Underwater collapse of a loosely packed granular column on an inclined plane: Effects of the Darcy number

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 ocean¹ 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, $St=ρsg′d3/182ρfνf$⁠, 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 ΔL − St relationships, where ΔL is the runout distance. For the initially loosely packed columns, the experimental and the numerical results of Lee, Huang, and Yu⁷ indicate that smaller particles have longer runout distances, indirectly suggesting a negative ΔL − St relationship. At present, it is not understood why St has different effects on ΔL for the different packing columns.

This study speculates that the possible reason for the discrepancy in the ΔL − 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.

Fig. 1 schematically presents the experimental set up, where ΔL 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/m³, and the angle of repose is θ_r ≈ 31°. The liquid is water with ρ_f = 1000 kg/m³ and ν_f = 10⁻⁶ m²/s. The other particle properties and experimental conditions are listed in Table I. This study newly defines $Da=ρfνf/ρsg′d3$ 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).

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 Aussillous⁶ 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.

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 simulation⁷). 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 simulation⁷). 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 Aussillous⁶ to represent the collapse duration by t₉₅, which is the time it takes for the sliding material to travel a distance equal to 95% of the runout distance ΔL, as shown in the inset of Fig. 2. It is more precise to determine t₉₅ than to determine the time for the entire process.

For the dry granular collapses, previous experimental results⁹ 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 Yu⁷ reveals that $t95∼(Hi/g′)1/2$ when A < A_H and L_i have an effect on t₉₅ 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: $(Hi/g′)1/2$ and $(Li/g)1/2$⁠. It can be seen in the insets of Figs. 3 (a)–(c) that $t95/(Li/g′)1/2∼A1/2$⁠, suggesting that t₉₅ depends linearly on $(Hi/g′)1/2$⁠. However, the computed results⁷ suggest that t₉₅ 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 $t95/(Hi/g′)1/2$⁠. Changing Da from 3.62 × 10⁻³ to 16.6 × 10⁻³ causes 52.2% difference in $t95/(Hi/g′)1/2$⁠. Increasing θ from 0° to 10° leads to approximately 30% difference in $t95/(Hi/g′)1/2$⁠.

Regarding the spreading velocity, the two-phase simulation of Lee, Huang, and Yu⁷ suggests that $uf∼(g′Hi)1/2$ when A < 3 but $uf∼(g′)0.5(Hi)0.3(Li)0.2$ when A ≥ 3 for underwater collapses. The insets of Figs. 3 (d)–(f) show that $uf/(g′Li)1/2∼A1/2$ for A ranging from 0 to 6; this means that $uf∼(g′Hi)1/2$⁠, 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, $uf/(g′Hi)1/2$ increases by 12%. Changing θ from 0° to 10° increases $uf/(g′Hi)1/2$ by 36%. Additionally, the effect of Da becomes more significant for higher θ.

This section studies the deposit morphology in terms of the runout distance (ΔL) and the finial deposit height (H_f). Previous experimental results^6,7 show a linear relationship between ΔL/L_i and A for a flat bed. Figs. 4 (a)–(c) show the measured ΔL/L_i as a function of A with the three different slopes. The linear relationship between ΔL/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 ΔL/L_i. ΔL/L_iA increases by 50% when Da increases from 3.62 × 10⁻³ to 16.6 × 10⁻³. ΔL/L_iA 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_iA^−1/2 by 16%.

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 t₉₅, u_f, ΔL, 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 $t95=3.3(Hi/g′)1/2$ and Eq. (3) ΔL = 1.23AL_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.

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 Lacaze⁸ 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 Homsy⁹ provide a simplified dynamic model to explain ΔL 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_sch_tL_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 relationship¹³ 

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 αd² 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 studies⁶ and multi-phase flow simulations.¹⁴ It is remarked that |tan Ψ| decreases in the dilation/contraction process¹³ 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 Δu_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, Δu_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.

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.

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).