# Run-out distance of initially fluidized, collapsing granular columns with different aspect ratios: Constraints and volcanological implications from experiments and 2D incompressible simulations

Alvaro Aravena, Laurent Chupin, Thierry Dubois, Olivier Roche

<figure><img src="image_1.png" /></figure>

## To cite this version:

Alvaro Aravena, Laurent Chupin, Thierry Dubois, Olivier Roche. Run-out distance of initially fluidized, collapsing granular columns with different aspect ratios: Constraints and volcanological implications from experiments and 2D incompressible simulations. Bulletin of Volcanology, 2024, 86, 10.1007/s00445-024-01778-w. hal-04771089

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Distributed under a Creative Commons Attribution 4.0 International License
Run-out distance of initially fluidized, collapsing
granular columns with different aspect ratios:
Constraints and volcanological implications from
experiments and 2D incompressible simulations

Alvaro Aravena¹,²,³*, Laurent Chupin⁴, Thierry Dubois⁴,
Olivier Roche²

¹Facultad de Ciencias Básicas, Universidad Católica del Maule, Av. San Miguel 3605, Talca, Chile.

²Laboratoire Magmas et Volcans, Université Clermont Auvergne, CNRS, IRD, OPGC, 6 Avenue Blaise Pascal, Clermont-Ferrand, France.

³Millennium Institute on Volcanic Risk Research – Ckelar Volcanoes,
Avenida Angamos 0610, Antofagasta, Chile.

⁴Laboratoire de Mathématiques Blaise Pascal, Université Clermont
Auvergne, CNRS, 3 Place Vasarely, Clermont-Ferrand, France.

*Corresponding author(s). E-mail(s): aaravena@ucm.cl;
Contributing authors: laurent.chupin@uca.fr; thierry.dubois@uca.fr;
olivier.roche@uca.fr;

Abstract

We investigate granular flows generated by collapse of an initially fluidized column into a horizontal channel in order to evaluate the factors controlling the efficiency of fluidization in increasing the run-out distance of pyroclastic density currents. This configuration is analogous to flows generated by collapse of volcanic domes or lava flow fronts. We use an incompressible two-phase numerical model able to simulate dam-break experiments, and we compare the numerical results with experimental data. This model permits us to describe depth-dependent variations of flow properties and the effect of pore pressure on the rheology of the granular material. We show that the interplay between timescales of column collapse and of flow front propagation plays a primary role in determining the effective influence of fluidization on run-out distance. For columns with high aspect ratio (i.e.
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initial height/initial width), the collapse velocity decreases abruptly after reach-
ing its peak, a significant portion of the collapse has occurred when the flow front
has travelled a long distance from the reservoir and, importantly, the decrease of
basal pore pressure with time in the reservoir translates into a reduced velocity of
the granular material entering into the propagation channel during final phases of
collapse. Thus, at some point, the collapsing material is not able to significantly
affect the flow front dynamics. This behaviour contrasts with that of low aspect
ratio collapsing columns. These results are consistent with complementary ana-
logue experiments of high aspect ratio collapsing columns, which show that the
granular material at the front of the deposit originates from lower levels of the
column. Comparison with new experimental data also reveals that the effective
pore pressure diffusion coefficient in the propagating flow is an increasing function
of column height, and it can be considered as proportional to a weighted aver-
age of flow thickness during propagation. This is consistent with experiments on
static defluidization columns, but had not been tested in dam-break experiments
until this study. Considering this type of dependency, under our experimental
and simulation conditions, the non-dimensional run-out distance presents a rela-
tive maximum for initial aspect ratios between 1 and 2, and beyond this critical
range the non-dimensional run-out distance decreases abruptly.

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**Keywords:** Fluidization. Dam-break experiments. Pore pressure diffusion coefficient.
Pyroclastic density currents.

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# 1 Introduction

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Pyroclastic density currents (PDCs) are high-speed flows composed of pyroclasts, lithic fragments and gas that propagate laterally on flanks of volcanoes due to the effect of gravity (Druitt 1998; Freundt et al. 2000; Branney and Kokelaar 2002; Dufek et al. 2015; Brand et al. 2023). Most PDCs result from total or partial collapse of an eruptive column, volcanic dome, unstable lava flow front or of an accumulation of pyroclastic material on steep slopes (Druitt 1998; Branney and Kokelaar 2002). Two end-members for the physical regime of PDCs, which often coexist, are described in literature: dilute turbulent suspensions and dense granular flows with variable amounts of gas pore pressure (Breard et al. 2016; Lube et al. 2020; Brand et al. 2023). Understanding the factors controlling the generation of PDCs, their propagation dynamics and the resulting run-out distance (RD) is a major objective in volcanology due to their potentially devastating consequences both for people and infrastructure (Baxter et al. 1998; Doronzo and Dellino 2011; Cole et al. 2015).

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Fluidization by gases has long been proposed as one of the main factors controlling PDC propagation (Sparks 1978; Wilson 1980). Fluidization is the result of the differential motion between solid particles and interstitial gas, whose interplay is able to generate pore pressure, counterbalance partially or entirely the solid particle weight, and thus reduce particle friction (Wilson 1980, 1984; Breard et al. 2018; Gravina et al. 2004). In the context of PDCs, fluidization is mainly generated at the impact zone

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of a collapsing mixture of solid particles and gas (Valentine 2020), while the temporal evolution of fluidization in currents generated from various mechanisms (e.g. dome collapse) is the result of the coupled effect of diffusion, advection, dilatancy, compaction and air entrainment (Iverson and Vallance 2001; Bouchut et al. 2021; Breard et al. 2023). These processes are in turn influenced by the grain-size distribution of the solid particles (Druitt et al. 2007), its variations during flow propagation (Breard et al. 2023), and the underlying topography (Aravena et al. 2021; Brand et al. 2023).

Fluidized granular flows have been largely studied through analogue experiments (Roche et al. 2008; Rowley et al. 2014) which have served as benchmarks for the development and calibration of numerical models (Breard et al. 2019; Aravena et al. 2021). A widely used experimental setup to study granular column collapse is the dam-break configuration (Thompson and Huppert 2007; Roche et al. 2008, 2010; Aravena et al. 2021), which consists in the release of an initially static column of granular material that spreads freely on a flat surface. In this configuration, fluidized granular flows are generated by injecting an air flow through the base of the granular column in order to generate pore pressure and induce fluidization conditions (Roche et al. 2010; Roche 2012). This simple configuration reveals complex feedback mechanisms between pore pressure diffusion and flow propagation, which are thought to be important in nature. For instance, Roche et al. (2008) showed that a slight initial expansion of a few percent of the collapsing granular material is sufficient to confer an inertial fluid-like behaviour to the resulting flow. Roche et al. (2010) described the evolution of the basal pressure during flow propagation and analysed the deposition dynamics of the granular material. Breard et al. (2019) showed by means of a 2D model that pore pressure in granular flows is modulated by dilation and compaction processes, which permitted them to explain the basal pore pressure signals measured in analogue experiments. Aravena et al. (2021) presented a non depth-averaged model built on the formulation of Chupin et al. (2021) to address the internal dynamics of experimental fluidized granular flows. They considered the temporal evolution of flow front position, run-out distance, thickness of final deposits, sedimentation dynamics and temporal evolution of pore pressure, showing that the latter could be successfully described considering a constant, effective pore pressure diffusion coefficient (Aravena et al. 2021).

In this study, in order to deepen our understanding of the influence of gas pore pressure on the mobility of PDCs and to obtain constraints to numerically describe these systems and scale results derived from analogue experiments, we adopted the model presented in Aravena et al. (2021, 2022). This model permits us to address the behaviour of collapsing columns with variable aspect ratios (AR) and to test different assumptions to set the effective pore pressure diffusion coefficient under variable setups. With the aim of confirming our model, we compare numerical results with experimental data of dam-break experiments performed using different aspect ratios and fluidization conditions (i.e. fluidized and non-fluidized flows), and we include an analysis on the factors controlling the efficiency of fluidization in facilitating granular material propagation as a function of the length scale of the flow. Finally we discuss the implications of our results in light of recent advances in our understanding of the influence of fluidization in the transport dynamics of PDCs (Brand et al. 2023; Breard
et al. 2019, 2023; Lube et al. 2020; Roche et al. 2021). Note that the new experimen-
tal results presented in this paper (in particular, the dependency between aspect ratio
and run-out distance for fluidized and non-fluidized granular flows) can be used as ref-
erence data to benchmark or validate other numerical models aimed to describe the
effect of pore pressure in granular flows and to scale results derived from the analysis
of analogue experiments.

2 Methods

2.1 Experimental configuration

The dam-break laboratory experiments presented in this paper were performed with
the experimental device described by Roche et al. (2010) (Fig. 1). It includes a reservoir
containing a column of granular material, separated by a sluice gate from a 3 m-long,
10 cm-wide horizontal channel where the granular material propagates laterally. The
sluice gate device includes a counterweight to ensure a rapid and uniform gate opening
phase. The reservoir base consists of a porous plate, through which a vertical air flow
can be injected from below in order to induce fluidization conditions (Roche et al.
2010; Roche 2012). Filters to dry the injected air and a series of manometers are also
included in the experimental device in order to control the imposed air flow and ensure
reproducibility of experiments. In the channel, no air flux is provided during flow
propagation. The granular material used in the experiments is constituted of spherical
glass beads with a density of ρ<sub>s</sub> = 2500 kg/m<sup>3</sup> and grain size between 60 and 90 μm
(mean of 75 μm).

<figure><img src="image_3.png" /><figcaption>Fig. 1 Simplified scheme of the experimental setup. Modified from Roche et al. (2010).</figcaption></figure>

We performed two sets of experiments (Table 1): (a) non-fluidized flows, where
no air flow is imposed on the particle column, and (b) fluidized flows, which are
characterized by an air flow supplied from below at a rate equivalent to the *minimum*
*fluidization velocity*, that is, the minimum value that guarantees that the bed weight
is counterbalanced by the drag generated by the air flowing between the particles, with negligible expansion of the granular material. For each set of experiments, we considered variable conditions for collapsing column width ($l_0$) and column height ($h_0$), giving rise to aspect ratios ($h_0/l_0$) between 0.5 and 8.0 (Table 1). Due to experimental limitations, high-aspect ratio experiments were only performed with relatively low values of column width ($l_0$), as given in Table 1. For each experiment, at $t = 0$, the reservoir gate is opened rapidly ($< 0.2$ s) allowing the granular material to spread laterally (note that air is still injected in the reservoir after gate opening). The resulting deposit is photographed and analysed to obtain the relationship between distance and thickness, and their dependency on flow scale and fluidization conditions.

## 2.2 Mathematical model

We performed simulations using the mathematical model presented in Aravena et al. (2021). The computational domain $\Omega$ is a 2D channel including the reservoir. At any time $t \ge 0$, the computational domain is decomposed as shown on Fig. 2, into $\Omega_s(t)$ which corresponds to the granular mass and $\Omega_f(t)$ which contains only (surrounding) air, called ambient air in the sequel. Note that at $t = 0$, $\Omega_s(t = 0)$ corresponds to the reservoir. The granular phase, in $\Omega_s(t)$, is a mixture of solid particles and air and is described by a two-phase model detailed in Section 2 of Chupin and Dubois (2024). It is based on the conservation of momentum equations for the two phases (Jackson 2000) involving forces between the two components, in particular the drag force which is proportional to the relative velocity. By comparing the order of magnitude of the different terms, the momentum equation of the fluid phase reduces to a simple equation similar to the Darcy law (Chupin and Dubois 2024). As a consequence, the mass conservation equation of the fluid phase leads to a linear advection-diffusion equation for the pore pressure (Iverson and Vallance 2001). We assume that the volume fraction of the particles remains constant so that the solid phase is incompressible. Finally, by using the $\mu(I)$ rheology (Jop et al. 2006), the propagation of the granular material is described by the following 2D incompressible model:

$$ \phi \rho_s (\partial_t \mathbf{u}_s + \mathbf{u}_s \cdot \nabla \mathbf{u}_s) + \nabla (p_s + p_f) = \phi \rho_s g + \text{div } \mathbf{T}_s \quad (1) $$

$$ \text{div } \mathbf{u}_s = 0 \quad (2) $$

$$ \partial_t p_f + \mathbf{u}_s \cdot \nabla p_f = \kappa \Delta p_f \quad (3) $$

where $\phi$ is the solid particles volume fraction, $\rho_s$ is the solid particles density, $\mathbf{u}_s$ is the material velocity, $p_s$ and $p_f$ are the solid (effective) pressure and pore pressure, respectively, $g$ is gravity and $\kappa$ is the effective pore pressure diffusion coefficient.

The deviatoric stress tensor $\mathbf{T}_s$ in equation (1) is defined by (Jop et al. 2006):

$$ \left\{ \begin{array}{ll} \mathbf{T}_s = \mu(I)p_s \frac{D(\mathbf{u}_s)}{|D(\mathbf{u}_s)|} & \text{if } \mathbf{D}(\mathbf{u}_s) \neq 0 \\ |\mathbf{T}_s| \le \tan(\alpha)p_s & \text{if } \mathbf{D}(\mathbf{u}_s) = 0 \end{array} \right. \quad (4) $$
<table><caption>**Table 1** Initial conditions and results of dam-break experiments.</caption><thead><tr><th rowspan="3">Experiment</th><th colspan="6">Non-fluidized</th></tr><tr><th colspan="3">Inputs</th><th colspan="3">Outputs</th></tr><tr><th>l<sub>0</sub> [cm]</th><th>h<sub>0</sub> [cm]</th><th>AR<sup>a</sup></th><th>max(h(t<sub>f</sub>))<sup>b</sup> [cm]</th><th>RD<sup>c</sup> [cm]</th><th>RD'<sup>d</sup> [cm]</th></tr></thead><tbody><tr><td>D1</td><td>5</td><td>10</td><td>2.00</td><td>5.2</td><td>23.4</td><td>15.7</td></tr><tr><td>D2</td><td>5</td><td>20</td><td>4.00</td><td>7.6</td><td>42.9</td><td>31.4</td></tr><tr><td>D3</td><td>5</td><td>30</td><td>6.00</td><td>8.8</td><td>60.6</td><td>47.0</td></tr><tr><td>D4</td><td>5</td><td>40</td><td>8.00</td><td>9.5</td><td>78.0</td><td>62.8</td></tr><tr><td>D5</td><td>10</td><td>10</td><td>1.00</td><td>8.3</td><td>25.8</td><td>17.7</td></tr><tr><td>D6</td><td>10</td><td>20</td><td>2.00</td><td>11.9</td><td>55.0</td><td>42.0</td></tr><tr><td>D7</td><td>10</td><td>30</td><td>3.00</td><td>13.9</td><td>76.3</td><td>61.5</td></tr><tr><td>D8</td><td>10</td><td>40</td><td>4.00</td><td>15.4</td><td>89.9</td><td>77.5</td></tr><tr><td>D9</td><td>15</td><td>10</td><td>0.67</td><td>12.2</td><td>24.9</td><td>17.1</td></tr><tr><td>D10</td><td>15</td><td>20</td><td>1.33</td><td>14.7</td><td>56.5</td><td>43.8</td></tr><tr><td>D11</td><td>15</td><td>30</td><td>2.00</td><td>17.5</td><td>83.8</td><td>67.3</td></tr><tr><td>D12</td><td>15</td><td>40</td><td>2.67</td><td>20.4</td><td>102.0</td><td>92.7</td></tr><tr><td>D13</td><td>20</td><td>10</td><td>0.50</td><td>9.9</td><td>22.3</td><td>16.3</td></tr><tr><td>D14</td><td>20</td><td>20</td><td>1.00</td><td>16.5</td><td>49.6</td><td>39.8</td></tr><tr><td>D15</td><td>20</td><td>30</td><td>1.50</td><td>19.9</td><td>81.7</td><td>66.6</td></tr><tr><td>D16</td><td>20</td><td>40</td><td>2.00</td><td>21.8</td><td>101.1</td><td>85.1</td></tr></tbody></table><table><thead><tr><th rowspan="3">Experiment</th><th colspan="6">Fluidized</th></tr><tr><th colspan="3">Inputs</th><th colspan="3">Outputs</th></tr><tr><th>l<sub>0</sub> [cm]</th><th>h<sub>0</sub> [cm]</th><th>AR<sup>a</sup></th><th>max(h(t<sub>f</sub>))<sup>b</sup> [cm]</th><th>RD<sup>c</sup> [cm]</th><th>RD'<sup>d</sup> [cm]</th></tr></thead><tbody><tr><td>F1</td><td>5</td><td>10</td><td>2.00</td><td>2.1</td><td>35.7</td><td>26.0</td></tr><tr><td>F2</td><td>5</td><td>20</td><td>4.00</td><td>3.0</td><td>63.5</td><td>53.6</td></tr><tr><td>F3</td><td>5</td><td>30</td><td>6.00</td><td>3.9</td><td>74.6</td><td>62.3</td></tr><tr><td>F4</td><td>5</td><td>40</td><td>8.00</td><td>4.1</td><td>90.3</td><td>80.2</td></tr><tr><td>F5</td><td>10</td><td>10</td><td>1.00</td><td>3.2</td><td>43.2</td><td>30.2</td></tr><tr><td>F6</td><td>10</td><td>20</td><td>2.00</td><td>4.5</td><td>85.2</td><td>66.9</td></tr><tr><td>F7</td><td>10</td><td>30</td><td>3.00</td><td>5.9</td><td>100.6</td><td>82.9</td></tr><tr><td>F8</td><td>10</td><td>40</td><td>4.00</td><td>7.8</td><td>109.9</td><td>92.8</td></tr><tr><td>F9</td><td>15</td><td>10</td><td>0.67</td><td>4.2</td><td>44.9</td><td>33.9</td></tr><tr><td>F10</td><td>15</td><td>20</td><td>1.33</td><td>5.5</td><td>106.1</td><td>84.9</td></tr><tr><td>F11</td><td>15</td><td>30</td><td>2.00</td><td>7.6</td><td>127.5</td><td>107.6</td></tr><tr><td>F12</td><td>15</td><td>40</td><td>2.67</td><td>8.5</td><td>154.6</td><td>137.4</td></tr><tr><td>F13</td><td>20</td><td>10</td><td>0.50</td><td>5.4</td><td>41.8</td><td>32.1</td></tr><tr><td>F14</td><td>20</td><td>20</td><td>1.00</td><td>7.1</td><td>96.2</td><td>74.3</td></tr><tr><td>F15</td><td>20</td><td>30</td><td>1.50</td><td>8.6</td><td>137.6</td><td>122.0</td></tr><tr><td>F16</td><td>20</td><td>40</td><td>2.00</td><td>10.9</td><td>153.2</td><td>139.5</td></tr></tbody></table>

$$ \mu(I) = \tan(\alpha) + \frac{(\mu_{\infty} - \tan(\alpha))I}{I + I_0}, \quad \text{with} \quad I = \frac{2d|D(u_s)|}{\sqrt{p_s/\rho_s}} \qquad (5) $$

<footnote><sup>a</sup>AR: aspect ratio. <sup>b</sup>max(h(t<sub>f</sub>)): maximum thickness of the resulting deposit. <sup>c</sup>RD: run-out distance. <sup>d</sup>RD': corrected run-out distance, considering a thickness threshold equivalent to the grid dimension used in numerical simulations.</footnote>

<footnote n="172">172 where $D(u_s) = \frac{1}{2}(\nabla u_s + \nabla u_s^t)$, $\alpha$ is the static internal friction angle of the granular material, and $\mu(I)$ is defined as follows:</footnote>

<footnote n="174">174 where $\mu_{\infty}$ is an asymptotic friction coefficient, $I_0$ is a dimensionless number and $d$ is the diameter of the solid particles.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 2 Computational domain adopted to describe numerically the collapse of a granular column.</figcaption></figure>

Following Chupin et al. (2021), the deviatoric stress tensor equation can be rewritten as follows:

$$ T_s = 2\eta_s(|D(\mathbf{u}_s)|, p_s)D(\mathbf{u}_s) + \tan(\alpha)p_s \frac{D(\mathbf{u}_s)}{|D(\mathbf{u}_s)|} \quad (6) $$

where $\eta_s(|D(\mathbf{u}_s)|, p_s)$ is defined by:

$$ \eta_s(|D(\mathbf{u}_s)|, p_s) = \frac{(\mu_\infty - \tan(\alpha))p_s}{2|D(\mathbf{u}_s)| + \frac{I_0}{d}\sqrt{p_s/\rho_s}} \quad (7) $$

Note from this formulation that the granular flow is described by a viscoplastic rheology with viscosity variable in time and space and a Drucker-Prager plasticity criterion (Jop et al. 2006; Ionescu et al. 2015). The numerical scheme adopted in the code includes a projection strategy (Chalayer et al. 2018; Chupin et al. 2021) to treat the singularity of $T_s$ in the absence of strain rate, dispensing with the application of any regularization technique.

In $\Omega_f(t)$, the ambient gas dynamics (i.e. the air outside the granular flow) is described with a Newtonian rheology:

$$ \rho_f(\partial_t \mathbf{u}_f + \mathbf{u}_f \cdot \nabla \mathbf{u}_f) + \nabla p_f = \rho_f g + \eta_f \Delta \mathbf{u}_f \quad (8) $$

$$ \mathrm{div} \, \mathbf{u}_f = 0 \quad (9) $$

where $\rho_f$ is air density, $\mathbf{u}_f$ is air velocity and $\eta_f$ is air dynamic viscosity.
Systems of equations (1)–(3) and (8)–(9) can be unified by using a level-set formulation (Osher and Fedkiw 2001), which writes:

$$ \partial_t \Phi + \mathbf{u} \cdot \nabla \Phi = 0 \qquad (10) $$

$$ \rho(\Phi)(\partial_t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla(p + p_f) = \rho(\Phi)\mathbf{g} + \operatorname{div}(2\eta(\Phi)\mathbf{D}(\mathbf{u})) + \operatorname{div}(\tau(\Phi)\mathbf{\Sigma}) \quad (11) $$

$$ \operatorname{div} \mathbf{u} = 0 \qquad (12) $$

$$ \partial_t p_f + \mathbf{u} \cdot \nabla p_f = \kappa(\Phi) \Delta p_f \qquad (13) $$

where $\Sigma$ is the plastic tensor (see Chupin et al. (2021) for details). The level-set function $\Phi$, which is transported by the velocity field, is used to describe the interface between the granular material and the ambient gas. Initially, the level-set function $\Phi$ is defined as the signed distance to the interface ($\Phi < 0$ for the granular flow and $\Phi > 0$ for the ambient gas). A redistancing algorithm is included in the code (Min 2010) and applied periodically in order to keep $\Phi$ sufficiently close to the signed distance function. The dependence of density $\rho(\Phi)$, viscosity $\eta(\Phi)$ and yield strength $\tau(\Phi)$ on $\Phi$ is given explicitly in Chupin et al. (2021). The diffusion coefficient $\kappa(\Phi)$ is constant and equal to $\kappa$ in granular material ($\Phi < 0$) and is fixed at $10^{16} \text{ m}^2/\text{s}$ in air (i.e. where $\Phi > 0$) ensuring that the pore pressure $p_f$ acts only on the granular flow (i.e., $p_f = 0$ where $\Phi > 0$).

Regarding the boundary conditions, Coulomb friction conditions (see Chupin et al. (2021)) with a constant basal friction angle $\alpha_b$ are imposed to the granular flow, while Navier (base), Free (top and reservoir wall), and outflow (wall opposite to the reservoir) boundary conditions are imposed to the air velocity field. Additional details of the model and numerical schemes used to discretise equations (10)–(13) are given in Chupin et al. (2021) and Aravena et al. (2021, 2022).

## 2.3 Input parameters

As discussed in Aravena et al. (2021), a key model parameter controlling the propagation dynamics of granular flows is the effective pore pressure diffusion coefficient ($\kappa$), which determines the rate at which pore pressure ($p_f$) diffuses within the granular material (see equation (3)). The benchmarking exercise presented by Aravena et al. (2021) allowed us to constrain the value of $\kappa$ in the reference experiment ($\kappa = 0.015 - 0.035 \text{ m}^2/\text{s}$), which was performed considering a collapsing column with a height of $h_0 = 40 \text{ cm}$ and a width of $l_0 = 20 \text{ cm}$, resulting in an aspect ratio of 2.0. However, it is important to highlight that the values of $\kappa$ obtained from the simulations are one order of magnitude smaller than the theoretical value and that extrapolating our estimates of $\kappa$ to different aspect ratio collapsing columns is particularly challenging (Montserrat et al. 2012; Aravena et al. 2021). Therefore, we made two assumptions for the present study, which are justified below:

(a) The effective pore pressure diffusion coefficient is independent on flow scale, which is consistent with the theoretical definition of this parameter, i.e. based exclusively on physical properties of the granular material (hydraulic permeability ($k$), pore volume fraction ($\varepsilon$)), and on viscosity ($\mu$) and compressibility ($\beta$) of the interstitial gas. Based on Aravena et al. (2021) and considering the experimental results presented in
this work, we consider an effective coefficient $\kappa = 0.025 \text{ m}^2/\text{s}$, and we also included for completeness the theoretical value of $\kappa$ for the benchmark experiment conditions (i.e. $\kappa = k/(\varepsilon\mu\beta) = 0.130 \text{ m}^2/\text{s}$, for which we have adopted $k \sim 10^{-11} \text{ m}^2$, $\varepsilon \sim 0.42$, $\mu \sim 1.8 \cdot 10^{-5} \text{ Pa s}$, and $\beta \sim 10^{-5} \text{ Pa}^{-1}$).

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(b) There is a linear relationship between the effective pore pressure diffusion coefficient and the initial height of the collapsing column (i.e., $\kappa = c \cdot h_0$, where c is a constant parameter). This assumption is consistent with experimental data from static defluidizing columns (Montserrat et al. 2012) but it has not been verified in dam-break experiments yet. Based on the results of Aravena et al. (2021) for the benchmark experiment, we considered $c = 0.0625 \text{ m/s}$.

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Note that both assumptions coincide in the value of $\kappa$ computed for the benchmark experimental conditions (i.e. $h_0 = 40 \text{ cm}$ and $l_0 = 20 \text{ cm}$), that is, $0.025 \text{ m}^2/\text{s}$. For each assumption, we developed a set of simulations of fluidized flows considering variable collapsing column aspect ratios (ranging from 1 to 8) and a fixed initial column width ($l_0$) of 20 cm (i.e., as in the benchmark experiment). A complementary set of simulations for non-fluidized granular flows with equivalent geometric characteristics was also performed. Note that, although the variation ranges of aspect ratio are equivalent, when compared to our numerical simulations, experimental data of high aspect ratio collapsing columns are associated with narrower columns due to limitations to reproduce experimentally collapse processes from too much height. The validity of this comparison scheme is supported by our experimental results of non-dimensional run-out distance as a function of aspect ratio (Fig. 3a), which show a well-defined, continuous dependency pattern between the mentioned variables even when experiments are associated with different values of $l_0$.

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The set of fixed input parameters adopted in our simulations is presented in Table 2. We stress that the inputs of the model (in particular, the Smagorinsky constant and time step) are slightly different from those presented in Aravena et al. (2021) for the benchmark experiment in order to reduce the computational time. These modifications are manifested in differences of the order of 5% in the simulated run-out distances for the benchmark experiment, much smaller than changes associated with the studied variation range of the effective pore pressure diffusion coefficient.

<table><caption>**Table 2** Constant input parameters used in numerical simulations.</caption><thead><tr><th>Input parameter</th><th>Symbol</th><th>Units</th><th>Value</th></tr></thead><tbody><tr><td>Particle density</td><td>ρ<sub>s</sub></td><td>kg/m<sup>3</sup></td><td>2500</td></tr><tr><td>Volume fraction of particles in the granular flow <sup>a</sup></td><td>φ</td><td>—</td><td>0.58</td></tr><tr><td>Internal friction angle of the granular material</td><td>α</td><td>°</td><td>27</td></tr><tr><td>Basal friction angle of the granular material</td><td>α<sub>b</sub></td><td>°</td><td>15</td></tr><tr><td>Air density</td><td>ρ<sub>f</sub></td><td>kg/m<sup>3</sup></td><td>1</td></tr><tr><td>Air viscosity</td><td>η<sub>f</sub></td><td>Pa s</td><td>2 · 10<sup>-5</sup></td></tr><tr><td>Collapsing column width</td><td>l<sub>0</sub></td><td>m</td><td>0.2</td></tr><tr><td>Smagorinsky constant</td><td>C<sub>s</sub></td><td>—</td><td>0.2</td></tr><tr><td>Computational cell size</td><td>Δx</td><td>cm</td><td>0.5</td></tr><tr><td>Time step</td><td>Δt</td><td>s</td><td>2 · 10<sup>-5</sup></td></tr></tbody></table>

<footnote><sup>a</sup> In non-fluidized simulations we considered $\phi = 0.62$, which is based on experimental constraints.</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Non-dimensional run-out distance (RD/h₀) versus aspect ratio for fluidized and non-fluidized dam break analogue experiments (panel a) and for numerical simulations (panels b, c and d), considering different assumptions to set the dependency between the effective pore pressure diffusion coefficient (κ) and collapse height (h₀; see titles). To simplify comparison with numerical results, the experimental data are presented in terms of the observed run-out distance (upper limit of bars) and we also include two corrected values of run-out distance, defined as the distance from the reservoir to the position at which the deposit thickness reaches thresholds equivalent to the cell size used in numerical simulations (lower limit of bars) and equivalent to half the cell size used in numerical simulations (filled markers). Note that the analogue experiments of high aspect ratio collapsing columns are associated with lower values of l₀ due to experimental limitations to reproduce collapse processes from too much height.</figcaption></figure>

## 3 Results

The deposits of experimental fluidized granular flows are characterized by surface slopes that gently decrease downstream, with the maximum thickness observed in the channel near the reservoir (Fig. 4; Roche et al. 2010; Aravena et al. 2021). In contrast, experimental non-fluidized flow deposits present a monotonically decreasing thickness downstream and have significantly larger surface slopes (Fig. 4). The effect of fluidization in increasing the run-out distance is evident in all the tested experimental setups, although the magnitude of this increase is also strongly controlled by the initial collapsing column aspect ratio (Table 1 and Fig. 3a). While fluidization is able to double the run-out distance for aspect ratios of about 1–2, the relative influence of fluidization is significantly reduced for higher aspect ratios. In fact, the
non-dimensional run-out distance of fluidized granular flows reaches a peak of about
~ 4 – 5 for an aspect ratio of ~ 1 – 2 and then it decreases monotonically and is
only slightly larger than that of non-fluidized flows at high aspect ratios (AR > 5).
Instead, non-fluidized granular flows present non-dimensional run-out distances weakly
controlled by aspect ratio (Fig. 3a), with values around RD/h₀ = 2 for the entire
variation range of aspect ratio used in experiments, with a relative maximum for aspect
ratios of ~ 1 – 3.

<figure><img src="image_6.png" /><figcaption>**Fig. 4** Surface profiles of deposits for dam-break experiments of fluidized and non-fluidized granular flows (see titles), considering different geometries of the collapsing columns (see legend and titles).</figcaption></figure>

Numerically simulated granular flows present an initial, short acceleration phase
followed by a period of flow propagation at nearly constant velocity and finally flow
front deceleration and stopping (Fig. 5). Froude numbers computed from numerical
simulations during the constant velocity phase are of the order of 0.86 – 1.59 (non-
fluidized flows, with supercritical conditions for AR > 2), 1.21 – 5.91 (fluidized flows
with κ = 0.025 m²/s), 1.54 – 2.69 (fluidized flows with κ = 0.0625 m/s · h₀) and
1.08 – 1.95 (fluidized flows with κ = 0.130 m²/s). The relative duration of each prop-
agation phase is similar in all the simulations presented here, with most of the flow
propagation time corresponding to the phase of deceleration (Fig. 6). The significant
increase in the non-dimensional run-out distance due to the effect of fluidization is
a consequence of the flow dynamics during early phases of propagation (Fig. 6). In
fact, front acceleration of non-fluidized flows is of the order of 0.7g, while that of flu-
idized flows is about 2.0 – 2.3g. Regarding variations in κ for fluidized flows, they are
manifested in differences in the late stage of acceleration and in the duration of the
constant front velocity phase (Fig. 6). On the other hand, the rates of deceleration of
fluidized and non-fluidized granular flows are remarkably similar (~ 0.22g), regardless
of the value of κ. This suggests that initially fluidized flows behave as non-fluidized
flows at late stages of propagation as a consequence of pore pressure diffusion during flow propagation, as discussed in Aravena et al. (2021).

When $\kappa$ is assumed independent on flow scale and equal to 0.025 m²/s (i.e. as determined in Aravena et al. (2021) to fit a benchmark experiment), the simulated non-dimensional run-out distance is an increasing function of aspect ratio up to AR ≈ 3 above which it is equal to about 4.2 – 4.3 (Fig. 3b). Instead, when $\kappa$ is assumed proportional to $h_0$, the non-dimensional run-out distance presents a relative maximum for an aspect ratio of about 1.5 and then it decreases rapidly (Fig. 3c). Finally, when $\kappa$ is assumed equal to the theoretical value (i.e. 0.130 m²/s), the effect of fluidization in increasing the simulated run-out distance is strongly reduced and a behaviour similar to that of non-fluidized flows is observed (Fig. 3d). The comparison between our simulations and experimental results (Fig. 3a) indicates that a positive dependency between the effective pore pressure diffusion coefficient and reservoir column height is needed to describe successfully the behaviour of the studied granular flows. This is also consistent with results for defluidizing static columns (Montserrat et al. 2012) and further demonstrates that the theoretical value of $\kappa$ fails in predicting the behaviour of the studied analogue experiments of fluidized flows (Figs. 3d and 5c). On the other hand, non-fluidized granular flow simulations present non-dimensional run-out distances weakly controlled by aspect ratio (Fig. 3b-c), with values around 2.0 and a relative maximum at an aspect ratio of ~ 1.5, in fair agreement with experimental data, which further confirm the validity of our model. According to the results described above, the following analysis of our numerical results is focused on non-fluidized flows and on fluidized flows with $\kappa$ assumed proportional to $h_0$.

The temporal evolution of the reservoir column collapse velocity for both fluidized and non-fluidized flows is characterized by an initial rapid acceleration phase that lasts about $\sqrt{h_0/g}$ and later a gradual deceleration stage, which is significantly steeper for high aspect ratio collapsing columns (Fig. 7). This abrupt reduction in the non-dimensional collapse velocity of high aspect ratio collapsing columns limits the rate at which granular material is introduced in the propagation channel and thus the effective influence of fluidization in favoring flow mobility during the final phases of collapse. This effect is enhanced (1) by the decrease of basal pore pressure with time in the reservoir (Fig. 8), which translates into an increase of the friction acting on the granular material during propagation; (2) by the efficient diffusion-driven decrease of pore pressure along the channel, and (3) by the rapidly increasing distance between new arrivals of granular material into the propagation channel and the flow front (Fig. 9). Therefore, the collapsing material at late stages is not able to reach the flow front and to promote increase of run-out distance, as shown by data in Fig. 3c. Complementary analogue experiments of initially fluidized high-aspect ratio collapsing columns with stratigraphic markers support this conclusion (Fig. 10). In fact, in these experiments, the granular material reaching the flow front originates from lower levels of the collapsing column. In contrast, the granular material collapsed during late stages of the experiments is deposited in proximal to intermediate domains. This observation represents an additional feature of analogue experiments of potential utility for benchmarking numerical models aimed at describing the effect of fluidization in the propagation dynamics of granular flows.
<figure><img src="image_7.png" /><figcaption>**Fig. 5** Non-dimensional position of the flow front as a function of non-dimensional time for numerical simulations of fluidized and non-fluidized granular flows (see titles). Note that numerical results further demonstrate that the theoretical value of $\kappa$ (panel c) completely fails in predicting the non-dimensional run-out distance measured in analogue experiments (see Fig. 3, i.e., $\text{RD}/h_0 \approx 4$ for an AR of 2.0).</figcaption></figure>

Following Roche (2012), Chupin et al. (2021) and Aravena et al. (2021), we investigated the sedimentation dynamics of granular flows by considering the area of material deposited as a function of time in our numerical simulations ($A_d$, Fig. 11). To produce results comparable with experiments, we assumed that a given cell of granular material in the computational domain is part of the deposited material if its velocity magnitude was smaller than an arbitrary threshold value of 0.1 m/s, following Aravena et al. (2021). In general, deposition of the granular material occurs at late stages of flow emplacement. For aspect ratios between 3 and 8, most of the deposition occurs during the final 40% of simulations (when the flow has already travelled > 80% of the run-out distance; Fig. 11b-c), while deposition occurs at even later stages of flow propagation ($t/t_f > 0.8$ and $L/L_f > 0.95$; Fig. 11b-c) for collapsing columns with an aspect ratio of about 1. These results are consistent with Roche (2012) and Aravena et al. (2021), and highlight the capability of our model to capture different aspects of granular flows propagation at experimental scale. Note that sedimentation is here described as a consequence of the momentum decrease of solid particles, which differs from other interpretations of sedimentation that consider this process as caused by hindered settling of particles (Girolami and Risso 2020; Shimizu et al. 2023; Roche et al. 2024).
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Non-dimensional velocity of the flow front as a function of non-dimensional time for numerical simulations of fluidized and non-fluidized granular flows (see titles).</figcaption></figure>

# 4 Discussion

In this work, we adopted a 2D multi-phase model to study granular flows in the dam-break configuration, considering in particular high aspect ratio collapsing columns. Our simulations permitted us to address how the effect of fluidization in increasing the run-out distance varies with flow dimensions, to analyse the propagation dynamics of fluidized granular flows and to describe sedimentation processes.

## 4.1 Relationship between pore pressure diffusion coefficient and column height

Comparison of numerical results with experimental data suggests that the effective pore pressure diffusion coefficient is an increasing function of the height of the granular material column. This confirms previous observations that the theoretical definition of pore pressure diffusion coefficient does not allow simulation of the behaviour of fluidized granular flows and is consistent with experiments of static defluidizing granular columns (Montserrat et al. 2012; Roche 2012). These experiments, which were performed by measuring the decay in time of basal pore pressure in an initially aerated granular mixture, show a linear relationship between the effective pore pressure diffusion coefficient and reservoir column height. These observations had not been tested in simulations of fluidized granular flows until this study. With the assumption of a linear dependency between $\kappa$ and $h_0$, the results of our simulations are close to the experimental data in terms of non-dimensional run-out distance (Fig. 3), even though $h_0$ can be considered an oversimplified parameter to characterize pore pressure diffusion given that flow height, both in the reservoir and the propagation channel, varies in position and time in dam-break experiments. This agreement between experiments
<figure><img src="image_9.png" /><figcaption>Fig. 7 (a,c) Non-dimensional velocity of column collapse in the reservoir as a function of non-dimensional time for numerical simulations of fluidized and non-fluidized granular flows (see titles). (b,d) Peak of non-dimensional velocity of column collapse in the reservoir as a function of aspect ratio.</figcaption></figure>

and simulations performed assuming a linear dependency between $\kappa$ and $h_0$ could be a consequence of the good correlation recognized in numerical results between $h_0$ and the average value of flow height in the propagation channel ($\bar{h}$), weighted in time and space by the basal pore pressure as follows:

$$ \bar{h} = \frac{\int_{0}^{l_x} \int_{0}^{t_f} (h(x, t) \cdot p_f(x, y = 0, t)) \, dt \, dx}{\int_{0}^{l_x} \int_{0}^{t_f} p_f(x, y = 0, t) \, dt \, dx} $$

where the horizontal limits of the computational domain are given by ($-l_0$) and $l_x$, $t_f$ is the simulation time, $h(x, t)$ is flow height at position $x$ and time $t$, and $p_f(x, y = 0, t)$ is pore pressure at the base of the propagation channel at position $x$ and time $t$. The parameter $\bar{h}$ likely represents a more informative parameter to describe the diffusion process of pore pressure in a granular flow. As expected, this results in only small differences among the estimates of $\kappa$ based on $h_0$ ($\kappa \propto h_0$) and those based on the parameter $\bar{h}$ ($\kappa \propto \bar{h}$) for the aspect ratio range considered in this study (Fig. 12), which translates in turn into slight variations in the non-dimensional run-out distances derived from the application of both the assumptions to set $\kappa$, as shown in Fig. 13. However, please note that $\bar{h}$ is not an input parameter of our model, and thus its use to compute $\kappa$ involves the application of an iterative procedure that significantly increases the computational time. Moreover, it is worth highlighting that our model considers a constant pore pressure diffusion coefficient, which may not describe correctly the behaviour of fluidized flows due to the effect of dilatancy/compaction (Breard et al. 2019; Bouchut et al. 2021). Based on our results and recent advances in
<figure><img src="image_10.png" /><figcaption>**Fig. 8** Temporal evolution of non-dimensional basal pore pressure at two different distances along the channel (see titles) as a function of non-dimensional time for fluidized granular flows (with $\kappa = 0.0625 \text{ m/s} \cdot h_0$), considering different aspect ratio collapsing columns (see legends).</figcaption></figure>

the research field of granular flows (Breard et al. 2019, 2023; Bouchut et al. 2021), we believe that consideration of dilatancy/compaction in future studies based on numerical modelling is crucial to deepen our understanding of the relationship between pore pressure diffusion and flow dimensions.

## 4.2 Efficiency of fluidization in increasing run-out distance and volcanological implications

The analysis of analogue experiments and their numerical description has been widely adopted in volcanology for understanding the dynamics of granular flows such as PDCs (Roche et al. 2010; Roche 2012; Breard et al. 2019; Aravena et al. 2021). However, scaling these results to natural systems is not straightforward and raises still unsolved aspects (Lube et al. 2004, 2005; Delannay et al. 2017; Man et al. 2021). We adopted a numerical modelling-based approach to scale, from a benchmark experimental setup, the behaviour of fluidized granular flows and, from this, we have evaluated the factors controlling the efficiency of fluidization in increasing the mobility of granular flows. We have shown that the interplay between column collapse and flow front velocities determines the relationship between column aspect ratio and run-out distance, and that this interplay is critically influenced by initial fluidization. In high aspect ratio
<figure><img src="image_11.png" /><figcaption>**Fig. 9** Flow front position when the peak of collapse velocity is reached as a function of aspect ratio, for simulations of fluidized and non-fluidized granular flows (see legend).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10** Results of complementary analogue experiments of an initially fluidized collapsing column, showing that the material reaching the flow front comes from lower levels of the column.</figcaption></figure>

collapsing columns as those modelled here, the capacity of fluidization to increase the flow run-out distance is hindered by three factors: (1) the abrupt decrease of reservoir column collapse velocity at late stages for high aspect ratios; (2) the rapidly increasing distance between the material entering the propagation channel and the flow front; and (3) the decrease of pore pressure in the reservoir with time and the corresponding increase of friction of the material entering the channel at late stages. Moreover, these factors reinforce the negative feedback mechanism between flow acceleration and fluidization described by Aravena et al. (2021), i.e. that the fluidization-induced velocity increase of the flow reduces its thickness, which in turn promotes rapid pore pressure diffusion and limits the rheological effect of fluidization.

The dam-break configuration can be considered as an analogue of PDCs generated by collapse of lava domes, lava flow fronts or accumulated pyroclastic material. Simulation of initially fluidized conditions applies better to the collapse of pressurized lava domes, and thus our experiments and numerical simulations are expected to be informative of this type of PDCs. The described results show that, in absence of
<figure><img src="image_13.png" /><figcaption>**Fig. 11** Deposition dynamics of particles in numerical simulations of fluidized granular flows ($\kappa = 0.0625 \text{ m/s} \cdot h_0$) and experiments. (a) Diagram showing the definitions used to describe the deposition dynamics. $A_d$: material deposited in the channel. $A_f$: material flowing. $L_d$: distance from the reservoir to the sedimentation front. (b) $A_d/A_d(t_f)$ as a function of $t/t_f$, where $t_f$ is the final time. (c) $A_d/A_d(t_f)$ as a function of $L/L_f$, where $L_f$ is the final run-out distance.</figcaption></figure>

secondary sources of fluidization during granular flow propagation, the positive effect of initial fluidization on non-dimensional run-out distance is maximum for an aspect ratio of 1–2 and it decreases beyond this critical value, suggesting that estimating the initial aspect ratio of a dome susceptible to collapse can be relevant for hazard assessment. Note that our simulations describe the collapse of a homogeneous column characterized by a constant pore pressure diffusion coefficient and uniform density (i.e. incompressibility conditions), while variations in the grain size distribution and density of the granular material are likely to occur during flow propagation in nature. These variations may induce a more complex dynamics of pore pressure diffusion and thus affect the resulting run-out distance, which is not captured by the presented numerical model. In fact, different authors have addressed the influence of PDC compressibility on pore pressure generation and diffusion (Breard et al. 2019, 2023; Walding et al. 2023). Breard et al. (2019) showed how pore pressure can be modulated by dilation and compaction of granular flows and that, although external sources of gas may enhance the effect of pore pressure in flow mobility, they are not required to generate fluidization conditions in pyroclastic mixtures. Breard et al. (2023) invoke a relevant effect of
<figure><img src="image_14.png" /><figcaption>**Fig. 12** Effective pore pressure diffusion coefficient as a function of aspect ratio, considering two assumptions to set the dependency between $\kappa$ and thickness of the granular material: as a function of initial column height ($h_0$) and as a function of the flow thickness in the channel, weighted in space and time by pore pressure ($\bar{h}$; see legend).</figcaption></figure>

compressibility and changes in the grain size distribution of the granular mixture during its propagation, which is able to increase the effective pore pressure, reduce basal friction and thus explain the high mobility of some block-and-ash flows. These studies further reinforce the need to incorporate variability in the granular material pore volume and permeability in numerical models aimed at understanding quantitatively the behaviour of pore pressure during propagation of granular flows and its implications on volcanic hazards.

## 5 Concluding remarks

The analysis of dam break experiments and their numerical simulations, which can be considered as analogues of PDCs generated by collapse of lava domes, lava flow fronts or accumulated pyroclastic material, reveals that:

(a) The relationship between the timescale of column collapse and that of flow front propagation controls the efficiency of initial fluidization in increasing the run-out distance of granular flows.

(b) In absence of secondary sources of fluidization and/or significant effects of compressibility/dilatancy during flow propagation, the effect of initial pore pressure of a collapsing column on the non-dimensional run-out distance presents its maximum for aspect ratios between 1 and 2, and this influence decreases rapidly for higher aspect ratios. This is due to the abrupt reduction of collapse velocity recognized in
<figure><img src="image_15.png" /><figcaption>**Fig. 13** Non-dimensional run-out distance versus aspect ratio for numerical simulations of fluidized and non-fluidized granular flows, considering different assumptions to set the dependency between $\kappa$ and thickness of the granular material (see legend).</figcaption></figure>

numerical simulations, the decrease of basal pore pressure with time in the reservoir and the rapidly increasing distance between the flow front position and material incorporated into the propagation channel.

(c) Comparison of numerical results with analogue experiments suggest a positive correlation between column height and effective pore pressure diffusion coefficient. This has been previously demonstrated for defluidization of static columns, but this is the first time this relationship is evaluated in dam-break experiments. Further investigation is required to understand the relationship between column height and diffusion coefficient.

Thereby, our study stresses the relevance of having reference experimental data to benchmark and confirm numerical models of granular flows, especially when the physics of these systems is poorly understood and subjected to several sources of uncertainty. We have presented a series of features of fluidized and non-fluidized experimental granular flows that can be potentially useful for the development of benchmarking exercises of numerical models, such as: dependency between non-dimensional run-out distance and aspect ratio (Fig. 3); sedimentation dynamics (Fig. 11); and stratigraphic relationship between the collapsing column and the resulting deposit (Fig. 10).

## Acknowledgements

This research was financed by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25). This is Laboratory of Excellence ClerVolc contribution number 669. The numerical simulations have been performed on a DELL cluster with 32 processors Xeon E2650v2 (8 cores), 1 To of total memory, and an infiniband (FDR
56Gb/s) connecting network. This cluster has been financed by the French Government Laboratory of Excellence initiative n°ANR-10-LABX-0006. Alvaro Aravena has been supported by the Millennium Institute on Volcanic Risk Research – Ckelar Volcanoes ANID/MILENIO (grant no. ICN2021_038). We thank the editor and reviewers of this manuscript, Dr. Greg Valentine, Dr. Eric Breard and Dr. Hiroyuki Shimizu, for their valuable contributions and suggestions, which significantly improved this paper.

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