Crossover phenomena in viscous fingering with nonviscous-flow threshold

Models for viscous fingering are presented to show a crossover between the diffusion-limited-aggregation (DLA) fractal and the dense pattern by interplay between nonviscous and viscous forces. The velocity has a characteristic property: If the velocity is less than the critical value ${\mathit{v}}_{\mathit{c}}$, the displaced fluid behaves as a nonviscous flow, and if the velocity is larger than ${\mathit{v}}_{\mathit{c}}$, the displaced fluid flow is viscous. By using the dimensional analysis, a dimensionless parameter is found to govern the crossover between the DLA fractal and the dense pattern. It is shown that the crossover length from the DLA fractal to the dense pattern increases with the applied pressure. At a very low flow rate, the pattern becomes the dense structure. At a high flow rate, the pattern becomes the DLA fractal. The similarity of the result with the experiment of the viscous fingering through porous media is discussed. The model can also be applied to the viscous fingering under the gravitational field.