Viscous instabilities induced by an initially piecewise linear concentration profile with a discontinuity

Abstract

In this study, we consider a viscous instability induced by injecting a fluid, containing a heterogeneously distributed species, into a two-dimensional porous medium. The initial concentration of the species is a piece wise linear function with a discontinuity. Using the quasi-steady-state approximation, the stability of the system can be reduced to a coupled system of ODEs. A dispersion equation for the initial configuration was obtained. As the species diffuses, the instantaneous growth rates evolve in time. The stability of the system is obtained numerically. Although the initial discontinuity destabilises the system, eventually, after a finite amount of time, the system becomes stable. The system becomes stable even when non-monotonic viscosity profiles are present. The linear stability predictions, around the time when the product of the instantaneous growth rate and time equal one, yields wave numbers around 17% smaller than those found in the early stages of the nonlinear simulations. The nonlinear simulations show that when a local minimum in the viscosity profile is present, the fingers tend to be slightly longer, than when a local maximum in the viscosity profile is present.