Effect of sinusoidal injection velocity on miscible viscous fingering of a finite sample: Nonlinear simulation

Abstract

The effect of a sinusoidal injection on the fingering instability in a miscible displacement in the application of liquid chromatography, pollutant contamination in aquifers, etc., is investigated. The injection velocity, U(t) is characterized by its amplitude of Γ and time-period of T. The solute transport, flow in porous media, and mass conservation in a two-dimensional porous media is modeled by the convection-diffusion equation, Darcy's equation, and the continuity equation, respectively. The numerical simulation is performed in COMSOL Multiphysics utilizing a finite-element based approach. The fingering dynamics for various time-period have been studied for two scenarios namely, injection-extraction (Γ>1) and extraction-injection (Γ<−1). The onset of fingers and vigorous mixing is observed for Γ>1, whereas for Γ<−1, the onset gets delayed. The viscosity contrast between the sample and the surrounding fluid is characterized by the log-mobility ratio R. When R>0 the rare interface becomes unstable, while for R<0 the frontal interface deformed. In the case of R<0, the extraction-injection process attenuates the fingering dynamics, which is beneficial in chromatographic separations or pollutant dispersion in underground aquifers. The injection-extraction process is observed to have a longer mixing length, indicating early interaction between both interfaces. The degree of mixing χ(t) is more pronounced for injection-extraction scenario and least for extraction-injection R<0,Γ=−2. The average convective forces are more dominant for Γ>1,R=2 till the deformed rare interface interact with diffusive frontal interface. The average diffusive forces are significant for Γ<−1,R=−2 which can be helpful in separation of chemicals in chromatography. This study therefore provided new insights into the role of alternate injection-extraction injections in altering the fingering dynamics of the miscible sample.