Viscous Fingering Instability

Viscous fingering is an ubiquitous instability that occurs when a less viscous fluid displaces a more viscous fluid in a porous medium. The interface between the two fluids starts to deform, and finger-like patterns emerge and grow. This phenomenon can either increases the mixing in porous media, which is incredibly difficult because of the absence of turbulence that can actively stir the flow or be dramatic to some processes.

The typical example is secondary oil recovery, for which fingering from the injected aqueous solution pushing the more viscous oil in underground reservoirs of porous rocks reduces the sweep efficiency severely. Similarly, one solution to decrease the carbon dioxide emitted to the atmosphere is to capture it directly from the power plants and gas production sites, and stores in available reservoirs. The interaction between the supercritical $\text{CO}_2$ and the interstitial fluids, usually brine, is of interest. The resulting mixture from the carbon dioxide dissolution could undergo fingering and change radically the distribution of $\text{CO}_2$ in the reservoir. Viscous fingering is also detrimental in the case of chromatography, a technique used to separate and identify chemical compounds in a mixture flowing through a porous medium. The displacing fluid (the eluent) may be less viscous than the sample mixture. The initial planar interface will deform because of fingering, resulting in an inefficient separation. Last, viscous fingering can play a major role in soil contamination by enlarging considerably the polluted area. Hence the study of viscous fingering is essential in numbers of domains.

In my PhD project, I simulate the viscous fingering instability using the lattice Boltzmann method ¹. The previous picture shows the evolution of the molar fraction of a binary mixture. The core ingredients of the instability are the diffusion and the viscosity contrast between the species. The behavior of the instability can dramatically change for a mixture of three and more species compared to the binary case. For instance, viscous fingering could be induced by reverse diffusion ² despite having a stable initial flow configuration.