Localized fluidization in granular materials: Theoretical and numerical study

We present analytical and numerical results on localized fluidization within a granular layer subjected to a local injection of fluid. As the injection rate increases the three different regimes previously reported in the literature are recovered: homogeneous expansion of the bed, fluidized cavity in which fluidization starts developing above the injection area, and finally the chimney of fluidized grains when the fluidization zone reaches the free surface. The analytical approach is at the continuum scale, based on Darcy's law and Therzaghi's effective stress principle. It provides a good description of the phenomenon as long as the porosity of the granular assembly remains relatively homogeneous, i.e., for small injection rates. The numerical approach is at the particle scale based on the coupled discrete element method and a pore-scale finite volume method. It tackles the more heterogeneous situations which occur at larger injection rates. The results from both methods are in qualitative agreement with data published independently. A more quantitative agreement is achieved by the numerical model. A direct link is evidenced between the occurrence of the different regimes of fluidization and the injection aperture. While narrow apertures let the three different regimes be distinguished clearly, larger apertures tend to produce a single homogeneous fluidization regime. In the former case, it is found that the transition between the cavity regime and the chimney regime for an increasing injection rate coincides with a peak in the evolution of inlet pressure. Finally, the occurrence of the different regimes is defined in terms of the normalized flux and aperture.

Figure 1

Sample geometry and boundary conditions.

Figure 2

Evolution of the dimensionless pressure field $p^{*}$ for an injection aperture $a^{*}=0.1$ and two injection rates $q^{*}=0.00019$ (a) and $q^{*}=0.00067$ (b).

Figure 3

Evolution of the dimensionless effective stress field when the injected flux increases. Narrow aperture $\left(a^{*}=0.1\right)$ on the first line and wide aperture $\left(a^{*}=0.8\right)$ on the second line. The normalized fluxes are $q^{*}=0.00019$ (a),(d), $q^{*}=0.00105$ (b),(e), $q^{*}=0.00114$ (c), and $q^{*}=0.00126$ (f).

Figure 4

Evolution of the porosity for narrow aperture $\left(a^{*}=0.1\right)$ in a very detailed color scale. (a) Static regime, $q^{*}=0$. (b) Expansion regime, $q^{*}=0.00095$.

Figure 5

Evolution of the porosity for narrow aperture $\left(a^{*}=0.1\right)$. (a) Static regime, $q^{*}=0$. (b) Cavity regime, $q^{*}=0.00116$. (c) Chimney regime, $q^{*}=0.00145$. (d) Irreversible changes in porosity after reducing the flux back to $q^{*}=0$.

Figure 6

Evolution of the fluidized zone for narrow aperture $\left(a^{*}=0.1\right)$. Blue (light gray) zone represents nonfluidized zone $\left(\frac{\sigma }{{\sigma }_o}\ge 0.1\right)$. Red (dark gray) zone corresponds to fluidized zone $\left(\frac{\sigma }{{\sigma }_o}<0.1\right)$. (a) Cavity regime, $q^{*}=0.00116$. (b) Chimney regime, $q^{*}=0.00145$.

Figure 7

Fluidized zone for a wide aperture $\left(a^{*}=0.9\right)$ and $q^{*}=0.00156$; an important part of the sample has liquefied. Blue (light gray) zone represents nonfluidized zone $\left(\frac{\sigma }{{\sigma }_o}\ge 0.1\right)$. Red (dark gray) zone corresponds to fluidized zone $\left(\frac{\sigma }{{\sigma }_o}<0.1\right)$.

Figure 8

Normalized height of the cavity in the theoretical and numerical models, compared with the experimental data from [14].

Figure 9

Dimensionless pressure-flux, $p^{*}\text{−}{q}^{*}$, curve for an increasing-decreasing cycle of flux at the bottom of the sample. DEM-PFV simulation with an injection aperture $a^{*}=0.1$.

Figure 10

$p^{*}\text{−}{q}_k^{*}$ curves comparison at the center of the injection orifice with an injection aperture $a^{*}=0.1$.

Figure 11

Relative error between numerical and analytical pressure within the entire sample. (a) Expansion regime, $q^{*}=0.00095$. (b) Cavity regime, $q^{*}=0.00116$.

Figure 12

Evolution of the analytical and numerical effective stress fields for different flow rates (injection aperture $a^{*}=0.1$). (a) $q^{*}=0.00105$: cavity regime; (b) $q^{*}=0.00114$: chimney regime in the theoretical model; cavity regime in the numerical model. (c) $q^{*}=0.00120$: chimney regime.

Figure 13

Sensitivity analysis on pressure-flux curves with injection aperture $a^{*}=0.1$. Viscosity dependency with normalized $p^{*}\text{−}{q}^{*}$ variables (a) and physical units (b); diameter dependency with normalized $p^{*}\text{−}{q}^{*}$ variables (c) and physical units (d).

Figure 14

Aperture dependency. $p^{*}\text{−}{q}^{*}$ curves at the bottom of the sample with different injection apertures.

Figure 15

Occurrence of the different fluidization regimes (static or expansion, cavity, or chimney) depending on aperture $a^{*}$.

Figure 16

Theoretical prediction of the different regimes (static or expansion, cavity, or chimney regimes) in the dimensionless aperture-flux, $a^{*}\text{−}{q}^{*}$, plane.

Figure 17

Fluid flowing outwards in a 2D porous media due to a punctual source.

Figure 18

Pressure field for semi-infinite sources net.