Fingering instability and mixing of a blob in porous media

The curvature of the unstable part of the miscible interface between a circular blob and the ambient fluid in two-dimensional homogeneous porous media depends on the viscosity of the fluids. The influence of the interface curvature on the fingering instability and mixing of a miscible blob within a rectilinear displacement is investigated numerically. The fluid velocity in porous media is governed by Darcy's law, coupled with a convection-diffusion equation that determines the evolution of the solute concentration controlling the viscosity of the fluids. Numerical simulations are performed using a Fourier pseudospectral method to determine the dynamics of a miscible blob (circular or square). It is shown that for a less viscous circular blob, there exist three different instability regions without any finite $R$-window for viscous fingering, unlike the case of a more viscous circular blob. Critical blob radius for the onset of instability is smaller for a less viscous blob as compared to its more viscous counterpart. Fingering enhances spreading and mixing of miscible fluids. Hence a less viscous blob mixes with the ambient fluid quicker than the more viscous one. Furthermore, we show that mixing increases with the viscosity contrast for a less viscous blob, while for a more viscous one mixing depends nonmonotonically on the viscosity contrast. For a more viscous blob mixing depends nonmonotonically on the dispersion anisotropy, while it decreases monotonically with the anisotropic dispersion coefficient for a less viscous blob. We also show that the dynamics of a more viscous square blob is qualitatively similar to that of a circular one, except the existence of the lump-shaped instability region in the $R$-Pe plane. We have shown that the Rayleigh-Taylor instability in a circular blob (heavier or lighter than the ambient fluid) is independent of the interface curvature.

Figure 1

Schematic of the displacement of circular blob in two-dimensional homogeneous porous media.

Figure 2

Streamline distribution in the vicinity of the circular blob of initial radius $r=0.5$ at time $t=1,2,5,10$ (from top to bottom) for $R=1.25$: (a) $\mathrm{Pe}=500$, (b) $\mathrm{Pe}=900$, and (c) $\mathrm{Pe}=1000$. The red contours correspond to $c=0.01,0.50,0.99$ (from outside to inside).

Figure 3

(a) Schematic of the displacement of an initially squared miscible blob in a homogeneous porous medium. (b) Phase plot in $R$-Pe plane for $r=0.5$ shows two distinct instability regions: VF (I) and comet-shaped deformation (II). Streamlines in the vicinity of the blob (red contour lines) are presented for $\mathrm{Pe}=500,R=1.25$ (I), and $\mathrm{Pe}=900,R=2.5$ (II).

Figure 4

Spatial distribution of concentration in the moving frame of reference at times $t=0,2,5,10$ from top to bottom for $r=0.5$, (Pe, $R$) = (a) (300, 1.25), (b) (900, 1.25), and (c) (900, 2.5).

Figure 5

Spatial distribution of concentration at $t=1,2,4$ from top to bottom for Pe = $1000,r=0.5,\epsilon =1$, and $R$ = (a) $-0.5$, (b) $-1$, (c) $-1.5$.

Figure 6

Streamline distribution in the vicinity of a circular blob of radius $r=0.5$ for $R=-1.5$ and Pe $=1000$. From top to bottom: $t=1,2,4$.

Figure 7

(a) Phase plot in $R$-Pe plane for $r=0.5$ shows three distinct instability regions: comet shaped (I), lump shaped (II), and VF (III). Region II corresponds to transition zone between the VF and comet-shaped instability. The symbols ($\square$ or $◯$) correspond to some of the data points used to calculate the boundaries between two adjacent stability regions. (b) Snapshots of the representative spatial structure of solute concentration c corresponding to each instability region are shown: VF ($R=-1,\mathrm{Pe}={10}^3$), lump shaped $(R=-0.55,\mathrm{Pe}=625)$, and comet shaped $(R=-0.35,\mathrm{Pe}=500)$.

Figure 8

Spatial distribution of the solute concentration at time $t=1,2,4,10$ (from top to bottom) for $r=0.5$, (Pe, $R$) $=$ (a) (500, $-0.35$), (b) (625, $-0.55$), and (c) (500, $-1.5$).

Figure 9

Phase plot in the $R$-Pe plane for $r=0.25$ shows three distinct instability regions: comet shaped (I), lump shaped (II), and VF (III), similar to the case of $r=0.5$. The qualitative features remain unchanged by changing the blob radius. Quantitative behaviors of the three regions change: for decreasing $r$, region III shrinks, while regions I and II expand. The symbols $(\square$ or $◯$) correspond to some of the data points used to calculate the boundaries between two adjacent stability regions.

Figure 10

Temporal evolution of the axial spreading relative to the initial extent, ${\sigma }_x^2\left(t\right)/{\sigma }_{x0}^2$, for Pe $=1000,r=0.5$.

Figure 11

Temporal evolution of the transverse spreading relative to the initial extent, ${\sigma }_y^2\left(t\right)/{\sigma }_{y0}^2$, for $\mathrm{Pe}=1000,r=0.5$. Inset: At time $t\lesssim 4$, the transverse spreading for $R=0$ is more than the corresponding values of $R\ne 0$.

Figure 12

For $\mathrm{Pe}=1000$ and different values of $R$, the temporal evolution of the (a) normalized area $A\left(t\right)/A_0$ occupied by the blob fluid and (b) degree of mixing $\chi \left(t\right)$ of a circular blob of radius $r=0.5$.

Figure 13

Effect of anisotropic dispersion on the degree of mixing, $\chi \left(t\right)$, for $r=0.5,\mathrm{Pe}=1000$, (a) $R=1$, (b) $R=-1$. Inset: Variation of $\chi$ with $\epsilon$ at time $t=8$.

Figure 14

Streamlines in the vicinity of the blob for $Ra=1000$, and (a) ${\rho }_1<{\rho }_2$, (b) ${\rho }_1>{\rho }_2$. The blob is represented by isocontours of solute concentration $c$ (red). The contour values are $c=0.01,0.5,0.99$ (from outside to inside).