Gradient stabilized and destabilized invasion percolation

The effects of gravity stabilization and destabilization on the slow displacement of a wetting fluid by a non-wetting fluid in two-dimensional and three-dimensional porous media have been investigated experimentally. The characteristic features of the resulting displacement patterns can be reproduced quite well by invasion percolation models with a spatial gradient added to the usual random threshold distribution. In the case of destabilized displacement the patterns can be described in terms of a string of blobs of size ξ that form a directed walk. The internal structure of the blobs is like that of an invasion percolation cluster (with trapping in the two-dimensional case). In the stabilized case the structure is like that of a fractal invasion percolation cluster on short length scales (lengths ⪡ξ) and is uniform on longer lengths. The correlation length (ξ) also describes the maximum hole diameter. The invasion front is a self-similar fractal on length scales shorter than ξ and flat on longer length scales. In both the experiments and the simulations the correlation length ξ is related to the Bond number (B0, the ratio between buoyancy and capillary forces) by ξ ∼ |B0|−ʋ(ʋ+1) where ʋ is the ordinary percolation correlation length exponent) in accord with the theoretical arguments of Wilkinson (Phys. Rev. A 30 (1984) 520; 34 (1986) 1380).