uf\u223c(g\u2032)0.5(Hi)0.3(Li)0.2$ when *A* ≥ 3 for underwater collapses. The insets of Figs. 3 (d)–(f) show that $uf/(g\u2032Li)1/2\u223cA1/2$ for *A* ranging from 0 to 6; this means that $uf\u223c(g\u2032Hi)1/2$, which is inconsistent of the computed one.^{7} Different particle types may be responsible for the inconsistency. Figs. 3 (d)–(f) reveal positive effects of *Da* and *θ* on *u*_{f}. When changing *Da* from 3.62 × 10^{−3} to 16.6 × 10^{−3}, $uf/(g\u2032Hi)1/2$ increases by 12%. Changing *θ* from 0° to 10° increases $uf/(g\u2032Hi)1/2$ by 36%. Additionally, the effect of *Da* becomes more significant for higher *θ*.

### C. Deposit morphology

This section studies the deposit morphology in terms of the runout distance (Δ*L*) and the finial deposit height (*H*_{f}). Previous experimental results^{6,7} show a linear relationship between Δ*L*/*L*_{i} and *A* for a flat bed. Figs. 4 (a)–(c) show the measured Δ*L*/*L*_{i} as a function of *A* with the three different slopes. The linear relationship between Δ*L*/*L*_{i} and *A* can be identified in Figs. 4 (a)–(c) for various particles and slopes. Additionally, Figs. 4 (a)–(c) reveal that *Da* and *θ* have positive effects on Δ*L*/*L*_{i}. Δ*L*/*L*_{i}*A* increases by 50% when *Da* increases from 3.62 × 10^{−3} to 16.6 × 10^{−3}. Δ*L*/*L*_{i}*A* can grow by 36% when *θ* increases from 0° to 10°.

The dimensionless deposit height *H*_{f}/*H*_{i} as a function of *A* is presented in Figs. 4 (d)–(f) for the different particles and the different slopes. For *θ* = 0°, *H*_{f}/*H*_{i} remains unity for small *A* due to the trapezoid deposition shape. For relatively larger *A*, the deposit shape becomes triangular, *H*_{f}/*H*_{i} becomes less than unity and *H*_{f}/*H*_{i} ∼ *A*^{−1/2}. When the bed is inclined, *H*_{f}/*H*_{i} can exceed unity for small *A* because *H*_{i} is defined in the middle of the column. The relationship *H*_{f}/*H*_{i} ∼ *A*^{−1/2} for large *A* can also be identified for the inclined planes. It can be seen from Fig. 4 (d)–(f) that the Darcy number seems to have an insignificant effect on the *H*_{f}/*H*_{i} − *A* relationships. The bed slope (*θ*) has negative effects on *H*_{f}/*H*_{i}. Changing *θ* from 0° to 10° can reduce *H*_{f}/*H*_{i}*A*^{−1/2} by 16%.

### D. Predictive models

According to the experimental results presented in Sections III B and III C, the following predictive models are proposed for practical uses:

The normalized root mean square errors are 9%, 10%, 6%, and 4% for *t*_{95}, *u*_{f}, Δ*L*, and *H*_{f}, respectively. It is remarked that Eq. (4) is obtained only for *A* > 1. In the case of *Da* = 0, Eqs. (1) and (3) can reduce to the predictive models obtained for collapsing dry granular columns on slopes.^{11} When *Da* = 0 and tan *θ* = 0, Eq. (1) becomes $t95=3.3(Hi/g\u2032)1/2$ and Eq. (3) Δ*L* = 1.23*AL*_{i}, with the former revealed in Ref. 9 and the latter being similar to the predictive model of Lube *et al.*^{10} for collapsing dry granular columns in a flat bed.

The predictive models (Eqs. 1–4) exclude effects of the shear-induced instability due to limitation of the experimental scale. The vortices generated by the instability can strongly stir the particles and can dilute the sediment mass with surrounding water,^{12} enhancing the mobility. Therefore, Eqs. (1)–(3) may underestimate the collapse duration, the spreading velocity, and the finial runout distance in practical uses.

## IV. DISCUSSION

Section III concludes that larger *Da* and *θ* favor the submarine granular collapse in the loose packing, which is further examined in this section. The experimental results of Bougouin and Lacaze^{8} reveal that the larger *Da* disfavors the submarine granular collapse in the dense packing. Obviously, *Da* has different effects on the submarine granular collapse for the loose and dense packings, which is also examined in this section.

Lajeunesse, Monnier, and Homsy^{9} provide a simplified dynamic model to explain Δ*L* depending on *A*. Here, their model is extended by incorporating the pore-water effects on the collapses. The granular mass near the flow front (called the flow front mass) is considered, and it is approximated via a triangulation with *h*_{t} height and *L*_{t} width, as shown in Fig. 5. The mass of the flow front *m* = *ρ*_{s}*ch*_{t}*L*_{t}/2 is assumed not to vary with time as in Ref. 9. The present extended model focuses on the collapse dynamics in the early stage (acceleration stage). In the early stage, the potential energy of the column can largely transfer into the kinetic energy of the flow front mass. The more kinetic energy of the flow front mass leads to the longer spreading duration and the longer runout distance.

The forces applied on the flow front mass include the solid pressure force, gravitational force, drag force, bed friction, buoyancy, and force due to the pore pressure. In the early stage, the densely/loosely packed granular mass undergoes a dilation/contraction process where the velocity along the slope *u*_{s} and the velocity perpendicular the slope $v$_{s} have the following relationship^{13}

where Ψ is the dilatancy angle. The value of Ψ is positive/negative for the dilation/contraction process. The dilation/contraction process yields a relative motion between the fluid and the grains, which creates a pore pressure gradient across the layer. Following Pailha, Nicolas, and Pouliquen,^{13} the pressure gradient is given by a linear Darcian drag

where *y*_{s} is the axis perpendicular to the slope; *α* is the constant and *αd*^{2} is the permeability of the granular layer. The pressure gradient can change the normal force exerted from the flow front mass on the slope bed and the friction force. Eq. (6) suggests a negative/positive pore pressure gradient for the loose/dense packing, which is revealed in previous experimental studies^{6} and multi-phase flow simulations.^{14} It is remarked that |tan Ψ| decreases in the dilation/contraction process^{13} so that the magnitude of pressure gradient declining with time.

The momentum balance equation along the bed slope for the flow front can be expressed by

where Δ*u*_{s} is the velocity difference between the granular particles and its surrounding fluid, and *η* is the bed friction angle. The terms on the right hand side of Eq. (7) account for the solid pressure force (with the hydrostatic assumption), gravitational force (reduced by buoyancy), bed friction due to weight, change of bed friction force due to the pore pressure gradient, and the drag force in sequence. The orders of magnitude of tan Ψ and *η* are *O*(0.1) and *O*(1). According to the two-phase simulation of Lee and Haung,^{14} the velocity difference between the two phases is extremely small in the early stage; the ratio of the velocity difference to the solid velocity is smaller than 0.001 in the early stage. Accordingly, Δ*u*_{s}/*u*_{s} < < *η* tan Ψ, and then the last term in Eq. (7) can be ignored.

The dimensionless time and the dimensionless spreading velocity are defined as

Then, Eq. (7) becomes

It can be noted from Eq. (9) that the *θ* has a positive effect on the submarine granular collapse. Additionally, *Da* plays a role due to the formation of the excess pore pressure, and it can be physically inferred as the ratio of pore pressure force to the gravitational force. The larger *Da* can cause a large pressure gradient, which is observed in the two-phase simulations of Lee, Huang, and Yu,^{7} and the bed friction decreases. Therefore, the larger *Da* favors the collapse, confirming the results of the present experiment.

For the initially loose/dense packing condition, the granular mass undergoes a contraction/dilation behavior and tan Ψ is negative/positive. Increasing *Da* enhances/reduces mobility for the loose/dense packing. Eq. (9) explains the discrepancy in *Da* effect on the collapse between the present study (for the loose packing) and the previous study (for the dense packing).^{8} Eq. (9) also suggests that *η* has an effect on the collapse, which was observed experimentally for the densely packed column.^{8} However, this study does not systematically examine the effect of *η*.

## V. CONCLUSIONS

This study experimentally investigates the effects of *Da* and *θ* on the collapse of submerged granular column in cases of loose packing. The experimental results reveal that the larger *Da* yields a longer runout, spreading velocity, and spreading duration; in the other words, the larger *Da* favors the collapse. Increasing *θ* has positive effects on the collapse due to the increase in the driving force, in conjunction with the gravitational force. The predictive models for the spreading duration, spreading velocity, runout distance, and finial height are provided. A simple dynamic analysis is performed to discuss the effect of *Da* on the collapse. The larger *Da* can cause higher pore pressure and reduces friction, favoring the collapse. The simple dynamic analysis also explains the cause of different effects of *Da* on columns with different packing. For the dense packing, the larger *Da* can generate larger negative excess pore pressure and increase friction; these effects are contrary to effects on the loose packing.

## ACKNOWLEDGMENTS

This work is supported by the Ministry of Science and Technology, Taiwan under Grant Nos. 107-2221-E-032-018-MY3 and 108-2636-E-032 -001(Young Scholar Fellowship Program).