Pressure evolution and deformation of confined granular media during pneumatic fracturing

By means of digital image correlation, we experimentally characterize the deformation of a dry granular medium confined inside a Hele-Shaw cell due to air injection at a constant overpressure high enough to deform it (from 50 to 250 kPa). Air injection at these overpressures leads to the formation of so-called pneumatic fractures, i.e., channels empty of beads, and we discuss the typical deformations of the medium surrounding these structures. In addition we simulate the diffusion of the fluid overpressure into the medium, comparing it with the Laplacian solution over time and relating pressure gradients with corresponding granular displacements. In the compacting medium we show that the diffusing pressure field becomes similar to the Laplace solution on the order of a characteristic time given by the properties of the pore fluid, the granular medium, and the system size. However, before the diffusing pressure approaches the Laplace solution on the system scale, we find that it resembles the Laplacian field near the channels, with the highest pressure gradients on the most advanced channel tips and a screened pressure gradient behind them. We show that the granular displacements more or less always move in the direction against the local pressure gradients, and when comparing granular velocities with pressure gradients in the zone ahead of channels, we observe a Bingham type of rheology for the granular paste (the mix of air and beads), with an effective viscosity ${\mu }_B$ and displacement thresholds $\vec{\nabla }{P}_c$ evolving during mobilization and compaction of the medium. Such a rheology, with disorder in the displacement thresholds, could be responsible for placing the pattern growth at moderate injection pressures in a universality class like the dielectric breakdown model with $\eta =2$, where fractal dimensions are found between 1.5 and 1.6 for the patterns.

Figure 1

(a) Sketch of the experimental setup where the top-down view of the prepared cell shows that the granular medium is confined inside the cell by three impermeable boundaries and an air-permeable boundary on the outlet side. The granular medium is placed against the outlet boundary such that it has a linear interface against a region empty of beads on the sealed inlet side, where pressurized air can be injected. (b) The side view of the setup shows the high-speed camera placed above. The glass plates are clamped together with aluminum framing while the cell gap is controlled with 1-mm-thick spacers, which are not shown here.

Figure 2

Image processing examples for a snapshot in an experiment with injection pressure $P_{\mathrm{in}}=200$ kPa. (a) The grayscale snapshot of the initial granular medium ($t=0$) is compared with (b) the grayscale snapshot of the deformed granular medium at the beginning of channel formation ($t=200$ ms), where a region is emptied of beads (black), to produce (c) the binary image where the emptied structure is shown in white. We evaluate deformation with image correlation between frames to find (d) the compacted zone indicated in gray and (e) the total magnitude of bead displacement as found with DIC.

Figure 3

Profiles of pressure fields for an experiment with $P_{\mathrm{in}}=200$ kPa, where the profiles show the average pressure across the cell width (perpendicular to the flow direction). The profiles are the Laplace solution (blue solid line) and the diffusing pressure (red dashed line); curves from left to right for each method correspond to the times in the list.

Figure 4

Evolution of the characteristic depths $s_L$ for the Laplace solution (blue solid line) and $s_D$ for the diffusing pressure (red dashed line) as a function of time for four different cases: I is a 1D rigid medium reference solved analytically, II is an experiment with $P_{\mathrm{in}}=100$ kPa, III is an experiment with $P_{\mathrm{in}}=150$ kPa, and IV is an experiment with $P_{\mathrm{in}}=200$ kPa. The difference $s_L-s_D$ for the experiments (black dotted line) seems to follow the difference $s_L-s_D$ for the 1D rigid media reference (green dashed line).

Figure 5

Shown on the left is the average velocity magnitude $|{\vec{v}}_g|$ during time windows of $\mathrm{\Delta }t=10$ ms, centered on $t=100$, 200, 300, 400, 500, and 600 ms for snapshots (a)–(f), respectively. The injection pressure is $P_{\mathrm{in}}=200$ kPa. A zone of mobile beads builds up on the cell scale initially and later focuses onto the most advanced fingers as the medium compacts. Beads behind the most advanced channel tips are not significantly displaced [note that we have removed data at the channel, however noise from the erosion inside it appears in (c)–(e)]. Shown on the right is the absolute pressure gradient $|\vec{\nabla }P|=\sqrt{{(\partial P/\partial x)}^2+{(\partial P/\partial y)}^2}$ for both solutions of the pressure at the snapshots (a)–(f). The main difference between the diffusing and Laplacian pressure fields is that the diffusing one has higher pressure gradients close to the air-solid interface and fingertips, while it has lower pressure gradients near the outlet boundary. At snapshot (f), the gradients start to look similar in magnitude around the channel. In both solutions, the pressure gradient is screened behind the longest fingers and the highest magnitudes are on the longest fingertips.

Figure 6

Normalized correlation between the vector fields of the negative pressure gradient $-\vec{\nabla }P$ and granular displacements ${\vec{u}}_g$ for snapshots separated by 1 ms at different stages of the deformation during an experiment with $P_{\mathrm{in}}=250$ kPa. The data are superimposed on top of the corresponding experimental images (a) during the initial mobilization, (b) during the instability stage just after the compaction front has reached the outer boundary, (c) during the instability stage with a more compacted medium, and (d) in the compacted stage. The correlation coefficient is the cosine of the angle between vectors and is close to 1 in all snapshots, meaning that the beads are generally displaced in the direction against the pressure gradient, i.e., in the direction of the driving force. Displacement magnitudes less than one-tenth of a pixel size (approximately equal to $70\mu \mathrm{m}$) are considered noise and are not included (gray). The bottom plot shows the average correlation coefficient in areas with displacement magnitude above the noise level plotted as a function of time for the experiments analyzed, showing that the normalized correlation coefficient between $-\vec{\nabla }P$ and ${\vec{u}}_g$ is close to 1 over time in all cases.

Figure 7

Shown on top is the depth of the compacted zones into the medium plotted as a function of time for experiments with different injection pressure. The inserted slopes (dashed lines) indicate velocities for reference. Typically, the compacted zone reaches the outer boundary between $t=150$ and 250 ms after the start of injection; however, for the lowest pressure it takes around 380 ms. The propagation of the compacted front has a common behavior for all experiments initially. First there is a 10-ms delay after the start of injection until any visible deformation is observed; then the growth is typically faster (approximately 8 m/s) over the first 20–30 cm before crossing over to a slower growth rate. The compacted zone in the 50-kPa experiment grows at a rate averaging around 1.5 m/s, while in the 250-kPa experiment the rate fluctuates around 3 m/s. In the 100-, 150-, and 200-kPa experiments the compacted zones grow at a rate similar to the 250-kPa experiment until a depth of $x=50–60$ cm, where they slow down, probably due to jamming of the system, but continue to propagate at an average rate around 1.5 m/s as the pressure gradients increase. The inset shows the time $t_c$ when the front hits the outlet as a function of $P_{\mathrm{in}}$. Shown on the bottom is a visual representation of the evolution of the compaction zones where the color code represents the time of the snapshot.

Figure 8

Shown on top is the volumetric strain rate ${\dot{\varepsilon }}_v=\vec{\nabla }·{\vec{v}}_g$ in snapshots centered at $t=100$, 200, 300, 400, 500, and 600 ms for snapshots (a)–(f) respectively, for an experiment with $P_{\mathrm{in}}=200$ kPa. Most of the compaction (negative ${\dot{\varepsilon }}_v$) occurs in front of the main channel, but some compaction at a lower rate happens on the sides of it. Shown on the bottom is the shear rate ${\dot{\gamma }}_{xy}$ during snapshots (a)–(f). The sheared regions coincide with the compacting regions, while there is little shear strain behind the longest fingers. We see lines out from the most advanced channel tips separating regions where the shear has opposite signs. These lines can be interpreted as follows: If the shear changes sign from positive to negative across a line (going in the positive $y$ direction), it means that the displacement is higher along the line than in the surrounding medium. If the change is from negative to positive, the medium is displaced less along the line than in the surrounding medium. The legend on the bottom relates flow behavior with the colors in the deformation map (the shear strain seen here is due to flow from left to right in front of the channel, and/or flow out from the sides of the main channel, i.e., towards the top or bottom in the figure).

Figure 9

Deformations during a stick-slip event in an experiment with $P_{\mathrm{in}}=250$ kPa. The snapshots show deformations in a zoomed-in region on successive 1-ms intervals, where the time increases from top to bottom. The columns from left to right show volumetric strain ${\varepsilon }_v$, displacement magnitude $|{\vec{u}}_g|$, shear strain ${\gamma }_{xy}$, and $C_{\mathrm{dir}}$, the normalized correlation coefficient between displacements and negative pressure gradient ${\vec{u}}_g$ and $-\vec{\nabla }P$. The thumbnail on the top indicates the location of the zoomed-in region. A stick-slip event typically lasts between 5 and 10 ms and the fingertips may expand up to 3–5 mm. The deformations shown in the snapshots are typical: Initially there is little deformation (compacted stage) until beads in a region in front of the channel rearrange and compact. Consequently, beads in a zone adjacent to the fingertips decompact and the fingers propagate as this zone recompacts, i.e., a decompaction-compaction front moves from the initially rearranged zone towards the channel and the fingers expand as the front reaches them such that the closest fingers expand first. The displacements are in the direction of $-\vec{\nabla }P$ and the medium is also sheared slightly in front of the channel.

Figure 10

Average velocity profiles $v_x$ plotted as a function of distance ahead of the channel for different snapshots (a) before and (b) after the compaction zone reaches the outlet boundary. The insets show the corresponding profiles in semilogarithmic plots. In (a) the decay of $v_x$ with increasing $x$ is more or less linear in the bulk of the compaction zone, while in (b) $v_x$ seems to have an exponential decay.

Figure 11

Average velocity profiles $v_y$ at 1.5 cm ahead of the channel tip are plotted as a function of distance from the centerline of the cell, for snapshots during channel growth. The magnitude of the velocity decays with time, but the profiles show that the medium flows away from the channel tip, with positions indicated by open circles.

Figure 12

Evolution with time of the fit parameter $\alpha$, proportional to the inverse of the Bingham effective viscosity of the granular medium, describing the average relationship $|{\vec{v}}_g|\approx \alpha ·|\vec{\nabla }P|+c$ between the velocity magnitude $|{\vec{v}}_g|$ and the pressure gradient magnitude $|\vec{\nabla }P|$. The plots of $\alpha$ are for the experiments with $P_{\mathrm{in}}=150$, 200, and 250 kPa, where the data are collapsed along the time axis by $t_c$, the time when the compacted zone reaches the outlet boundary. Before the compaction front reaches the outlet boundary ($t<t_c$), $\alpha$ increase linearly with time [the linear fit equals $9.8(t/t_c){\mathrm{cm}}^2$/($\mathrm{kPa}\mathrm{s}$)], while for $t>t_c$, $\alpha$ has an exponential decay with time [the exponential fit equals $41.68e^{-1.61(t/t_c)}{\mathrm{cm}}^2$/($\mathrm{kPa}\mathrm{s}$)]. The inset shows examples of linear fits to the average velocity magnitude as a function of pressure gradient. Note that over time there is an increasing offset from the ordinate axis, corresponding to the threshold $\vec{\nabla }{P}_c=-c/\alpha$. For the experiments with $P_{\mathrm{in}}=150$, 200, and 250 kPa, the value of $t_c$ is 230, 250, and 170 ms, respectively.

Figure 13

Evolution of the thresholds $\vec{\nabla }{P}_c$ as a function of normalized time $t/t_c$ for the experiments with $P_{\mathrm{in}}=150$, 200, and 250 kPa, where $t_c=230$, 250, and 170 ms, respectively. For $t/t_c\le 1$, the thresholds decrease from 10–15 kPa/cm initially to between 1 and 2 kPa/cm (indicated by the horizontal lines) at $t/t_c=1$. The power-law fit (black solid curve) suggests that the decrease in $\vec{\nabla }{P}_c$ is inversely proportional to the square root of time. After the compacted zone has reached the outlet boundary, for $t/t_c>1$, $\vec{\nabla }{P}_c$ begins to increase with time, faster for higher injection pressure. Then, at $t/t_c$ around 2.5–3, the increase of the threshold with time slows down. At this point, $\vec{\nabla }{P}_c$ is approaching the pressure gradient values $|\vec{\nabla }P|$ in the zone surrounding the most advanced fingertips.

Figure 14

Steps to obtain compacted zones from total displacement DIC data. (a) In front of the region invaded by air in a $P_{\mathrm{in}}=100$ kPa experiment at $t=80$ ms, we obtain (b) the total bead displacement with DIC, which is then converted to (c) a binary image showing the total bead displacements above $70\mu \mathrm{m}$ in white. (d) The smoothened compacted zone is obtained after image treatment of (c).