Viscous fingering near the percolation threshold: Double-crossover phenomena

Viscous fingering at a nonzero viscosity ratio on percolating clusters is considered to study morphological changes of patterns formed by the injected fluid in porous media. A fraction P of bonds is filled by the displaced fluid, while the others (1-P) are blocked, where P is the usual percolation probability. Fluid with a low viscosity is injected into the percolating cluster filled by the displaced fluid with high viscosity. Morphological changes of patterns of the injected fluid are described in terms of crossover phenomena by making use of a four-parameter position-space renormalization-group method. It is found that when ${\mathrm{\mu }}_{\mathit{I}}$/${\mathrm{\mu }}_{\mathit{D}}$≪(P-${\mathit{P}}_{\mathit{c}}$)≪1 the double crossover occurs from the diffusion-limited aggregation (DLA) on an incipient percolation cluster through the DLA on the perfect lattice to the dense structure, and when 1≫${\mathrm{\mu }}_{\mathit{I}}$/${\mathrm{\mu }}_{\mathit{D}}$≫(P-${\mathit{P}}_{\mathit{c}}$) the other double crossover appears from the DLA on an incipient percolation cluster through the invasion percolation to the dense structure, where ${\mathrm{\mu }}_{\mathit{I}}$/${\mathrm{\mu }}_{\mathit{D}}$ is the viscosity ratio and ${\mathit{P}}_{\mathit{c}}$ the critical percolation probability.