Kinetics of gravity-driven water channels under steady rainfall

We investigate the formation of fingered flow in dry granular media under simulated rainfall using a quasi-two-dimensional experimental setup composed of a random close packing of monodisperse glass beads. Using controlled experiments, we analyze the finger instabilities that develop from the wetting front as a function of fundamental granular (particle size) and fluid properties (rainfall, viscosity). These finger instabilities act as precursors for water channels, which serve as outlets for water drainage. We look into the characteristics of the homogeneous wetting front and channel size as well as estimate relevant time scales involved in the instability formation and the velocity of the channel fingertip. We compare our experimental results with that of the well-known prediction developed by Parlange and Hill [D. E. Hill and J. Y. Parlange, Soil Sci. Soc. Am. Proc. 36, 697 (1972)]. This model is based on linear stability analysis of the growth of perturbations arising at the interface between two immiscible fluids. Results show that, in terms of morphology, experiments agree with the proposed model. However, in terms of kinetics we nevertheless account for another term that describes the homogenization of the wetting front. This result shows that the manner we introduce the fluid to a porous medium can also influence the formation of finger instabilities. The results also help us to calculate the ideal flow rate needed for homogeneous distribution of water in the soil and minimization of runoff, given the grain size, fluid density, and fluid viscosity. This could have applications in optimizing use of irrigation water.

Figure 1

Diagram of the quasi-2D experimental setup used to visualize the formation of water channels during steady rain. A rain source built with equally spaced capillary tubes provides a constant rainfall rate, $Q$, on the sample cell of cross section $A$. The model soil packing occupies a length, $l$, and a width, $w$. The sample cell is filled with monodisperse glass beads as model soil and is suspended under the rain source. We can modify the distance, $h$, between the capillary and model soil surface to vary the free-fall height of the raindrops. The cell is oriented vertically in the direction of gravity, $g$.

Figure 2

Sequence of images showing formation of water channels under steady rain. (a) Illustration of the formation of water channels in initially dry granular medium. This is preceded by a slow development of a homogeneous wetting front and growth of instabilities. The water channel serves as a preferential path for water drainage and is defined by a certain channel width, $d$. (b) Experimental images on infiltration of water channels for particle size $D=300\mu \mathrm{m}$, viscosity $\mu =1$ mPa s, and rain rate $Q=14.5$ cm/h. The first water channel appears just after $t=180$ s after the formation of a homogeneous wetting front and only two water channels are formed. (c) Experimental images on the infiltration of water-glycerol solution in $D=1$ mm, $\mu =4$ mPa s, and $Q=96.0$ cm/h. At higher viscosities and higher flow rates, multiple water channels appear quickly and simultaneously.

Figure 3

Experimental images showing the evolution of the homogeneous wet zone, $z_{\text{wet}}$, with time during the early moments of rainfall. This is for the following experiment: droplet impinging speed, $U_T=1.0$ m/s, particle size $D=300\mu \mathrm{m}$, viscosity $\mu =1$ mPa s, and rain rate $Q=14.5$ cm/h.

Figure 4

(a) Plot of the velocity of the wetting front, $v_{\text{wet}}$, as a function of bead diameter, $D$, at constant $U_T$, $Q$, and $\mu$. The graph shows that water moves faster in larger pores than in smaller ones. (b) Average channel front velocity of the first and second channels as function of bead diameter, $D$, at constant $U_T$, $Q$, and $\mu$. (c) Experimental data on the average channel front velocity, $v$, as a function of flow rate $Q$ also for different fluid viscosities, $\mu$, at constant $D$ and $U_T$.

Figure 5

(a) Plot of the time of formation of the first channel, $t_{C_1}$, and second channel, $t_{C_2}$, as well as the time it takes for the front to become homogeneous, $t_h$, all as a function of bead size diameter. A clear trend exists during the formation of the first channel, meaning water channels form much later in smaller particles where liquid flow is much slower. However, the formation of the second channel seems to be conditioned not just by particle size but also by other parameters, which are yet to be fully determined. A third quantity $t_h$ is the time when the front becomes homogeneous. Owing to the unique design of the rain source, it takes time for neighboring droplet impact sites to coalesce and form a continuous front. (b) Experimental data of average time of formation of the first water channel, $t_{C_1}$, as a function of flow rate $Q$ for different fluid viscosities, $\mu$, but at constant particle size diameter $D$ and droplet impinging speed, $U_T$. There appears to be a decreasing trend with respect to flow rate $Q$ initially at low flow rates but this trend slowly increases at higher flow rates and higher viscosities.

Figure 6

(a) Illustration of the first few moments of rainfall. Due to experimental design, there is a time scale, $t_h$, at which the front becomes homogeneous. As the droplets impact the granular medium of an initial volume, successive drops will increase this volume as a function of time until neighboring impact sites coalesce to form a homogeneous wetting front. We suppose that this coalescence is achieved when the depth of the front is equal to the spacing between adjacent capillary tubes. (b) At constant $U_T$, $Q$, and $\mu$, experimental data on time of appearance and formation of the first water channels as function of bead size, $D$. The time of formation of the first water channel, $t_{C_1}$ (circle), fits well with a model [Eq. (18)] derived from the linear stability analysis of a stable front. (c) At constant $U_T$ and $D$, we plot the experimental data for different $\mu$ as a function of $Q$ and show agreement with the model [Eq. (18)], particularly at low flow rates. The model predicts an initial decrease in $t_{C_1}$ at low $Q$, but will gradually increase at higher $Q$, especially for higher values of $\mu$. The three curves in this figure are calculated from the model [Eq. (18)] using different viscosity values.

Figure 7

(a) Experimental data on the average channel front velocity, $v$, as a function of $D$ at constant $Q$, $\mu$, and $U_T$. Model fit shown by the solid line is Eq. (15). (b) Experimental data on the average channel velocity, $v$, rescaled with viscosity $\mu$, as a function of flow rate $Q$ also for different fluid viscosities using Eq. (16). The relationship between $\mu v$ and $\mu Q$ is further emphasized in Eq. (17), where the dependence of $\left(\mu v\right)$ on $\sqrt{\mu Q}$ is demonstrated. These are results from experiments performed at constant $D$ and $U_T$. (c) Experimental data of number of channels, $N$, as function of $D$ at constant $Q$, $\mu$, and $U_T$. (d) Experimental data of number of channels observed, $N$, as a function of flow rate $Q$ and for different fluid viscosities, $\mu$, rescaled using Eq. (21) at constant $D$ and $U_T$.

Figure 8

(a),(b),(c) Plots showing optimal values of $\mu Q$ for minimizing channel formation time. (a) 3D plot of $\mu Q$ as a function of fluid density, $\rho$, and particle size, $D$, as calculated from Eq. (22). The color bar shows a range of $\mu Q$ values with higher values at the top (white) and lower values at the bottom (black). (b) 2D plot of $\mu Q$ as a function of density for different values of particle size as indicated by the color bar. Higher $D$ values correspond to upper curves with $D=1$ mm corresponding to the topmost curve (black online), while lower $D$ values are the lower curves. (c) 2D plot of $\mu Q$ as a function of particle size for different values of fluid density as indicated by the color bar. Higher $\rho$ values correspond to the upper curves, with $\rho =10000$ corresponding to the topmost curve (black online), while lower $\rho$ values are the lower curves. (d),(e),(f) Plots showing optimal values of $\mu Q$ for maximizing the number of channels. (d) 3D plot of $\mu Q$ as a function of fluid density, $\rho$, and particle size, $D$, as calculated from Eq. (23). The color bar shows a range of $\mu Q$ values with higher values at the top (white) and lower values at the bottom (black). (e) 2D plot of $\mu Q$ as a function of density for different values of particle size as similarly assigned by the color bar with $D=1$ mm being the topmost curve (black online). (f) 2D plot of $\mu Q$ as a function of particle size for different values of fluid density as also similarly described by the color bar with $\rho =10000$ being the topmost curve (black online).