Prebifurcation enhancement of imbibition-drainage hysteresis cycles

The efficient transport of fluids through disordered media requires a thorough understanding of how the driving rate affects two-phase interface propagation. Despite our understanding of front dynamics in homogeneous environments, as well as how medium heterogeneities shape fluid interfaces at rest, little is known about the effects of localized topographical variations on large-scale interface dynamics. To gain physical insights into this problem, we study here oil-air displacements through an “imperfect” Hele-Shaw cell. Combining experiments, numerical simulations, and theory, we show that the flow rate dramatically alters the interface response to a porous constriction as one approaches the Saffman-Taylor instability, strictly under stable conditions. This gives rise to asymmetric imbibition–drainage hysteresis cycles that feature divergent extensions and nonlocal effects, all of which are aptly captured and explained by a minimal free boundary model.

Figure 1

Setup and sample images. (a) Schematic $y\text{−}z$ side view of our tilted HS cell near the back-end and front-end of the mesa (light orange). Red (blue) arrow indicates the direction of oil propagation in imbibition (drainage). (b1,b2) Sample images from two separate experiments, captured from above the back-end or front-end of the mesa.

Figure 2

Oil-air displacements through a mesa constriction at increasing flow rates. Top views of the experimental and computational interface evolution during imbibition (red, advancing from left to right) and drainage (blue, advancing from right to left). Interface profiles are presented in intervals of $\mathrm{\Delta }t=100\mathrm{s}$ for low velocities (left panels) and $\mathrm{\Delta }t=10\mathrm{s}$ for high velocities (right panels). We define the deformation measure $\eta$ as shown in the bottom right experimental panel, with $d=9\mathrm{mm}$.

Figure 3

Asymmetric imbibition-drainage hysteresis cycles. Top left: The deformation amplitude $\eta =h\left(0\right)-\overline{h}(\pm d)$ is plotted with confidence $\left|h\right(d)-h(-d\left)\right|$ against the interface reference position, $\overline{h}(\pm d)=(h\left(d\right)+h(-d))/2$, for drainage or imbibition experiments (blue or red curves). Lighter colors correspond to faster speeds. Increasing the displacement rate in drainage (imbibition) gives rise to stronger (weaker) deformations. Experimental drainage speeds are $V_0=\{-216\pm 14,-139\pm 9,-10\pm 1\}\mu \mathrm{m}/\mathrm{s}$ about the front-end of the mesa and $V_0=\{-201\pm 14,-126\pm 10,-9\pm 2\}\mu \text{m/s}$ about the back-end. Experimental imbibition speeds are $V_0=\{8\pm 2,51\pm 4,205\pm 14\}\mu \mathrm{m}/\mathrm{s}$ about the back-end of the mesa and $V_0=\{8\pm 1,52\pm 4,215\pm 14\}\mu \mathrm{m}/\mathrm{s}$ about the front-end. Bottom left: Same plots for FE simulations with drainage speeds $V_0=\{-210,-130,-8\}\mu \mathrm{m}/\mathrm{s}$ (light to dark blue) and imbibition speeds $V_0=\{8,130,210\}\mu \mathrm{m}/\mathrm{s}$ (dark to light red). Right boxes: Zooms highlighting different corners of the trajectories, excluding intermediate velocities for clarity.

Figure 4

Impact of flow rate on steady deformations. Measured and computed values of the deformation amplitude $\eta$ as a function of the driving speed $V_0$. Green line marks the threshold of the Saffman-Taylor instability, where $\text{Ca}=-\text{Bo}$.

Figure 5

Extreme front deformations and nonlinear depinning from the back-end. (a1,a2) Experimental images separated by $10\text{s}$, captured in the course of oil withdrawal with $V_0=-255\pm 17\mu \mathrm{m}$. (b1,b2) Mesh and interface of a moving computational domain separated by $10\text{s}$, taken from a drainage simulation with $V_0=-260\mu \mathrm{m}$. (c) Thin filament produced by quasistatic oil withdrawal through a wider mesa ($w=9\text{mm}$) with a lower inclination of the cell ($\alpha =2^{\circ }{21}^{\prime }$), meaning lower Bo. This picture was captured manually from an oblique angle using a different type of illumination.

Figure 6

Mesh and pressure in FE simulation. Snapshots from a typical drainage simulation ($V_0=-210\mu \mathrm{m}/\mathrm{s}$) showing the adaptive mesh (black) and the resulting pressure field (color-coded contour plot). The mesa constriction is either fully embedded, partly embedded, or out of the oil domain $\mathrm{\Omega }$, depending on the progression of the free interface ${\mathrm{\Gamma }}_1$.

Figure 7

Linear steady state. We plot the linear profile $h\left(x\right)$, defined in Eq. (4), for different values of $V_0$. Drainage speeds were set to $V_0=\{-200,-160,-120,-80,-40\}\mu \mathrm{m}/\mathrm{s}$ (light to dark blue) and imbibition speeds were set to $V_0=\{40,80,120,160,200\}\mu \mathrm{m}/\mathrm{s}$ (dark to light red). The $V_0=0$ state is plotted in black. Axes ticks are in mm.

Figure 8

Invalidity of the linear curvature assumption. The maximal slope of the linear and nonlinear steady state is plotted as a function of the driving speed. Other parameters are fixed at their experimental values. Green line marks the threshold of the Saffman-Taylor instability, where $\text{Ca}=-\text{Bo}$.

Figure 9

Steady state profiles. Top panels: The nonlinear steady-state solution (black, obtained via our “shooting method”) and its explicit linear approximation [dashed, Eq. (4)] are compared with the maximal interface deformation in drainage or imbibition simulations (blue or red). Confidence levels on the first represent the deviation due to cumulative uncertainty in our measured parameters (not including error in $V_0$). Axes ticks are in mm. Bottom panels: Comparison of the profile derivatives.