Morphodynamics of Fluid-Fluid Displacement in Three-Dimensional Deformable Granular Media

We study experimentally the displacement of one fluid by another in a granular pack to uncover relationships between fluid invasion and medium deformation. We develop an experimental setup that allows us to reconstruct the coupled invasion-deformation dynamics in 3D. We simultaneously characterize the fluid invasion pattern and document a transition from fluid-fluid displacement in pores to the formation of conduits by grain motion. We rationalize the findings in terms of a simple poromechanics model that indeed captures this transition as a result of the balance between viscous and frictional forces. These results contribute to elucidating the role of three dimensionality in the timing, mode, and morphology of fluid-fluid displacement and injection-induced deformation in porous media.

Figure 1

Image taken by the camera during a direct-imaging experiment in a cell with horizontal dimension $L=6\mathrm{cm}$ and beads of diameter $d=3\mathrm{mm}$. The dark pattern is the injected silicone oil.

Figure 2

Schematic of the experimental setup for PLIF imaging. The cubic cell is filled with borosilicate glass beads. We first inject glycerol, which is the defending fluid. Then we inject a mix of Dow Corning silicone oil at a constant injection rate $q$ with a syringe pump. We shine a laser sheet through the cell that excites the defending and invading fluids, and we image using a camera filtering out the light emitted directly by the laser. The beads appear black, the defending fluid gray, and the invading fluid white. The laser and the camera are mounted on a motorized stage, which allows scanning of the entire cell.

Figure 3

Image from the goniometer during measurement of the contact angle. We deposit a droplet of glycerol (dark gray) on a glass bead immersed in silicone oil.

Figure 4

Behavior of the solution [Eq. (9)] to the poroelastic diffusion IBVP [Eqs. (5)–(8)]. (Top panel) Piezometric head $h(r,t)$. (Bottom panel) Time derivative $\dot{h}(r,t)$. See the text for details on the parameters used.

Figure 5

Mohr-circle interpretation of the Mohr-Coulomb failure criterion [Eq. (13)]. Because the diagram is expressed in terms of effective stresses, ${\sigma }_n^{\prime }={\sigma }_n-p_f$, an increase in fluid pressure ($\delta p_f>0$) results in a decrease in the effective normal stress ${\sigma }_n^{\prime }$, shifting the Mohr circle to the left, towards the Mohr-Coulomb failure line.

Figure 6

Displacement of the top lid of the cell as a function of time, $H(t)-H_0$. The solid line represents the experimental data, whereas the dashed line is a linear fit. The data correspond to an experiment with $L=6\mathrm{cm}$, $d=1\mathrm{mm}$, $W=18.83\mathrm{N}$, $H_0=51\mathrm{mm}$ and $q=45\mathrm{ml}{\min }^{-1}$.

Figure 7

Mean squared error (MSE) for $\dot{H}$ in $(\overline{r},{\alpha }_f)$ space, illustrating that there is a Pareto curve that minimizes the MSE. Shown are the three sets of experimental data: (a) $L=6\mathrm{cm}$, $d=1\mathrm{mm}$; (b) $L=2\mathrm{cm}$, $d=1\mathrm{mm}$; and (c) $L=6\mathrm{cm}$, $d=3\mathrm{mm}$.

Figure 8

Mean squared error for $t_f$ in $(K_d,{\alpha }_f)$ space, where the relationship ${\alpha }_f={\alpha }_f^{\mathrm{opt}}(\overline{r})$ from the Pareto curve in Fig. 7 is already incorporated. The plots illustrate that $K_d$ is not well constrained by the data. Shown are the three sets of experimental data: (a) $L=6\mathrm{cm}$, $d=1\mathrm{mm}$; (b) $L=2\mathrm{cm}$, $d=1\mathrm{mm}$; and (c) $L=6\mathrm{cm}$, $d=3\mathrm{mm}$.

Figure 9

Model fit for lid displacement rate $\dot{H}$ [Eq. (21)]. The red solid line indicates the $1:1$ line denoting the perfect match between theory and experiment. Shown are the three sets of experimental data: (a) $L=6\mathrm{cm}$, $d=1\mathrm{mm}$; (b) $L=2\mathrm{cm}$, $d=1\mathrm{mm}$; and (c) $L=6\mathrm{cm}$, $d=3\mathrm{mm}$.

Figure 10

Model fit for time of failure $t_f$ [Eq. (18)]. The red solid line indicates the $1:1$ line denoting the perfect match between theory and experiment. Shown are the three sets of experimental data: (a) $L=6\mathrm{cm}$, $d=1\mathrm{mm}$; (b) $L=2\mathrm{cm}$, $d=1\mathrm{mm}$; and (c) $L=6\mathrm{cm}$, $d=3\mathrm{mm}$.

Figure 11

Model prediction and experimental data for lid displacement rate $\dot{H}$ as a function of injection rate $q$ [Eq. (21)]. Different symbols represent different confining weights $W$. Shown are the three sets of experimental data: (a) $L=6\mathrm{cm}$, $d=1\mathrm{mm}$; (b) $L=2\mathrm{cm}$, $d=1\mathrm{mm}$; and (c) $L=6\mathrm{cm}$, $d=3\mathrm{mm}$.

Figure 12

Fluid invasion pattern. (a) Injected volume as a function of time. The blue solid line represents the volume reconstructed from 3D dynamic imaging; the red dashed line is a fit with an imposed slope equal to the injection rate $q$. (b) Reconstructed 3D injection pattern at $t=16.2\mathrm{s}$. The base of the cube is of size $L=6\mathrm{cm}$. Experiment with $q=10\mathrm{ml}{\min }^{-1}$, $d=3\mathrm{mm}$, and $W=1.47\mathrm{N}$.

Figure 13

Fractal dimension. (a) $V_s/V_p$ as a function of $r_s/d$, where $V_s$ is the volume of the injected liquid inside a half sphere of radius $r_s$ centered at the injection point. $V_p$ is the characteristic volume of a single pore, and $d$ is the nominal bead diameter. The different colors correspond to scans at different times $t$. The solid line represents a linear fit for $r_s<18\mathrm{mm}$, which we consider to be unaffected by boundary effects and has, in this case, a slope of 2.50. The plot corresponds to an experiment with $d=3\mathrm{mm}$, $L=6\mathrm{cm}$, $W=1.47\mathrm{N}$, and $q=36\mathrm{ml}{\min }^{-1}$. (b) Fractal dimension $D_f$ estimated for different experiments with a differing injection rate $q$ and confining weight $W$, with $L=6\mathrm{cm}$ and $d=3\mathrm{mm}$.

Figure 14

Position of the glass beads at the beginning of fluid injection. Experiment with $L=6\mathrm{cm}$ and $d=3\mathrm{mm}$.

Figure 15

Injection pattern (in white) superimposed on the conduit volume (in red) near the time of breakthrough for experiments with $L=6\mathrm{cm}$, $d=3\mathrm{mm}$, and $W=1.47\mathrm{N}$. (a) $q=2\mathrm{ml}{\min }^{-1}$ at $t=108\mathrm{s}$ and $V=2.1\mathrm{ml}$. (b) $q=5\mathrm{ml}{\min }^{-1}$ at $t=61\mathrm{s}$ and $V=4.7\mathrm{ml}$. (c) $q=10\mathrm{ml}{\min }^{-1}$ at $t=29\mathrm{s}$ and $V=4.7\mathrm{ml}$. (d) $q=20\mathrm{ml}{\min }^{-1}$ at $t=12.8\mathrm{s}$ and $V=3.8\mathrm{ml}$. (e) $q=27\mathrm{ml}{\min }^{-1}$ at $t=9.6\mathrm{s}$ and $V=3.6\mathrm{ml}$. (f) $q=36\mathrm{ml}{\min }^{-1}$ at $t=12.7\mathrm{s}$ and $V=6.8\mathrm{ml}$.

Figure 16

Fraction of the fluid invasion volume that corresponds to the volume involving bead displacement, for different injection rates $q$.