Stability of miscible displacements in porous media with nonmonotonic viscosity profiles

Abstract

The effect of a nonmonotonic viscosity profile on the stability of miscible displacements in porous medium is studied. A linear theory using the quasi-steady-state approximation is employed to find the growth rate of the perturbations in the flow. The resulting eigenvalue problem has both a discrete and continuous spectrum. It is possible to obtain an analytical solution for a step base state concentration profile. The step profile result indicates that a nonmonotonic viscosity profile can be stable even when the viscosity contrast, measured by the end-point viscosities, is unfavorable, and vice versa. Asymptotic expansions of the growth rate for short times and for small wave numbers of the disturbances are obtained. The short-time expansion shows that the diffusion of the base state does not always mitigate the instabilities and the small wave-number expansion gives a sufficient condition for the flow to be unstable. Finally, the eigenvalue problem is solved numerically for diffused concentration profiles using finite difference methods. A model viscosity profile is chosen to parametrically study the stability of nonmonotonic profiles. The parametric study shows that the diffusion of the base state can have a destabilizing effect, with the surprising result that the nonmonotonic profiles that are predicted to be stable by the step profile analysis eventually become unstable by the destabilizing effect of diffusion. The ‘‘critical’’ time at which a stable flow turns unstable is obtained as a function of the parameters of the problem. A physical mechanism is proposed to explain the effects of diffusion on the stability of the flow.