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.