E3S Web of Conferences 415, 01030 (2023)
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https://doi.org/10.1051/e3sconf/202341501030

Fines-controlled drainage in just-saturated, inertial column
collapses
William Webb1*, Charles Heron1, and Barbara Turnbull1
1Faculty of Engineering, University of Nottingham, Nottingham, NG7 2RD, UK

Abstract. The wide particle size distributions, over several orders of magnitude, observed in debris flows
leads to a diverse range of rheological behaviours controlling flow outcomes. This study explores theinfluence of different scale grains by conducting subaerial, fully saturated granular column collapse
experiments with extreme, bimodal particle size distributions. The primary particles were of a size where
their behaviour was controlled by their inertia while a suspension of kaolin clay particles within the fluid
phase acts at spatial scales smaller than the pore space between the primary particles. The use of ageotechnical centrifuge allowed for the systematic variation of gravitational acceleration, inertial particle
size and the degree of kaolin fines. Characteristic velocity- and time-scales of the acceleration phase of thecollapse were quantified using high-speed cameras. Comparing tests containing fines to equivalent collapses
with a glycerol solution mimicking the enhanced viscosity but not the particle behaviour of the fines, it was
found that all characteristic dynamic quantities were dependent on the degree of fines, the system size, thegrain fluid-density ratio and the column– and grain-scale Bond and Capillary numbers. We introduce a finescale Capillary number showing that, although surface tension effects at the column scale are negligible,
fines do control the movement of fluid through the pore spaces.

1 IntroductionThe influence of a wide particle size distribution on thedynamics and outcomes of naturally occurring debris
flows is still disputed. Field studies [e.g. 1] and large-scaletesting [e.g. 2] have shown that events containing highquantities of fine granular material, like clays and silts,
often achieve increased mobility through the developmentof significant excess pore pressures which reduce thefrictional energy losses between the bulk flow and theterrain bed.
Recent two-phase shallow water models [3-5] haveattempted to capture the temporal and spatial variation of
the grain size distribution by modelling aspects of
unsteady debris flow behaviour. However, there is still noconsensus on the most appropriate, computationallyefficient way to implement this complexity. This suggeststhat a more refined understanding of the mechanisms
controlling grain-scale flow dynamics is required to allow
conclusions to be made on what physical processes aremost influential on global flow outcomes.
Given the small length and time scales associated withthese effects, it seems pertinent to analyse them further byconducting laboratory-scale experiments where it ispossible to fully control the initial and boundary
conditions of the flow [6]. A notable study [7] utilised adrum centrifuge configuration to evaluate the influence ofthe particle size distribution on steady-state flow
dynamics. They found that mixtures that contained higher
percentages of fine granular material exhibited moresignificant and prolonged excess pore pressures which

reduced bulk flow resistance. While this is encouraging,further parametric investigations are required toquantitatively evaluate the influence of the inclusion offines on flow dynamics and understand the mechanisms
at play.
A recent study [8] attempted to isolate the influence ofinertial grains on flow dynamics over a wide parameter
space by conducting g-elevated, fluid-saturated granularcolumn collapse experiments where water-glycerol
mixtures were used as a pseudo-fluid. The current study
attempts to build on this work and introduce an extreme
bimodal grain size distribution by using a fluid phase
comprised of kaolin clay particles and water. As such, by
varying gravitational acceleration, the coarse grain
diameter and the percentage of fines, the influence of thefine length- and time-scales on acceleration phasecollapse dynamics is evaluated and compared within thepreviously established parameter space.

2 Methodology
2.1 Experimental setup
The experiment (see Figure 1) consists of rapidly
releasing a granular-fluid mixture, which is initially
accommodated within a partially filled steel cylinder, overa horizontal plane where it is allowed to spread under theinfluence of a prescribed gravitational acceleration 𝐺. By
spinning the apparatus at the end of a geotechnical beam
centrifuge, the collapse can occur at an elevated* Corresponding author: evyww@nottingham.ac.uk© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 01030 (2023)
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gravitational acceleration 𝐺 = 𝑁𝑔 where 𝑁 is dependent
on the effective radius and rotational speed of the model,and 𝑔 = 9.81 m s-2.
The current study focused on the collapse of fluid
saturated granular columns with an initial height 𝐻0 = 50mm and a column radius 𝑟0 = 54 mm. The granular phase
was comprised of soda lime glass spheres with a density
𝜌𝑝 = 2642 ± 27.4 kg m-3 and a mean packing fraction
𝜙𝑝 = 0.61 ± 0.01. The fluid phase consisted of kaolin clay
particles suspended in water. Tests were conducted over awide parameter space by varying 𝐺 = [9.81, 45.22,137.64, 275.45] m s-2, the coarse grain diameter 𝑑𝑝 = [4,8] mm, and the degree of fines suspended within the fluidby mass 𝐶𝑘 = [0.1, 0.2]. Assuming a characteristiccollapse strain rate 𝛾 = √𝐺/𝐻0 , the density, viscosity andsurface tension of the fluid phase was varied between
𝜌𝑠 = [1165, 1330] kg m-3, 𝜇𝑠 = [7.65, 108.30] cP, and𝜎𝑠 = [92.59, 98.96] mN m-1, respectively.
The evolution of the collapsing mixture was recordedby two Go-Pro cameras and the temporal evolution of thecollapse front was obtained through the image analysis
procedure detailed in [8].

where 𝑓2 is an unknown function, 𝐿 is the characteristic
length scale of interest and 𝜌 is the granular-fluid mixture
effective density 𝜌 = 𝜙𝑝 𝜌𝑝 + (1 − 𝜙𝑝 )𝜌𝑠 . The parameteron the left-hand side of Equation (2) is the flow front
Froude number Fr𝐿 which is the ratio of inertial andgravitational forces over a characteristic length scale 𝐿.
The first two parameters on the right-hand side ofEquation (2) are the dimensionless parameters 𝜙𝑝 and 𝐶𝑘
which remain unchanged from Equation (1). The thirdparameter, 𝑟 ∗ is the ratio between 𝑟0 and 𝑑𝑝 . The fourthparameter, 𝑡 ∗ , compares 𝑡 against the length scaledependent characteristic inertial timescale √𝐿𝐺 −1 . The
fifth parameter is the relative granular-fluid density ratio
accounting for acceleration-scale buoyancy effects 𝜌∗ .
The sixth and seventh parameters are referred to as thecolumn- and grain-scale Bond numbers, Bo and Bo𝑘 ,respectively. The first parameter quantifies the relative
influence of inertial forces at the column scale against
capillary forces at the grain scale while the latter is ananalogous quantity relating grain scale inertial forces tokaolin scale capillary effects. The final two parameters are
scale-relative Capillary numbers. The first relates column
scale viscous forces to grain scale capillary effects Ca
while the second compares grain scale viscous forces tokaolin scale capillary effects Ca 𝑘 .
As stated in [10], given that Π groups can be recast
through multiplication, the dimensionless parameter set inEquation 2 is not a unique solution to Equation (1). Thisset was deemed suitable as it has significant overlap withthe dimensionless set used in the previous scale analysis
study [8]. The included force ratio terms also allow forcomparison across all three length scales of interest (𝐻0 ,𝑑𝑝 and 𝑑𝑘 ) which is critical in interpreting thecontribution of the fine particulate to flow dynamics
throughout the acceleration phase.

Fig. 1. Schematic of the initial column configuration prior tocollapse initiation.

3 Results and Discussion

2.2 Dimensional analysis

The acceleration phase of each collapse was characterisedby the maximum velocity of the flow front 𝑢𝑚 , and thetime elapsed between collapse initiation and the instance
where 𝑢𝑚 was achieved 𝑡𝑚 . The two camera angles
allowed averaged values of the two quantities to be taken
which reduces the impact of external forces resulting from
centrifuge modelling like the Coriolis force.
Given that 𝜙𝑝 remains approximately constant acrossall tests, Equation (2) can be simplified as follows

Buckingham’s  theorem [9] can be used to examine how
different test parameters may impact the propagation of
the collapse. As such, a relationship between the flow
front velocity 𝑢 at time 𝑡 with the collapse test parameters
can be hypothesised as
𝑢 = 𝑓1 (𝐻0 , 𝑟0 , 𝑑𝑝 , 𝜌𝑝 , 𝜙𝑝 , 𝐶𝑘 , 𝑑𝑘 , 𝜌𝑠 , 𝜇𝑠 , 𝜎𝑠 , 𝑡, 𝐺)(1)

where 𝑓1 is an unknown function and 𝑑𝑘 is a reference
length scale for the kaolin clay particles taken as 5 m.
The theorem then states that the 11 dimensional quantities
in Equation (1), which are functions of mass, length andtime scales, can be interrelated by 8 dimensionless groups. Following the methodology described in [10],Equation (1) can be transformed into the following form𝑢√𝐺𝐿𝑟𝑡= 𝑓2 (𝜙𝑝 , 𝐶𝑘 , 0 ,𝑑𝑝 √𝐿𝐺 −1,(𝜌𝑝 −𝜌𝑠 )𝑁𝜌𝑠,𝜎𝑠,𝜎𝑠 𝑑𝑝,𝜎𝑠 𝑑𝑘)(3)∗
where Frm,L and 𝑡𝑚,𝐿are equivalent to FrL and 𝑡𝐿∗
evaluated at 𝑢𝑚 and 𝑡𝑚 , respectively. The characteristic
length scales of interest 𝐿 are 𝑑𝑝 and 𝐻0 meaning that
there are four quantities of interest 𝑄 related to the∗acceleration phase of the collapse; Fr𝑚,𝑑𝑝 , Fr𝑚,𝐻0 , 𝑡𝑚,𝑑,𝑝∗and 𝑡𝑚,𝐻0 .
To evaluate the effect of the inclusion of fines on theacceleration phase, a single dataset containing thequantities collected from the current study and the same,33𝜌𝐺𝑑𝑝 𝐻0 𝜌𝐺𝑑𝑝 𝑑𝑘 𝜇𝑠 √𝐺𝐻0 𝜇𝑠 √𝐺𝑑𝑝𝜎𝑠∗Fr𝑚,𝐿 = 𝑓3 (𝐶𝑘 , 𝑟 ∗ , 𝑡𝑚,𝐿, 𝜌∗ , Bo, Bo𝑘 , Ca, Ca 𝑘 )(2)2E3S Web of Conferences 415, 01030 (2023)
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Fig. 2. (a) 𝐅𝐫𝒎,𝒅𝒑 , (b) 𝐅𝐫𝒎,𝑯𝟎 , (c) 𝒕𝒎,𝒅𝒑 and, (d) 𝒕𝒎,𝑯𝟎 normalised by 𝑫 = (𝐁𝐨𝒌 𝒌𝟐 𝑪𝒌𝐂𝐚𝒌)(𝐂𝐚𝒌 )𝒌𝟑𝑪𝒌 (𝝆∗ )𝒌𝟒 (𝒓∗ )𝒌𝟓 against 𝐁𝐨/𝐂𝐚 for 𝑪𝒌 =0 (●), 𝑪𝒌 = 0.1 (△) and 𝑪𝒌 = 0.2 (▲). The black line defines the power law fit described by Equation (4) using the exponents
provided in Table (1). Error bars define the absolute error of the independent variable.

quantities published in [8] for the Newtonian fluid test
cases, where 𝐶𝑘 = 0, and Bo𝑘 and Ca 𝑘 values are notconsidered, was developed. The influence of every
parameter in Equation (3) was then evaluated using a nonlinear regression scheme and, as was found for theNewtonian fluid dataset, the measured quantities ofinterest can be described by empirical power law fits thatare independent of the other quantities of interest. For thecurrent combined dataset, this relationship is defined as
Bo 𝑘1

Bo𝑘 𝑘2 𝐶𝑘

on surface-tension effects emanating from the kaolinscale. In contrast, as demonstrated by Equation (5),surface tension effects were not pertinent to the behaviourof the Newtonian fluid test case.
Table 1. Summary of exponents for Equation (4) for all 𝑄
along with the normalised root mean squared error RMSEN ofeach fit.
𝐅𝐫𝒎,𝒅𝒑𝐅𝐫𝒎,𝑯𝟎∗
𝒕𝒎,𝒅𝒑𝒕∗𝒎,𝑯𝟎(4)

𝒌𝟏

-0.04

-0.04

0.09

0.11

where 𝑘1 – 𝑘5 are constants summarised in Table 1.
Figure 2 details the finalised fits for each 𝑄. The structureof Equation (4) allows Newtonian- and non-Newtonianfluid tests to be fitted to the same power law as terms thatinclude parameters associated with the kaolin-scale can beneglected as 𝐶𝑘 = 0.
Equation (4) is very similar in structure to theempirical fit found for 𝑄 in the case of the purely
Newtonian fluid dataset

𝒌𝟐

0.07

0.15

-0.48

-0.63

𝒌𝟑

0.18

0.27

-0.40

-0.55

𝒌𝟒

0.10

0.11

0.22

0.23

𝒌𝟓

0.30

-0.19

-0.48

-0.07

𝐑𝐌𝐒𝐄𝐍

0.14

0.19

0.11

0.10

𝑄=( )Ca(Ca𝑘)Bo 𝛼(Ca 𝑘 )𝑘3𝐶𝑘 (𝜌∗ )𝑘4 (𝑟 ∗ )𝑘5

𝑄 = ( ) (𝜌∗ )𝛽 (𝑟 ∗ )𝛾Ca(5)The relative importance of each force ratio term inEquation (4) associated with the grain- and kaolin-lengthscales can be quantified by comparing their magnitude
and sign to the magnitude and sign of (Bo/Ca)k1 . Figure(3) shows the plots of (Bo𝑘 /Ca 𝑘 )𝐶𝑘 𝑘2 (Bo/Ca)−𝑘1 ,(Ca 𝑘 )𝐶𝑘 𝑘3 (Bo/Ca)−𝑘1(Bo𝑘 /Ca 𝑘 )𝐶𝑘 𝑘3 (Bo𝑘 /and∗Ca 𝑘 )−𝐶𝑘 𝑘2 against Bo/Ca for Fr𝑚,𝐿 and 𝑡𝑚,𝐿. The figure∗
shows that for Fr𝑚,𝐿 and 𝑡𝑚,𝐿 , the trend for bothcharacteristic length scales is similar for all values of 𝐶𝑘 .

where 𝛼, 𝛽, and 𝛾 are constants. It is reassuring to see thatthe general trend and, the coefficients of the terms that are
independent of kaolin-scale phenomena (i.e. 𝑘1 , 𝑘4 and𝑘5 ), are largely comparable in both magnitude and signfor Equation (4) and Equation (5) for all 𝑄. The
normalised root mean squared error RMSEN has also
reduced for every 𝑄.
The most significant outcome from this study is theappearance of the Ca 𝑘 term in Equation (4). This suggeststhat the inclusion of fine granular material results in all
acceleration phase quantities of interest being dependent3

E3S Web of Conferences 415, 01030 (2023)
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https://doi.org/10.1051/e3sconf/202341501030∗Fig. 3. Comparison of the force ratio influence for 𝐅𝐫𝒎,𝑳 and 𝒕𝒎,𝑳for 𝑪𝒌 = 0.1 (△) and 𝑪𝒌 = 0.2 (▲), and 𝑳 = 𝒅𝒑 (black) and 𝑳 =𝑯𝟎 (grey).

The relative influence of grain- and kaolin-scale
phenomena increases with Bo/Ca for Fr𝑚,𝐿 while the∗influence of both scales reduces with Bo/Ca for 𝑡𝑚,𝐿.
More significantly, Figure 3(b) suggests that kaolin-scalesurface tension effects are more influential to Frm,d andFrm,H0 than the driving forces at the column scale.
Furthermore, Figure 3(c) shows that for Fr𝑚,𝐿 , kaolinscale phenomena are more influential than grain-scale∗
phenomena while Figure 3(f) shows that for 𝑡𝑚,𝐿, grainscale phenomena are more influential than kaolin-scale
phenomena. Logically, these effects are exaggerated byincreasing 𝐶𝑘 .

fluid test case, the inclusion of kaolin particles appears tocouple all 𝑄 with kaolin-scale surface tension effects
which, for the range of the Bo/Ca values tested, was more
influential to Frm,d and Frm,H0 than column-scale inertial
and viscous forces. Future work involves considering theinfluence of a bimodal grain size distribution on themaintenance of excess pore pressures by conducting
similar analyses for the dissipation of fluid pressure at thebase of the column over the duration of the collapse.

References
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B.W. McArdell, P. Bartelt, J. Kowalski. Geophys.
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4. R.M. Iverson, D.L. George. Proc. R. Soc. A 470(2014)
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8. W. Webb, C. Heron, B. Turnbull. PREPRINT
available at Research Square.
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4 Conclusions and Further work
A series of fully saturated, axisymmetric granular column
collapse experiments using a fluid phase comprised ofkaolin particles suspended in water were conducted toinvestigate the effects of a bimodal grain size distribution
on acceleration phase drainage behaviour. The use of ageotechnical centrifuge allowed for a wide test parameter
space where gravitational acceleration, inertial grain
diameter, and the percentage of fines within the fluid
phase could be varied during testing. A dimensionlessparameter space consisting of 8 parameters, obtainedusing Buckingham’s  theorem, was postulated tocharacterise acceleration phase collapse dynamics. The
data gathered from the current series of tests wascombined with the Newtonian fluid tests from a previous
study and a non-linear, least squares fitting analysis wasundertaken to investigate the influence of eachdimensionless parameter on the four measured quantities∗∗
of interest 𝑄: Frm,d , Frm,H0 , 𝑡𝑚,𝑑and 𝑡𝑚,𝐻. As was found
0for the Newtonian fluid test cases, all values of 𝑄 scaled
independently of each other and scaled withBo 𝑘1

𝑄=( )Ca(Bo𝑘 𝑘2 𝐶𝑘Ca𝑘)(Ca 𝑘 )𝑘3𝐶𝑘 (𝜌∗ )𝑘4 (𝑟 ∗ )𝑘5 where 𝑘1 -𝑘5 are constants dependent on 𝑄. Unlike the Newtonian4