Interface evolution during radial miscible viscous fingering

We study experimentally the miscible radial displacement of a more viscous fluid by a less viscous one in a horizontal Hele-Shaw cell. For the range of tested injection rates and viscosity ratios we observe two regimes for the evolution of the fluid-fluid interface. At early times the interface length increases linearly with time, which is typical of the Saffman-Taylor instability for this radial configuration. However, as time increases, the interface growth slows down and scales as $\sim t^{\frac{1}{2}}$, as one expects in a stable displacement, indicating that the overall flow instability has shut down. Surprisingly, the crossover time between these two regimes decreases with increasing injection rate. We propose a theoretical model that is consistent with our experimental results, explains the origin of this second regime, and predicts the scaling of the crossover time with injection rate and the mobility ratio. The key determinant of the observed scalings is the competition between advection and diffusion time scales at the displacement front, suggesting that our analysis can be applied to other interfacial-evolution problems such as the Rayleigh-Bénard-Darcy instability.

Figure 1

(a)–(c) Concentration fields of the injected fluid displacing a more viscous one (viscosity contrast $M=10$) at the center of a Hele-Shaw cell at three different flow rates ($Q=0.004$, 0.008, and 0.02 mL/min, at $t=5508$, 2416, and 1214 s after the start of injection, respectively). (d), (e) Magnifications of the concentration field and its gradient, respectively. The black line represents the location of the fluid-fluid interface, defined as the locus of the local maxima of the concentration gradient.

Figure 2

Time evolution of the interface $\left(M=10\right)$. There exist two regimes, the first of rapid growth due to active viscous fingering, and the second of slower growth that corresponds to radial expansion of a frozen finger pattern. The crossover into the second regime occurs at earlier times for faster injection rates. Inset: Crossover time ${\tau }_{\times }$ between regimes (blue triangles) and total experimental time $T$ (red circles) for the $M=10$ experiments. The estimate of ${\tau }_{\times }$ is obtained from the theoretical model developed below. $T$ decreases with increasing injection rates faster than ${\tau }_{\times }$—an indication of the decreasing duration (and eventual absence) of the second regime for increasing injection rates.

Figure 3

Determination of finger-tip positions and number of fingers. (a) At each time step, the Cartesian coordinates $(x,y)$ of the interface $\gamma$ are converted into a radial-curvilinear coordinate system $(r,s)$, where $r$ is the radial distance from the injection point (center of the image) and $s$ is the distance along the interface $\gamma$. For each value of $s$, a single value of $r$ is defined, and thus the pattern is “unravelled.” (b) We define the finger tips as the local maxima (brown diamonds) of the curve $r\left(s\right)$, which allows for a robust determination of the number of fingers. (c) The finger-tip positions can be mapped onto the $(x,y)$-coordinate system, and tracked as a function of time. The orange circle in (a)–(c) indicates the same finger through the steps of image processing used to determine the number of fingers.

Figure 4

Experimental measurements of number of fingers $n_f$ as a function of injection rate $Q$ for all mobility ratios $M$ tested. The number of fingers scales as $Q^{2/3}{M}^{1/2}$ (the dashed line denotes a $Q^{2/3}$ scaling). Inset: Evolution of $n_f$ with time for $M=10$ and five different values of $Q$. For all injection rates, $n_f$ first increases as a result of tip splitting, but then stabilizes around the crossover time (marked with a red cross for each experiment).

Figure 5

Rescaled interface length vs rescaled time, for $M=10$. By rescaling each data set with the crossover time [Eq. (1)] and the interface length expected at this crossover time [Eq. (3)], the data sets collapse onto a single curve, and the crossover into the second regime occurs at $t/{\tau }_{\times }\approx 1$. Inset: The same rescaling applied to all the experimental data: $M=5,10,15,20,25$ (different colors) for the five different injection rates (different symbols).