Viscous fingering with partially miscible fluids

Viscous fingering—the fluid-mechanical instability that takes place when a low-viscosity fluid displaces a high-viscosity fluid—has traditionally been studied under either fully miscible or fully immiscible fluid systems. Here we study the impact of partial miscibility (a common occurrence in practice) on the fingering dynamics. Through a careful design of the thermodynamic free energy of a binary mixture, we develop a phase-field model of fluid-fluid displacements in a Hele-Shaw cell for the general case in which the two fluids have limited (but nonzero) solubility into one another. We show, by means of high-resolution numerical simulations, that partial miscibility exerts a powerful control on the degree of fingering: fluid dissolution hinders fingering while fluid exsolution enhances fingering. We also show that, as a result of the interplay between compositional exchange and the hydrodynamic pattern-forming process, stronger fingering promotes the system to approach thermodynamic equilibrium more quickly.

Figure 1

Displacement of a gas band through liquid phase in a Hele-Shaw cell: (a) initial setup and (b) the displacement leads to viscous fingering due to viscosity contrast. Meanwhile, compositional exchange occurs along the fingering interface if the two fluids are out of thermodynamic equilibrium. For example, in this sample image, the liquid phase is initially supersaturated with respect to gas and will swell the gas fingers as they evolve (see Sec. 4a). The blue dashed box indicates the area of study in our discussions.

Figure 2

Common tangent construction: Here we assign the parameters in Eq. (10) as ${\alpha }_l=2\times {10}^{-7},{\beta }_l=2\times {10}^4,{\alpha }_g=200,{\beta }_g=2\times {10}^{-4}$ so that the common tangent construction yields the equilibrium composition of the two fluids: $c_g^{\text{eq}}\approx 0.89,c_l^{\text{eq}}\approx 0.33$.

Figure 3

The coupling between different viscosity contrast and compositional effects leads to a rich set of viscous fingering patterns. Here we show snapshots of $c$ at $t=50$ for six different $R$ values (across each row). Each of the three rows correspond to different $c_l^0$ values: defending liquid is (a) undersaturated $\left(c_l^0=0.05\right)$, (b) near saturated $\left(c_l^0=0.33\right)$, and (c) oversaturated $\left(c_l^0=0.5\right)$. Note that the color map differs between each row to reveal the detailed structures in the concentration field.

Figure 4

Defending fluid is (a) undersaturated or (b) supersaturated. Top: snapshots of $\varphi$ at $t=46$. Middle: snapshots of $c$ at $t=46$. Bottom: horizontal transects of $\varphi$ (dashed) and $c$ (solid) at $t=46$ along the dashed lines indicated in the 2D plots in the top and middle rows. Note that regions where $c\approx 0.33$ indicate areas in which local thermodynamic equilibrium is in place.

Figure 5

Normalized interfacial length, $\mathcal{L}/L_y$, as a function of time for three compositional scenarios with (a) $R=5$ and (b) $R=1$. In both plots, colored insets show the traced outline of the fingering front at $t=50$ for oversaturated, near-saturated, and unsaturated defending liquid (left to right).

Figure 6

$\mathrm{\Delta }\overline{{\varphi }_g}\left(t\right)$ for $R=0,1,2,3,4,5$ (arrows indicating increasing order) when the defending phase is undersaturated (left) and supersaturated (right). The insets in both plots show snapshots of $\varphi$ at $t=50$ for the different values of $R$.

Figure 7

(Left) ${\overline{c_l}}_y\left(x\right)$ at $t=0,10,20,30,40$, and 50. Blue arrow indicates direction of time. Red dashed line indicates thermodynamic equilibrium: $c_l^{\text{eq}}\approx 0.33$. Red shaded area indicates metastability. (Right) Snapshots of $c$ for the corresponding times. The defending liquid is initially undersaturated in panel (a) and supersaturated in panel (b), and all simulations correspond to $R=5$.

Figure 8

Impact of Cahn number at $\text{Ma}=1/200$ and other parameters kept the same as in Sec. 3. We simulate for four values of Ch differing by a factor of two each. Shown here are snapshots of $c$ at $t=40$. The color-map range is $[0.05,0.9]$ for all figures.

Figure 9

Impact of solutal Marangoni number at $\text{Ch}=1/2000$ and other parameters kept the same as in Sec. 3. We simulate for five values of Ma differing by a factor of two each. Shown here are snapshots of $c$ (top) and $\varphi$ (bottom) at $t=50$.

Figure 10

Concentration variance ${\sigma }^2$ as a function of time for different values of Ma. The enlarged portion on the left corresponds to the early time regime when the fluid interface has not been significantly destabilized. The insets in the main figure correspond to snapshots of $c$ for the $\text{Ma}=1/25$ simulation.

Figure 11

Impact of miscibility gap in the bulk-mixing energy curves. (Left): $f_l\left(c\right)$ and $f_g\left(c\right)$ and their common tangent constructions. (Right) Snapshots of $c$ from simulations using the bulk-mixing energy described in the left column. Note that the color-map ranges are different and correspond to the initial composition (slightly wider than miscibility gap).