Viscosity scaling in hydrodynamic instabilities in porous media

The importance of viscosity scaling in the context of viscous fingering in a finite slice with viscosity dependent diffusivity is investigated theoretically. Choosing the characteristic viscosity classically as either the displacing or displaced fluid viscosity for both more and less viscous slice leads to inappropriate theoretical predictions, which do not support the physics. With an appropriate choice of the characteristic viscosity, we show that the onset of instability and the initial dynamics of the finger patterns are the same for both more and less viscous slices. Our analysis will be helpful in the theoretical understanding of buoyancy-driven convection in a variable viscosity layer in vertical porous media or VF with non-monotonic viscosity profiles.