Viscous fingering in square-lattice models with two types of bonds

Immiscible two-fluid displacement was studied experimentally and numerically in inhomogeneous porous models. Square-lattice models were designed using bond percolation. A fraction f of the bonds was randomly given a high permeability, set equal to 1. The rest (1-f) have a low permeability (=0.004). The patterns formed by the displacement of glycerol by air injected at the center of the model network were studied and compared with the results of diffusion-limited-aggregation (DLA) simulations on model bond networks that correspond precisely to the experimental models. We studied models with fractions of good bonds close to the threshold value for bond percolation fc=0.50. Experimental results are well represented by simulations and can be characterized by the same effective fractal dimension Dc1.5. We find that the model geometry strongly influences formation of the displacement pattern, and that the results are independent of f. Unlike usual DLA aggregates, which grow mainly at the tips, the geometry of our models causes growth in the interior, yielding a gradual increase of the effective fractal dimension as the aggregate grows.