We examine rectilinear two-dimensional miscible viscous fingering in response to both injection and withdrawal of low-viscosity fluid using numerical simulations. We consider sinusoidal injection-withdrawal cycles. In the injection period, the input flow rate increases initially before decreasing; we perform a linear stability analysis and examine the development of nonlinear fingering regimes. The instability can progress through the regimes of independent finger growth, coalescence and interaction, the emergence of a single finger, and finally shutdown. Whether or not substantial fingering occurred during the injection controls the dynamics in the withdrawal period. The amount by which fingers are reversed is somewhat insensitive to the specific nonlinear fingering regime that was attained. At the end of injection, the fingers generally have a tip with a localized peak in concentration of the low-viscosity fluid. This region then changes direction upon withdrawal. The low viscosity tip retraces the original finger because flow there is preferential as the viscosity is lower relative to the ambient fluid. However, a “reverse viscous finger” emerges because this tip has even lower viscosity than the fluid within the body of the original finger, leading to an unstable concentration gradient at the edge of the reverse finger. Where the original fingers merged, the reverse finger tends to split. This reversal can substantially reduce the mixing length during the withdrawal period. The analysis is extended to the case of multiple cycles.

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• Flow instability
• Flows in porous media
• Multiphase flows
• Viscosity