Fracturing in Wet Granular Media Illuminated by Photoporomechanics

We study fluid-induced deformation and fracture of granular media and apply photoporomechanics to uncover the underpinning grain-scale mechanics. We fabricate spherical photoelastic particles of 2-mm diameter to form a monolayer granular pack in a circular Hele-Shaw cell that is initially filled with a viscous fluid. The key distinct feature of our system is that, with spherical particles, the granular pack has a connected pore space, thus allowing for pore-pressure diffusion and the study of effective stress in coupled poromechanical processes. We inject air into the fluid-filled photoelastic granular pack, varying the initial packing density and confining weight. With our recently developed experimental technique, photoporomechanics, we find two different modes of fluid invasion: fracturing in fluid-filled elastic media (with strong photoelastic response) and viscous fingering in frictional fluids (with weak or negligible photoelastic response). We directly visualize the evolving effective stress field and discover an effective stress shadow behind the propagating fracture tips, where the granular pack exhibits undrained behavior. We conceptualize the behavior of the system by means of a mechanistic model for a wedge of the granular pack bounded by two growing fractures. The model captures the pore-pressure build-up inside the stress shadow region and the grain compaction in the annular region outside. Our model reveals that a jamming transition determines the distinct rheological behavior of the wet granular pack, from a friction-dominated to an elasticity-dominated response.

Figure 1

The experimental setup: a monolayer of photoelastic particles (diameter $d$, initial packing density ${\varphi }_0$) saturated by silicone oil is confined in a circular Hele-Shaw cell (internal diameter $L$). Vertical confinement is supplied by a weight, $W$, adding up the weights from a confining weight, a light panel, a polarizer, and a glass disk that rest on top of the particles. The disk is slightly smaller than the cell to allow the fluids (but not particles) to leave the cell. Air is injected at a fixed flow rate $q$ at the center of the cell with a coaxial needle, with the injection pressure monitored by a pore-pressure sensor. Three LVDTs are attached to the top of the light panel, capturing the vertical displacement of the top plate during the fracturing process. A white-light panel and right- and left-circular polarizers form a dark-field circular polariscope. Bright-field and dark-field videos are captured by cameras placed underneath the cell.

Figure 2

Bright-field (left) and dark-field (middle) images of the invading fluid morphology at breakthrough and a histogram (right) of the light intensity of the blue channel of the dark-field image before air injection (in gray) and at breakthrough (in black), corresponding to two different initial packing densities ${\varphi }_0$, with confining weight $W=25$ N. From the dark-field images that visualize the effective stress field, the invading morphology and rheology of the granular packs are classified as (a) fracturing in fluid-filled elastic media, with a strong photoelastic response ($I>0.65$), ${\varphi }_0=0.84$, or (b) viscous fingering in frictional fluids, with a weak or negligible photoelastic response ($I<0.65$), ${\varphi }_0=0.78$. Behind the propagating fracture tips, the effective stress field exhibits an evolving “effective stress shadow,” where the intergranular stress is low and the granular pack exhibits undrained behavior. For the evolution of the morphology in each regime, see the videos in the Supplemental Material [35].

Figure 3

(a)–(c) The time evolution of (a) the injection pressure $P_{\mathrm{inj}}$, (b) the normalized vertical displacement of the top plate $\delta h/d$, and (c) the averaged packing density ${\varphi }_{\mathrm{out}}$ in the annular region outside fractures for experiments with initial packing density ${\varphi }_0=0.84$ and $W=25$ N, 65 N, and 85 N. The insets of (b) and (c) show the time evolution of the normalized fracture radius $r_{\mathrm{frac}}/R$ and the averaged effective stress ${\sigma }_{\mathrm{out}}^{\prime }$ in the annular region outside fractures. The modeling results are plotted in dashed lines. (d)–(f) For the experiment with ${\varphi }_0=0.84$ and $W=25$ N, a sequence of snapshots shows the time evolution of (d) the interface morphology, (e) the packing-density field, and (f) the effective stress field, where the radius of the blue circle represents the fracture radius ($r_{\mathrm{frac}}$) averaged from three representative fracture tips.

Figure 4

The radial distribution of the packing fraction ($\varphi (r)$) for the fracturing experiment with ${\varphi }_0=0.84$ and $W=25$ N. The temporal evolution of $\varphi (r)$ is plotted at six time instances, $t_0$ at $t=0$ and $t_{\mathrm{i}}\sim t_{\mathrm{v}}$ in Fig. 3. The location of the fracture tip is indicated with the cross marker. The packing-fraction distribution behind the fracture tip is plotted in dashed lines and ahead of the fracture tip in solid lines. The red dashed line shows the packing fraction at the jamming transition, $\varphi ={\varphi }_c=0.85$.

Figure 5

A mechanistic model on fracturing that explains the effective stress shadow observed in experiments. (a) A schematic of the model setup for a fracture wedge with an angle $\theta ={60}^{\circ }$. The granular flow driven by the concentrated pore-pressure gradient within fracture tips keeps compacting particles in the annular region outside, leading to its increase in packing density and a rheological transition from frictional flow to elastic medium. (b) The modeled flow-velocity field at time instance (iii) in Fig. 3. (c) A sequence of snapshots showing the time evolution of the modeling pore-pressure field. Modeling conditions: ${\varphi }_0=0.84$, $W=25$ N, $q=100$ ml/min, and $V_0=15$ ml.

Figure 6

The jamming-transition analysis for the fracturing experiments (${\varphi }_0=0.84$, 0.82, and 0.80; $W=$ 25 N, 65 N, and 85 N). (a) The determination of the critical packing density and effective stress at jamming for the experiment $W=25$ N, ${\varphi }_0=0.84$. (b) ${\sigma }^{\prime }-{\sigma }_c^{\prime }$ plotted against $\varphi -{\varphi }_c$ for the fracturing experiments, which follows the power-law constitutive relationship ${\sigma }^{\prime }-{\sigma }_c^{\prime }=K⟨\frac{\varphi -{\varphi }_c}{{\varphi }_c}{⟩}^{\psi }$ [39, 40, 44, 45, 46].

Figure 7

A visual phase diagram of the bright-field (left) and dark-field (right) invading fluid morphology at breakthrough corresponding to different confining weights $W$ and initial packing densities ${\varphi }_0$. From dark-field images that visualize the effective stress field, the invading morphology and rheology of the granular packs is classified as fracturing in fluid-filled elastic media (with a strong photoelastic response, ${\varphi }_0=0.84$, 0.82, and 0.80), or viscous fingering in frictional fluids (with a weak or negligible photoelastic response, ${\varphi }_0=0.78$). Behind the propagating fracture tips, the effective stress field exhibits an evolving “effective stress shadow,” where the intergranular stress is low and the granular pack exhibits undrained behavior.

Figure 8

An annulus sector used to derive the pressure-diffusion equation in radial coordinates.

Figure 9

The numerical implementation scheme for the mathematical model. The fluid pressure is fully coupled with the granular mechanics by solving the unknown variables, $h(t)$ and $r_{\mathrm{frac}}(t)$, iteratively until convergence at each time step.