2.1. Experimental set-up

The experimental set-up, shown in figure 2(a), consists of a linear motor stage translating a porous material at a constant velocity $V_0$ into a water reservoir of large dimensions. The porous medium is formed by packing spherical glass beads into a transparent cell of length 40 cm, of variable width $W=5$, 10, or 20 cm and of thickness 12 mm, so that the configuration can be considered bidimensional, as shown in figure 2(b). The side and bottom walls of the cell are permeable thanks to a metal wire mesh with openings of 250 $\mathrm {\mu }$m (visible in figure 2a). The low thickness of the cell allows us to observe the dry and wet grains as dark and light, respectively, using the contrast of absorption of the light emitted by an LED panel placed behind the set-up (figure 2c). It also allows us to track the impregnation front separating the dry and wet regions.

Figure 2. (a) Photo of the porous medium made of grains (system W3). The front and rear are made of PMMA, and the sides and bottom of the cell consist of a metal wire mesh with openings of 250 $\mathrm {\mu }$m. (b) Schematic of the experimental set-up. A vertical linear motor imposes the speed of translation $V_0$. (c) Picture of the stationary impregnation front during the translation of the porous medium into the water bath. The system is illuminated by an LED panel placed behind the cell. The light part is fully saturated with water whereas the dark part is dry.

The porous medium is translated vertically into the water bath at a constant speed $V_0$ varying between 1 mm s$^{-1}$ and 150 mm s$^{-1}$. The impregnation dynamics is recorded at 30 frames per second (using a Nikon D-7100 with an F30 lens) and then analysed by image processing to extract the shape of the impregnation front and its time-evolution during the experiment.

2.2. Characterization of porous media

Each porous medium was prepared by filling the transparent cell with glass beads. Different systems of grains were used to tune the pore size and the wettability of the porous medium. We performed the experiments with the systems W1, W2, W3, where W stands for ‘wetting’, and the systems N1 and N2, where N stands for ‘non-wetting’ thanks to coated glass beads (Sigmund Lindner). The coating changes the surface properties of the beads by increasing the contact angle with the liquid. The beads are sieved to refine the size distributions, which are summarized in table 1 (see also Appendix A). For each system, the glass beads are cleaned with soap and thoroughly rinsed with de-ionized water to remove any dust on the grains.

Table 1. Physical properties of the grains composing the model porous media.

The contact angle between the grain and the water is measured by trapping a single bead at the surface of a pending drop of liquid (Timounay, Lorenceau & Rouyer Reference Timounay, Lorenceau and Rouyer2015). For each experiment, the granular packing is prepared by following a protocol detailed in Appendix A. Several taps are given to compact the granular packing until it reaches a constant volume fraction of $\phi \simeq 0.62$. The compacity $\phi$ and the porosity $\epsilon$ of the granular packing are estimated by measuring the mass and the volume occupied by the grains (see Appendix A). The permeability is measured by an experiment of filtration through the porous medium, following the procedure presented by Chopin & Kudrolli (Reference Chopin and Kudrolli2011). This measurement is performed at low Reynolds number, in the range of velocities compatible with Darcy's law. The measured permeability is denoted by $k_1$ in table 1. Another set of measurements of the permeability is performed at larger Reynolds numbers to determine the Forchheimer coefficient $\beta _2$, which is required in the Darcy–Forchheimer equation used in § 5. Furthermore, a third value for the permeability, denoted by $k$, is used to fit the theoretical prediction to the experimental data. Only the values $k$ and $\beta _2$ are used in this paper. The discrepancies between the measurements of the permeability are relatively small and remain less than $20\,\%$ for all experimental configurations. The small difference can be explained by experimental artifacts introduced by the glass tube used for the flow measurements through the packed beads. The physical characteristics of all model porous media used in this study are summarized in table 1, and all the experimental methods used to characterize the properties of the beads are detailed in Appendix A.

2.3. Phenomenology

An example of a time series extracted from an experiment is shown in figure 3. This experiment is performed with a porous medium made of glass beads from the system W1 in a cell 5 cm wide translating into a water bath at a constant velocity $V_0=30$ mm s$^{-1}$. When the porous medium plunges into the water, air is entrained within the porosity of the material, and liquid penetrates laterally and vertically into the pores, which leads to the $V$-shaped impregnating front visible in the last image. This front separates the part of the porous medium invaded by the liquid (the wet grains, light region in figure 3) from the dry part of the sample (the dark region), which contains only air in the pores. The dynamical impregnation is characterized by a transient phase for $t < \tau$, where $\tau$ is the time to reach a stationary impregnation profile. The transient phase is followed by a stationary regime for $t > \tau$. During the transient phase, the impregnation front evolves through different shapes until it reaches a steady $V$ shape in the stationary regime. The extent of the $V$ shape increases with the imposed velocity $V_0$, leading to sharper profiles with straighter sides.

Figure 3. Time series during the transient phase of a porous medium of width $W=10$ cm composed of the system W2 plunging into water at a speed $V_0 = 30$ mm s$^{-1}$.

The first row of images in figure 3 shows that initially the lower part of the front remains flat and parallel to the bottom of the cell during the transient regime. On the horizontal front and at the bottom, the pressure is constant, so that the streamlines are vertical. To model this transient phase, a 1D vertical model is sufficient. Later, a two-dimensional (2D) flow takes place inside the porous medium. However, in the vicinity of the free surface, one can foresee that at first order the flow could be governed only by the vertical boundaries and the shape of the top front, so that the bottom of the cell has no effect. We may even idealize the top corners of the system as a triangular shape, which will be more valid at large velocities, as we will see later. In this case, we again anticipate that a 1D lateral-flow model may be sufficient to account for the experimental results, while a full 2D numerical model will be needed for intermediate regimes.

In any case, the flow in the porous medium is driven by the capillary pressure and the hydrostatic pressure. The Darcy equation in a porous medium is expected to capture the situation. However, at early times, when the length of wet grains is small, inertia may be important; we will test the relevance of inertia in describing such flows when our data allow us to follow the entire time evolution of the front.

Finally, the wettability of the grains plays a significant role. For non-wetting grains, the impregnation does not start before the grains reach the depth at which the hydrostatic pressure compensates the withstanding capillary pressure. For wetting grains, the liquid invades the granular media more rapidly, even allowing the liquid to rise above the free surface of the water for very low driving velocity. It is worth noting that the wettability of the grid itself modifies the behaviour of the system. The liquid needs to overcome a small pressure to flow through the grid, corresponding to a depth $h_0$. However, this pressure is not always the same, because the holes are partially clogged by grains. The resulting pores of the grid in fact comprise both the grid and the grains of differing wettability, which leads to different $h_0$ for the start of impregnation.

3.1. Dynamics of impregnation

We study the transient regime by tracking the temporal evolution of the impregnation front. The shape of the impregnation front is reported at different times in figure 4(a) for an experiment performed with a porous medium of width $W=10$ cm composed of grains from the system W2 and plunged in the water bath at a constant velocity $V_0 = 30$ mm s$^{-1}$. The impregnation front evolves continuously from an initial flat profile to a $V$-shaped profile by keeping a constant slope on the side. The temporal evolution of the impregnation is characterized by measuring the vertical length of air entrained in the bath, which is denoted by $\ell _{dry}(t)$ (see figure 4a). The slope of the impregnation front remains constant throughout the process and will be characterized in the next section. The time evolution of $\ell _{dry}(t)$ is presented in figure 4(b). The length $\ell _{dry}(t)$ increases linearly with a velocity $V_f$ before saturating to a constant value, $\ell _{dry}(t \rightarrow \infty ) = h_{dry}$, which corresponds to the maximum length of air entrained by the porous material when the stationary regime is reached.

Figure 4. (a) Evolution of the impregnation front during the transient regime. The data are extracted from the experiment shown in figure 3. The air–water interface is located at $z=0$. (b) Evolution of the maximum length of air $\ell _{dry}(t)$ entrained under the liquid surface. The red circles correspond to the position of the bottom of the porous medium, which is translated at a constant velocity V ₀ under the surface.

Using $h_{dry}$ and $\tau = h_{dry}/V_f$ as the characteristic length scale and time scale, respectively, we can rescale the evolution of $\ell _{dry}(t)$ for all the experiments performed with the different model porous media. The results, reported in figure 5, show that all the rescaled data, which are measured mainly for $t/\tau >0.25$, seem to collapse on an empirical law with a single fitting parameter, $n$:

(3.1)\begin{equation} \frac{\ell_{dry}}{h_{dry}} = f \left( \frac{t}{\tau} \right) \quad \mbox{with } f(\xi) = ( 1 - \textrm{e}^{-\xi^n} )^{1/n} \quad \mbox{and} \quad n \simeq 4.5 . \end{equation}

We have few experimental data for $t/\tau <0.25$, where disturbances from the corners and from the initial impregnation are seen.

Figure 5. Rescaled length of air entrained under the free surface, $\ell _{dry}/h_{dry}$, as a function of the dimensionless time $t/\tau$. The figure summarizes the results of 30 sets of parameters varying the size of the grains, the width of the cell $W$ and the plunging speeds $V_0$. Experiments with beads of systems W1, W2 and W3 are plotted in blue, red and green, respectively. The red dashed line is the empirical fit given by (3.1).

The impregnation by the fluid from the bottom wall of the porous medium governs the evolution of $\ell _{dry}$ during the transient regime. In parallel, the liquid invades the porous medium from the lateral walls. The resulting impregnation profile exhibits a constant slope resulting from the balance between the liquid impregnation and the translation of the porous medium. The stationary regime is reached when the two impregnation fronts from each side meet to form the stationary $V$-shaped profile. Before that, the vertical impregnation length $\ell _{dry}(t)$ is limited by the vertical 1D flows coming from the bottom of the cell. Therefore, the transient regime and the linear time evolution of $\ell _{dry}(t)$ before saturation can be modelled by considering the vertical impregnation in a uniform 1D porous medium translating into a liquid bath.

3.2. 1D impregnation model

We consider a 1D porous medium with impermeable sidewalls and a porous bottom wall translated at a constant speed $V_0$ into a liquid bath (see figure 6). The porous material is characterized by its permeability $k$, its compacity $\phi$, the average pore diameter $\bar {d}_g$ and the contact angle $\theta _c$ of the three-phase contact line on the glass beads. The capillary pressure drop at the impregnation front, $p_c$, is given by Reyssat et al. (Reference Reyssat, Sangne, van Nierop and Stone2009) and Xiao et al. (Reference Xiao, Stone and Attinger2012):

(3.2)\begin{equation} p_c = \frac{4 \gamma \cos\theta_c}{ \bar{d}_g},\end{equation}

where $\gamma \simeq 70\ \textrm {mN} \ \textrm {m}^{-1}$ is the air–water surface tension. We also define the Reynolds number associated to the fluid displacement in the porous material as

(3.3)\begin{equation} Re_p = \frac{ \rho u \bar{d}_g }{\eta}, \end{equation}

where $\rho$ is the density of the fluid, $\eta$ the viscosity and $u$ its characteristic velocity in the porous medium. In the transient regime studied here, an order of magnitude of the velocity $u$ is about 10 mm s$^{-1}$, resulting in a Reynolds number $Re_p$ in the range 1–10, depending on the grain sizes. In this range of Reynolds numbers, the flow dynamics into the porous medium is modelled by Darcy's equation, which relates the average liquid velocity $\boldsymbol {u}_{Darcy}$ to the pressure gradient ${\boldsymbol {\nabla }} p$ (Mensire et al. Reference Mensire, Ault, Lorenceau and Stone2016):

(3.4)\begin{equation} \boldsymbol{u}_{Darcy} = (1-\phi)\boldsymbol{u}_{p} ={-}\frac{k}{\eta} ( {\boldsymbol{\nabla}} p - \rho \boldsymbol{g} ), \end{equation}

where $\boldsymbol {u}_{p}$ is the mean velocity of the liquid into the porous medium, and $\boldsymbol {g}$ is the gravitational acceleration. In this configuration, the liquid flow is driven both by the drop of capillary pressure at the interface and by the hydrostatic pressure.

Figure 6. Schematic of the impregnation of a 1D porous medium plunging in water at a constant velocity $V_0$ along the $z$-axis. The total length of the porous medium immersed in water is denoted by $\ell _0$, while $\ell _{dry}$ and $\ell _{wet}$ respectively correspond to the dry and wet length under the surface.

We denote by $\ell _0$ the position of the bottom of the porous medium, so that $\ell _0$ is the total length of the material immersed under the liquid surface. This length can be decomposed as the sum of the wet part $\ell _{wet}$ and the dry part $\ell _{dry}$, as shown in figure 6. The porous material is translated at a constant speed $V_0$, so that $\ell _0 = V_0 \,t = \ell _{dry} + \ell _{wet}$.

Darcy's equation (3.4) in one dimension, along the $z$-axis, is then

(3.5)\begin{equation} \frac{\mbox{d} \ell_{wet}}{\mbox{d} t} = \frac{k}{(1-\phi)\eta} \left( \frac{p_c + \rho g \ell_{0}}{\ell_{wet}} - \rho g \right), \end{equation}

where the fluid velocity at the pore scale is associated to the velocity of the impregnation front $u_p = \mbox {d}\ell _{dry}/\mbox {d}t$. We introduce $V^\star$ as the characteristic impregnation velocity associated to the gravity-driven flow:

(3.6)\begin{equation} V^\star{=} \frac{k \rho g}{(1-\phi)\eta}. \end{equation}

We also introduce the dimensionless length $L_{wet} = \ell _{wet}/h_J$ and the dimensionless time $T = t V^\star /h_J$, where $h_J$ is the Jurin height defined by the balance of the capillary forces and the liquid weight: $h_J = p_c/\rho g$. The equation (3.5) then becomes

(3.7)\begin{equation} \frac{ \mbox{d} L_{wet} }{\mbox{d} T} = \frac{ 1 + (V_0/V^\star)T}{L_{wet}} - 1. \end{equation}

This equation can be solved analytically, as detailed in Appendix B. The theoretical impregnation dynamics are computed for different values of $V_0/V^\star$ and reported in figure 7(a). For $V_0/V^\star = 0$, the dynamics reduces to the well-known Lucas–Washburn equation for the vertical impregnation of a liquid in a static porous medium under gravity (Lucas Reference Lucas1918; Washburn Reference Washburn1921). The fluid displacement in the porous medium is controlled by the capillary imbibition and follows a diffusion dynamics in $t^{1/2}$ before saturating under the effect of gravity at the Jurin height $h_J$ (Delker, Pengra & Wong Reference Delker, Pengra and Wong1996; Lago & Araujo Reference Lago and Araujo2001). For moderate values of $V_0/V^\star$, a capillary regime is observed at early times, followed by a pressure-driven regime exhibiting a constant impregnation velocity $V_{wet}$, as shown in figure 7(a). These front velocities are reported in figure 7(b) as a function of the dimensionless plunging speed $V_0/V^\star$. We can derive an analytical expression for the impregnation front velocity $V_{wet}$ by simplifying (3.7) for large time scales, i.e. for $(V_0/V^\star )T \gg 1$. In this regime, the capillary term can be neglected with respect to the hydrostatic term, and the equation becomes

(3.8)\begin{equation} \frac{ \mbox{d} L_{wet} }{\mbox{d} T} = \left( \frac{V_0}{V^\star} \right) \frac{T}{L_{wet}} - 1. \end{equation}

This equation has a solution of the form $L_{wet} = \nu T$, where $\nu$ is a constant (Mullins & Braddock Reference Mullins and Braddock2012). This kind of solution is consistent with the experimental observations, where the front velocity is constant for a long time in the transient regime before saturating when the stationary state is reached. Injecting the ansatz $L_{wet} = \nu T$ in (3.8), we obtain

(3.9)\begin{equation} \nu^{{\pm}} ={-} \frac{1}{2} \pm \frac{1}{2}\sqrt{1 + 4 \left( \frac{V_0}{V^\star} \right) }, \end{equation}

where only the positive root has a physical meaning, leading to

(3.10)\begin{equation} L_{wet} = \frac{T}{2}\,\left[\sqrt{1 + 4 \left( \frac{V_0}{V^\star} \right)} -1\right]. \end{equation}

We also derived this expression in Appendix B by considering the long-term behaviour of the implicit analytical solution. The solution (3.10) is plotted as a dashed line in figure 7(b) and captures the long-term behaviour of the implicit solutions of (3.7), confirming that the long-term dynamics is dominated by the pressure-driven flow and the capillary effects can be neglected.

Figure 7. (a) Evolution of $L_{wet}$ as a function of $T$ given by the numerical resolution of (3.7) for different values of $V_0/V^\star$. The slope gives the value of $V_{wet}/V^\star$. (b) Evolution of the constant impregnation velocity $V_{wet}/V^\star$ as a function of the dimensionless plunging speed $V_0/V^\star$ fitted from the curves in (a). The dashed line corresponds to the solution (3.10).

The velocity $V_{wet}$ corresponds to the impregnation front velocity with respect to the bottom of the porous medium and is thus related to the front velocity $V_f$ with respect to the free surface through

(3.11)\begin{equation} V_f = V_0 - V_{wet}. \end{equation}

The dimensionless front velocity $V_f/V^\star$ is reported as a function of the rescaled plunging velocity $V_0/V^\star$ in figure 8, where we also show the experimental data obtained for different systems of grains and cell widths. Additional data obtained from 1D impregnation experiments are also plotted in figure 8 for comparison with the 2D results in the transient regime. Both 1D and 2D experimental results collapse on a master curve and are well captured by the analytical model developed above. These results confirm that the transient regime is governed by the impregnation from the bottom of the porous medium.

Figure 8. Front velocity $V_f/V^\star$ as a function of the plunging speed $V_0/V^\star$ for different experimental systems and cell widths. The dashed line is obtained from (3.10).

We can formulate two comments. First, the data for the transient regime shown in figure 5 do not allow us to test the influence of the inertia that is expected to play a role at early times. Indeed, the experimental configuration introduces artifacts at the beginning of the impregnation (delay due to the grid and the presence of the spacers at the corners of the cell). Therefore we have only reported constant velocities after this very short regime (figure 8). Secondly, the collapse of the data observed in figure 5 at early times seems to be a coincidence: with wetting grains, the effect of the capillary pressure should drive the front faster and shift the curves toward lower values, but the delay of impregnation due to the grid roughly compensates this effect and leads to apparent curves going through the origin. For non-wetting grains, the curves would be shifted upward because the impregnation would start after overcoming the capillary pressure, and for wetting grains the curves would be shifted downward, even having a negative part close to the origin where spontaneous capillary rise may dominate the imposed velocity at early times.

4.1. Morphology of the impregnation front

The morphology of the stationary front, which separates the wet and dry grains, is characterized by a $V$-shaped profile, as shown in figure 9(a). The impregnation profile exhibits a shouldering of height $h_0$ in the vicinity of the surface of the bath and a local curvature at the tip of the profile. The maximum length of air entrained under the liquid surface in the bath is denoted by $h_{dry}$, as represented in figure 9(a). To characterize the impregnation front, we denote by $\theta$ the opening angle of the profile, so that $1/\tan \theta$ is the slope. We also denote by $R_{c}$ the radius of curvature of the front near the tip of the profile (see figure 9c). The measurements of the opening angle $\theta$ and the radius of curvature $R_c$ are described in figure 9(b,c).

Figure 9. (a) Picture of a stationary impregnation front obtained with the system W2, a cell of width $W = 10$ cm and a plunging speed $V_0 = 60$ mm s$^{-1}$. (b) Impregnation profile $z(x)$ extracted from the experiment shown in (a). The red dashed line corresponds to the parabola fitting the centre of the profile. (c) Close-up view of the bottom of the impregnation profile. The radius of curvature $R_c$ is evaluated by fitting the profile with a parabola.

The evolution of $\tan \theta$, measured on each stationary $V$-shaped profile, is reported in figure 10(a) as a function of the dimensionless plunging speed $V_0/V^\star$ for different systems of beads and cell widths $W$. The experimental data collapse well on a master curve, confirming that the $V$ shape of the impregnation profile results from the competition between the fluid penetration into the porous medium and the translation in the water bath. The experimental data are well fitted by a power law of the dimensionless velocity:

(4.1)\begin{equation} \tan\theta \propto \left( \frac{V_0}{V^\star} \right)^{\alpha} \quad \mbox{where} \ \alpha ={-}0.65. \end{equation}

Figure 10. Evolution of (a) the slope $\tan \theta$ and (b) the dimensionless radius of curvature $R_c/W$ of the stationary impregnation profile as a function of the dimensionless plunging speed $V_0/V^\star$ for several systems of wetting porous materials with different widths $W$. The dotted line in each figure shows the best-fitting power law.

Similarly, the radius of curvature $R_c$ at the tip of the profile as a function of $V_0/V^\star$ is reported in figure 10(b). A good collapse of the data is again observed, after rescaling the radius of curvature by the width $W$ of the porous medium. This result indicates that the curvature is invariant under a change of scale and depends only on the aspect ratio of the impregnation front. Moreover, the experimental results suggest that the radius of curvature at the tip evolves as the inverse of the dimensionless plunging velocity:

(4.2)\begin{equation} \frac{R_c}{W} \propto \left( \frac{V_0}{V^\star} \right)^\gamma \quad \mbox{where} \ \gamma ={-}1.14. \end{equation}

Therefore, the larger the speed of immersion, the smaller the curved area is.

4.2. Effect of wettability

We investigate the influence of the wettability of the grains composing the synthetic porous medium on the shape of the impregnation profile. The coated grains used for these experiments (systems N1 and N2 in table 1) have a contact angle of $\theta _c \simeq 75^\circ$. The grains can thus be considered non-wetting grains, for which the capillary pressure is neglected. Moreover, a contact angle larger than about $55^\circ$ prevents the spontaneous imbibition in a granular packing (Raux et al. Reference Raux, Cockenpot, Ramaioli, Quéré and Clanet2013). This phenomenon results from geometrical inaccessibility of the material porosity for the liquid menisci. The non-wetting porous medium is plunged in the water bath in the same conditions as in the previous experiments. After a similar transient regime, a stationary profile is again observed. In contrast to the situation with wetting grains (systems W1, W2 and W3), the profile is now shifted by a much larger length $h_0$ and does not present any shouldering in the vicinity of the sidewalls, as shown in the inset in figure 11(b).

Figure 11. Evolution of the slope $\tan \theta$ of the stationary profile of impregnation as a function of the dimensionless plunging speed $V_0/V^\star$ for two systems of non-wetting porous media (systems N1 and N2 of contact angle $\theta _c \simeq 75^\circ$) and a cell of width $W=5$ cm. (b) Evolution of the lateral entrance depth $h_0-h_0^{\star }$ as a function of the plunging speed $V_0$ for the systems N1 and N2. The dotted lines show the best-fitting power laws.

The evolution of $\tan \theta$ as a function of $V_0/V^\star$ for the non-wetting porous media is reported in figure 11(a). A transition is observed, characterized by a change of the exponent describing the evolution of $\tan \theta$ with $V_0/V^\star$:

(4.3)\begin{equation} \tan\theta \propto \left( \frac{V_0}{V^\star} \right)^{\alpha}, \quad \mbox{with} \ \left\{ \begin{array}{@{}ll} \alpha ={-} 0.64 & \mbox{for} \ V_0/V^\star \lesssim 5, \\ \alpha ={-} 0.5 & \mbox{for} \ V_0/V^\star \gtrsim 5. \end{array} \right. \end{equation}

For large plunging speeds ($V_0/V^\star \gtrsim 5$), the exponent $\alpha$ matches well with the prediction of Nasto et al. (Reference Nasto, Regli, Brun, Alvarado, Clanet and Hosoi2016), who neglected the capillary effects and assumed a horizontal impregnation in the porous material. For small plunging speeds ($V_0/V^\star \lesssim 5$), we find an exponent $\alpha \simeq - 0.64$ similar to the one reported for wetting systems in figure 10(a), which suggest that this exponent does not depend on the capillary effect. Instead, for these small plunging speeds, the impregnation front is likely shaped by the direction of the flow inside the porous medium.

The impregnation front is also characterized by an entrance depth above which the liquid does not penetrate the porous medium. This macroscopic length, $h_0$, corresponds to the entrance depth of the impregnation front and is also measured for different grain sizes and several plunging speeds $V_0$. The length $h_0$ increases with $V_0$ and typically varies from 10 to 200 mm in the range of velocities studied in this work with the systems N1 and N2 ($V_0=1\text {--}150 \ \mbox {mm} \ \mbox {s}^{-1}$).

In static situations, the entrance depth corresponds to the hydrostatic forcing required to impregnate the porous material and the metal wire mesh. The wire mesh holds together the grains, which have a contact angle larger than the critical angle of impregnation $\theta ^\star \simeq 55^\circ$ (Raux et al. Reference Raux, Cockenpot, Ramaioli, Quéré and Clanet2013). Using geometrical arguments, several authors have linked this depth to the accessibility of the pores by the liquid menisci (Bán, Wolfram & Rohrsetzer Reference Bán, Wolfram and Rohrsetzer1987; Lago & Araujo Reference Lago and Araujo2001; Shirtcliffe et al. Reference Shirtcliffe, McHale, Newton, Pyatt and Doerr2006; Raux et al. Reference Raux, Cockenpot, Ramaioli, Quéré and Clanet2013). We denote by $h_0^\star$ the entrance depth in the static case, i.e. for $V_0 = 0$. At constant contact angle $\theta _c$, $h_0^{\star }$ depends only on the size of the pores (different for each system) and the nature of the wire mesh. In all the experiments presented here, the opening of the wire mesh remains unchanged. The evolution of $h_0 - h_0^\star$ with the plunging speed $V_0$ is presented in figure 11(b) for the systems N1 and N2. The entrance depth $h_0-h_0^*$ increases as ${V_0}^{0.5}$ in a range of lengths too large to be consistent with a viscous entrainment of air along the surface of the porous material (Lorenceau, Restagno & Quéré Reference Lorenceau, Restagno and Quéré2003). A possible explanation is to consider the influence of the wire mesh and the first layer of grains just behind as an additional porous material presenting a lower permeability. Because there is little difference in size between the grid openings ($d_{grid} = 250 \ \mathrm {\mu } \mbox {m}$) and the grain diameters ($d_g = 140\text {--}320 \ \mathrm {\mu }\mbox {m}$ and $d_g = 280\text {--}420\ \mathrm {\mu }\mbox {m}$), the effective pore size in this layer is very small. This layer of small permeability delays the penetration of the liquid into the porous medium and leads to an increase of the length $h_0$. The influence of the opening size of the mesh and the physical mechanism responsible for this evolution will be discussed in § 7.2.

5.1. Forchheimer's equation

For large Reynolds numbers (typically $Re_p > 10$), the flow velocity in the porous material is not simply proportional to the pressure gradient. An additional correction is required to account for the inertial effects that appear in the inertial regime and add a dissipation at the pore scale. These effects are modelled by Forchheimer's equation, written as

(5.1)\begin{equation} \boldsymbol{\nabla}p ={-} \frac{\eta}{k} \boldsymbol{u}_{Darcy} \left( 1 + \frac{k \rho \beta}{\eta}|\boldsymbol{u}_{Darcy}| \right),\end{equation}

where $\beta$ is the Forchheimer coefficient (in m$^{-1}$) and is typically of the order of $1/\sqrt {k}$. For small Reynolds numbers, ${k \rho \beta } |\boldsymbol {u}_{Darcy}|/{\eta } \ll 1$, and Forchheimer's equation (5.1) reduces to Darcy's equation (3.4).

The Forcheimer coefficient $\beta$ is measured for the different model porous media using the methods presented in § 2 for the permeability measurement, but now imposing a larger pressure gradient $\boldsymbol {\nabla } p$. The details of the measurements are given in Appendix A, and the values for wetting beads (systems W1, W2 and W3) are reported in table 1.

5.2. Profile of the impregnation front

We model the shape of the impregnation front as the result of the lateral impregnation from the sidewalls of the porous material. Forchheimer's equation (5.1) reads

(5.2)\begin{equation} (1-\phi) \boldsymbol{u}_p \left[ 1 + (1-\phi) \frac{k \rho \beta}{\eta}|\boldsymbol{u}_p| \right] ={-} \frac{k}{\eta} \left( \boldsymbol{\nabla}p - \rho \boldsymbol{g} \right), \end{equation}

where $\boldsymbol {u}_p$ is the mean velocity of the liquid in the porous medium, so that $\boldsymbol {u}_{Darcy} = (1 - \phi ) \boldsymbol {u}_p$. We first assume that the streamlines are horizontal in the reference frame of the porous material. We further assume that the velocity field is uniform along the horizontal direction, so that $\boldsymbol {u}_p(x,z) = u_x(z)\boldsymbol {e}_x$, where the $z$-axis is oriented downward and the $x$-axis is perpendicular to it. The relevance of this assumption depends on the plunging velocity $V_0$ and the properties of the porous material. We shall discuss the limits of this assumption in the following sections.

Along the $x$-direction, the pressure gradient in the porous medium depends on the hydrostatic pressure, the capillary pressure and the position of the impregnation front $x_f(z)$. An additional length $\ell _g$ is added to $x_f$ to model the hydraulic resistance contributed by the metal wire mesh which holds the grains within the cell. The pressure gradient is thus given by

(5.3)\begin{equation} \boldsymbol{\nabla} p \boldsymbol{\cdot} \boldsymbol{e}_x ={-} \frac{p_c + \rho g z}{x_f(z) + \ell_g}. \end{equation}

The length $\ell _g$ is taken equal to 2 mm in all the calculations, leading to good agreement between the experimental profiles and the theoretical predictions, as we shall see later. Equation (5.2) written along the $x$-direction in the reference frame of the porous medium yields

(5.4)\begin{equation} u_x(z) \left[ 1 + (1-\phi) \frac{k \rho \beta}{\eta} u_x(z) \right] = \frac{k}{\eta (1-\phi)} \left[ \frac{p_c + \rho g z}{x_f(z) + \ell_g} \right]. \end{equation}

We denote by $V^\star$ the characteristic microscopic velocity of impregnation under gravity, by $\tilde {V}^\star$ the characteristic Forchheimer velocity and by $h_J$ the Jurin height:

(5.5a–c)\begin{equation} V^\star{=} \frac{k \rho g}{(1-\phi)\eta}, \quad \tilde{V} = \frac{1}{(1-\phi)}\sqrt{\frac{g}{\beta}}, \quad h_J = \frac{p_c}{\rho g}. \end{equation}

The equation (5.4) for the front profile thus becomes

(5.6)\begin{equation} \frac{u_x(z)}{V_0} \left( 1 + \frac{V^\star}{\tilde{V}^2} u_x(z) \right) = \frac{V^\star}{V_0} \left( \frac{h_J + z}{x_f(z) + \ell_g} \right).\end{equation}

Experimentally, $u_x{V^\star }/{\tilde {V}^2}$ varies from 0.1 to 10, which further justifies the use of the Forchheimer correction to accurately predict the front shape. Then (5.6) leads to

(5.7)\begin{equation} \frac{u_x(z)}{V_0} = \frac{\displaystyle -1 + \left[ 1 + 4 \left( \frac{V^\star}{\tilde{V}} \right)^2 \left(\frac{h_J + z}{x_f(z) + \ell_g} \right) \right]^{1/2} } {2 {V^\star V_0}/{\tilde{V}^2} }.\end{equation}

We validate this model by designing a complementary 1D experiment of impregnation in a granular medium. The details of this experiment are presented in Appendix C. The predictions of (5.7) are in good agreement with the experimental measurements. While validating the relevance of Forchheimer's formalism to describe these capillary flows, it also shows that this term is more important near the free surface of the bath, where the capillary pressure is predominant compared to the hydrostatic pressure, and where the pressure gradients are larger.

We can get additional insight into the phenomenon by looking at the long-term behaviour of (5.6), where we neglect the inertial correction. If we assume that the grains are moving downward at a constant speed $V_0$, then the last term of $z=V_0t$ becomes predominant at large times, $z\gg h_J$. A steady solution of this equation is given by a constant speed of the wetting front $x_f(t)=u_x(z\gg h_J) t$:

(5.8)\begin{equation} \frac{u_f(z\gg h_J)}{V_0} \approx \frac{V^\star}{V_0} \frac{V_0 t}{ u_f(t\gg h_J) t}.\end{equation}

This long-term behaviour leads to a straight interface between the wet grains and the dry grains whose scaling of the slope matches the scaling found by Nasto et al. (Reference Nasto, Regli, Brun, Alvarado, Clanet and Hosoi2016). The slope of the dry–wet interface is given by the following equation:

(5.9)\begin{equation} \tan\theta = \frac{\mbox{d} x_f}{\mbox{d} z} = \left( \frac{V_0}{V^\star} \right)^{{-}1/2}. \end{equation}

In the following, we compare the analytical model to the experimental results obtained with the 2D porous media.

5.3. Theoretical model and experimental 2D impregnation profile

The solution of (5.7) is now compared to the stationary impregnation fronts extracted from the experiments in the translated 2D porous medium described in § 4. Again, (5.7) is solved with a fourth-order Runge–Kutta method with the initial condition $x(t_0 = h_0/V_0) = 0$. The solutions are plotted for each system (W1, W2 and W3) in figure 12(a–c) for several cell widths ($W = 5$, 10 and 20 cm). The theoretical solution captures well the shape of the front profile in the first half of the profile, near the free surface. The discrepancy observed at the tip of the profile comes from the assumption of a semi-infinite material, which does not predict the junction between the profiles. In particular, the region at the tip, close to the junction, is more challenging to describe since the pressure field and the streamlines become complex. The focusing of the pressure gradients occurs near the tip of the impregnation profile and imposes the inflection of the streamlines. This inflection results from the isotropy of the porous material, which prevents the specific orientation of the fluid. The streamline reorientation and the bidimensional effects are more significant at low values of $V_0/V^\star$, particularly when $V^\star$ is large, i.e. for a porous medium made of large grains. This point is consistent with the limited agreement between experimental results and theoretical prediction reported for large pore sizes (system W3) and wider cells ($W = 20$ cm), as shown in figure 12(c). A better prediction is observed for narrow porous media with low permeability (figure 12a,b).

Figure 12. Profiles of the impregnation front for different experimental configurations: (a) system W1 and $W = 5 \ \mbox {cm}$, (b) system W2 and $W = 10 \ \mbox {cm}$, (c) system W3 and $W = 20 \ \mbox {cm}$. The red dashed lines show the theoretical predictions given by (5.7).

The analytical model described previously allows us to determine the shape of the first half of the impregnation profile when the flow remains mostly unidirectional and horizontal. Nevertheless, this model does not correctly predict the slopes reported experimentally in figure 10(a). The calculation presented in the previous section predicts an evolution of $\tan \theta$ as $(V_0/V^\star )^{-1/2}$ regardless of the grains’ wettability. However, the experimental results for wetting grains indicate an evolution of $\tan \theta \propto (V_0/V^\star )^{-0.64}$. This discrepancy comes both from the 2D effects in the flow through the porous medium, and from the influence of the grains’ wettability, since a transition in the scaling law is only observed for the non-wetting particles (see figure 11a). The complexity of the flow field around the tip of the impregnation profile suggests that numerical simulations should be used to determine the entire shape of the impregnation front.

6.1. The numerical method

We solve the pressure field $p(x,z)$ numerically in the liquid confined in the moving porous medium. The liquid–air interface is imposed by the shape of the impregnation front, which is related to the plunging velocity. Combining Darcy's equation and the incompressibility of the liquid, the associated pressure field is obtained from the classical Laplace equation ${\nabla }^2 {p} = 0$ with the following boundary conditions: a hydrostatic pressure along the vertical side and the capillary pressure on the liquid–air interface. The velocity field in the porous medium, $\boldsymbol {u}_p(x,z)$, is then derived from the pressure field $p(x,z)$. The numerical calculation is performed in the moving frame (see figure 13a) to obtain the steady state as in experiments. Numerically, a stationary shape of the impregnation front is found when the flow velocity remains constant along the front and is constant and equal to the plunging speed $V_0$. For the computation and in the following sections, the variables are made dimensionless:

(6.1a–e)\begin{equation} \tilde{\boldsymbol{u}} = \frac{\boldsymbol{u}_p}{V^\star}, \quad \tilde{p} = \frac{p}{\rho g W}, \quad \tilde{x} = \frac{x}{W}, \quad \tilde{z} = \frac{z}{W}, \quad \tilde{\boldsymbol{\nabla}} = \frac{\partial}{\partial \tilde{x}}\boldsymbol{e_x} + \frac{\partial}{\partial \tilde{z}}\boldsymbol{e_z}. \end{equation}

Figure 13. (a) Schematics of the numerical model of the stationary regime and the associated notation. (b) Examples of streamlines (left side) and velocity field (right side) observed in the stationary regime of impregnation for a non-wetting porous material ($p_c = 0$) and $V_0/V^\star =9.7$.

An example of a stationary fluid velocity field in the laboratory reference frame and the associated streamlines $\tilde {\boldsymbol {u}}(\tilde {x},\tilde {z})-\tilde {V}_0\boldsymbol {e}_z$ are shown in figure 13(b) for non-wetting grains ($\tilde {p}_c = 0$).

6.2. Non-wetting grains – $\tilde{p}_c = 0$

We first consider the case of a porous medium composed of non-wetting grains. The capillary pressure is thus equal to zero, $\tilde {p}_c = 0$. Therefore, the boundary condition imposed along the front profile simply becomes $\tilde {p}_{i_F, j_F} = 0$. The shape of the impregnation front is modelled in the half-domain [0, W/2] by the function

(6.2)\begin{equation} \tilde{z}(\tilde{x}) = \frac{H}{W} \left( \frac{1}{C^n+1} - \frac{(1-\tilde{x})^{n+1}}{C^n + (1-\tilde{x})^n} \right), \end{equation}

which depends on the three parameters $H$, $C$ and $n$. These parameters are chosen to have an impregnation profile along which the vertical projection of the fluid velocity remains constant to ensure the stationarity of the liquid–air interface in the laboratory reference frame. The parameter $H/W$ controls the aspect ratio of the profile with no curvature, the exponent $n$ adjusts the slope of the profile and the parameter $C$ modifies the curvature at the tip of the profile.

6.2.1. Slope of the stationary profile

The evolution of $\tan \theta$ and of the dimensionless radius of curvature at the tip of the front $R_c/W$ are reported in figures 14(a) and 14(b), respectively, for different stationary impregnation profiles computed numerically. The numerical results are in good agreement with the experimental data obtained with non-wetting particles (contact angle around $75^\circ$). The value of $\tan \theta$ decreases with $V_0/V^\star$ according to the scaling laws

(6.3)\begin{equation} \tan\theta \propto \left( \frac{V_0}{V^\star} \right)^{\alpha}, \quad \mbox{with} \ \left\{ \begin{array}{@{}ll} \alpha ={-} 0.66 & \mbox{for} \ V_0/V^\star \lesssim 5, \\ \alpha ={-} 0.50 & \mbox{for} \ V_0/V^\star \gtrsim 5. \end{array} \right. \end{equation}

The exponents of the power laws obtained as the best fit of the numerical results are similar to the experimental values previously reported in § 4.2. The transition between the two regimes is observed at approximatively the same value of $V_0/V^\star$. Therefore, the numerical simulations correctly reproduce the evolution of the slope of the stationary profile. In addition, the radius of curvature evaluated at the tip of the impregnation profile for the numerical simulation is also in fairly good agreement with the experiments. The dimensionless radius of curvature $R_c/W$ decreases with the plunging speed as

(6.4)\begin{equation} \frac{R_c}{W} \propto \left(\frac{V_0}{V^\star}\right)^{{-}0.95}. \end{equation}

Figure 14. (a) Evolution of $\tan \theta$ of the stationary profile of impregnation obtained numerically as a function of the dimensionless plunging speed $V_0/V^\star$ for $p_c = 0$. (b) Evolution of the rescaled radius of curvature $R_c/W$ measured at the centre of the profile as a function of the dimensionless plunging speed $V_0/V^\star$. In both panels, the numerical results are reported in black ($\blacksquare$) and the best fit is plotted in dashed lines. Numerical simulations are carried out with a mesh of width $N_x = 100$ and length varying between $N_z = 600$ and $N_z = 1300$ depending on the aspect ratio $H/W$ of the impregnation front. In each case, preliminary tests are performed to ensure that the domain is large enough not to influence the front velocity.

6.2.2. Velocity field

The numerical simulations allow us to obtain the velocity field in the porous medium and the associated streamlines. Figures 15(a) and 15(b) provide two examples of stationary profiles and associated flow fields for two different dimensionless plunging speeds $V_0/V^\star$. Figure 15(a) corresponds to an immersion velocity $V_0/V^\star = 0.65$, below the change of slope in the evolution of $\tan \theta$, whereas figure 15(b), obtained for $V_0/V^\star = 49.6$, reports the case beyond the transition. For $V_0/V^\star$ larger than 5, the streamlines are mostly horizontal. The region affected by the tip effect is small and located close to the tip. In this configuration, the assumption of a pure lateral flow in the porous medium is appropriate. Indeed, it correctly predicts the evolution of the slope $\tan \theta$ of the front profile with the dimensionless plunging speed. Conversely, for $V_0/V^\star$ smaller than 5, as illustrated in figure 15(a), streamlines are not horizontal but instead are bent around the front. The flow reorientation induces a flattening of the front profile and an increase of $\tan \theta = \mbox {d}x/\mbox {d}z$. The radius of curvature can reasonably be interpreted as a characteristic length of the disturbance generated by the tip of the impregnation front. The larger the radius of curvature, the broader the area of streamline deformation is.

Figure 15. Streamlines (left) and amplitude of the velocity field (right) for two stationary impregnation fronts determined numerically for a non-wetting porous material ($p_c = 0$) and a dimensionless plunging velocity (a) $V_0/V^\star = 0.65$ and (b) $V_0/V^\star = 49.6$.

The numerical simulations suggest that the transition observed for the slope of the impregnation profile results from a change of flow morphology inside the porous medium. The exponent $\alpha = - 0.66$ observed at low plunging speed is the result of the deformation of the streamlines of the flow on a characteristic length that increases as $1/V_0$. Below a rescaled critical radius of curvature $R_c/W \simeq 5\times 10^{-2}$, the change of direction of the streamlines inside the porous medium remains confined near the tip of the impregnation profile. It affects only the shape of the impregnation front in this region, not the slope. The critical radius of curvature corresponds to a slope $\tan \theta$ of about 0.5. This means that the transition between the two regimes occurs for impregnation fronts presenting an aspect ratio smaller than 0.5, which is consistent with the experimental observations.

6.3. Capillarity: wetting grains – $\tilde {p}_c > 0$

The previous numerical simulations were performed without capillary pressure and revealed an evolution of the flow morphology with the plunging velocity $V_0$. This observation is consistent with the experiments carried out with non-wetting grains (systems N1 and N2). However, for wetting grains (systems W1, W2 and W3), this transition is not observed in the same range of plunging speeds. To understand the difference, we now account for the capillary pressure along the impregnation front in the numerical simulation. The new boundary condition at the dry–wet interface is $\tilde {p}_{i_{F}, j_{F}}=-\widetilde {p_c}$, with $\widetilde {p_c}>0$ for wetting grains. We have observed experimentally that the shape of the stationary impregnation front exhibits two curvatures in opposite directions: one close to the free surface of the water bath and another near the tip of the front. To capture the numerical stationary fronts, we adjust the shape of the boundary in the porous material, now using the function

(6.5)\begin{equation} \tilde{z}(\tilde{x})=\frac{H}{W}\left(\frac{1}{{C_1}^{n}+1}- \frac{(1-\tilde{x})^{n+1}}{{C_1}^{n}+(1-\tilde{x})^{n}}+(1-\tilde{x}) \frac{\tilde{x}^{n+1}}{{C_2}^{n}+\tilde{x}^{n}}\right).\end{equation}

The last term, with the parameter $C_2$, is added to reproduce the upper part of the profiles. Again, the parameters $H$, $C_1$, $C_2$ and $n$ are optimized to get an impregnation profile along which the vertical projection of the fluid velocity remains constant. Note that at low plunging velocities $V_0$, the fluid may move upward close to the top corner because of the capillary forces. We adapted the numerical code by increasing the size of the domain above the free surface of the bath with a no-flux boundary condition at the material–air interface for $\tilde {z}<0$. The capillary effects are expected to be important when the capillary speed, $V_c^\star =k\, p_c/(\eta \, W)$, is comparable to the plunging speed $V_0$. The simulations are performed with $\tilde {p}_c=1$, corresponding to $p_c = \rho g W$. As a result, the capillary speed $V_c^\star$ is equal to the characteristic velocity $V^\star$. The upward capillary impregnation is only observed for $V_0 \ll V^\star$ ($V_0/V^\star < 50$ in practice).

An example of a velocity profile is reported in figure 16(a). From the stationary front, we extract the value of $\tan \theta$ at the middle depth for varying $V_0/V^\star$ as shown in figure 16(b). For wetting grains, we observe that $\tan \theta \propto (V_{0} / V^{\star })^{-0.66}$ over the whole range of $V_0/V^\star$ investigated here. This evolution is in quantitative agreement with the experimental measurements reported in figure 10. The conservation of the power law for high plunging speed $V_0/V^\star$ is thus related to the capillary effects, which constrain the shape of the impregnation front near the free surface.

Figure 16. (a) Streamlines and velocity field obtained numerically for the stationary impregnation front for a wetting porous material, $\tilde {p}_c = 1$ and $V_0/V^\star = 44.4$. (b) Evolution of the slope $\tan \theta$ measured on the numerical stationary profiles as a function of the imposed dimensionless velocity $V_0/V^\star$ for $\tilde {p}_c = 1$. The dotted line is the best fit of the numerical data ($\bullet$) by a power law of exponent 0.66.