Manipulation of viscous fingering in a radially tapered cell geometry

When a more mobile fluid displaces another immiscible one in a porous medium, viscous fingering propagates with a partial sweep, which hinders oil recovery and soil remedy. We experimentally investigate the feasibility of tuning such fingering propagation in a nonuniform narrow passage with a radial injection, which is widely used in various applications. We show that a radially converging cell can suppress the common viscous fingering observed in a uniform passage, and a full sweep of the displaced fluid is then achieved. The injection flow rate $Q$ can be further exploited to manipulate the viscous fingering instability. For a fixed gap gradient $\alpha$, our experimental results show a full sweep at a small $Q$ but partial displacement with fingering at a sufficient $Q$. Finally, by varying $\alpha$, we identify and characterize the variation of the critical threshold between stable and unstable displacements. Our experimental results reveal good agreement with theoretical predictions by a linear stability analysis.

Figure 1

(a) Schematic diagram of the side view of the experimental setup of immiscible fluid-fluid displacement in a radially converging passage, with a viscosity ratio $\lambda ={\mu }_2/{\mu }_1=8.8\times {10}^3$. (b) Snapshot of a classical viscous-fingering pattern obtained when air pushes oil in a flat radial Hele-Shaw cell with $h_0=1.2$ mm and $Q=40$ ml/min. (c) In contrast, snapshot of a stable interface with a complete sweep of oil by air in a radially tapered cell with $\alpha =-6.67\times {10}^{-2}, h_0=150\mu \text{m}$, and $Q=40$ ml/min. For the experiments in (b) and (c), $h_0$ are chosen so that both configurations have equal fluid volumes. The scale bars in (b) and (c) are 2 cm.

Figure 2

(Supplemental Material movie [24].) The dependence of sweep efficiency on flow rate $Q$: snapshots of two representative experiments [24] of air displacing oil for the same geometrical configuration with the gap gradient $\alpha =-4.75\times {10}^{-2}$ and $h_0=500\mu \text{m}$, but different flow rates: (a) stable displacement at $Q=40$ ml/min, whereas (b) viscous fingering at $Q=110$ ml/min.

Figure 3

(a) Stability diagram of stable vs unstable propagation front by varying the flow rate $Q$, for different gap gradients $\alpha$ (while $h_0=250\mu \text{m}$). The general trend shows that for each $\alpha$, stable and complete sweep occurs at a relatively small $Q$ (denoted by $•$), whereas unstable fingering propagation emerges at large $Q$ ($\circ$). For a radially tapered cell of $\alpha =-4.75\times {10}^{-2}, h_0=250\mu \mathrm{m}$, time evolution of top-view, stable interfaces with $Q=20$ ml/min in (b) [corresponding to $\square$ in (a)], while unstable interfaces in (c) for $Q=70$ ml/min [$◊$ in (a)]. The time steps are $\mathrm{\Delta }t=9$ and 1 s between each contour for (b) and (c), respectively.

Figure 4

(a) Variation of the critical capillary number ${\text{Ca}}^{*}$ separating stable vs unstable displacements for different depth gradients $\alpha$ and $h_0$. We compare the experimental values ($•$, $\blacksquare$, $⧫$) to the theoretical ${\text{Ca}}_{\text{th}}^{*}$ ($○$, $\square$, $◊$) derived from Eq. (2) with $\alpha , r$, and $N$ from our experimental results and parameters. (b) Surface plot of theoretical ${\text{Ca}}_{\text{th}}^{*}$ greatly depends on $r$ and $N$ using Eq. (2), for $\alpha =-8.66\times {10}^{-2}$ and $h_0=250\mu \text{m}$, showing a stable displacement when $\text{Ca}<{\text{Ca}}^{*}$, whereas an unstable one when $\text{Ca}>{\text{Ca}}^{*}$.