Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles

The nonlinear evolution of viscous fingering instabilities in miscible displacement flows in porous media with nonmonotonic viscosity profiles is investigated. The flow is accurately simulated using a Hartley transform based spectral method. A flow with nonmonotonic viscosity profile has an unstable region followed downstream by a stable region. Instabilities first begin in the unstable region and then grow and penetrate the stable region. The striking contrast between viscous fingering in flows with monotonic and nonmonotonic viscosity profiles is the direction of fluid penetration. The nonmonotonicity in the viscosity profile gives rise to a new phenomena of ‘‘reverse’’ fingering in which the displaced fluid fingers through the displacing fluid more readily than vice versa. A forward and a reverse mixing lengths are defined to characterize the growth of the mixing zone in the two directions. At large times, both the forward and reverse mixing lengths grow linearly in time. A model nonmonotonic viscosity profile is used to parametrically study the asymptotic mixing rates. The parametric study shows that for a given end point and maximum viscosities the growth rate of the mixing zone varies nonmonotonically with the length of the stable barrier in the viscosity profile. A physical mechanism is put forth to explain the observed phenomena of reverse fingering and its dependence on the parameters of the problem. Finally, a finger is isolated and its evolution is studied to understand the differences in the mechanisms that control the growth of a finger in flows with monotonic and nonmonotonic viscosity profiles.