Fractal dimension and avalanches of invasion percolation: the effect of aspect ratio

Abstract

Troubled by unreliable estimates of the fractal dimension from straightforward box-counting applied to invasion percolation in million site short-wide systems (i.e., the length in the average flow direction is the smaller dimension), we undertook a study of the effect of aspect ratio on fractal dimension determinations and on avalanche structure. In box-counting, we found evidence of a competition between the different singular behaviors associated with the bulk and the external hull (interface), which was most noticeable for the short-wide systems with a long external hull. Modifying the box-counting to exclude those boxes covering the external hull provides results for the short-wide systems which are consistent with results from the literature and from straightforward box-counting on long-narrow systems. Not surprisingly, we found that the avalanche size distribution was ‘cut off’ by the length in the short-wide systems; however, we also found that the distribution was cut off by the width in the long-narrow systems. Therefore, the smaller dimension served as a cut-off length for the distribution of avalanche sizes, so that in the long-narrow systems, the distribution of avalanche sizes collapses long before the injected fluid reaches the outlet. This results in fingering patterns in the long-narrow systems that are different from those in the short-wide systems where the avalanche size distribution is maintained all the way to the outlet. Determining the fractal dimension from the power-law dependence of mass upon a typical length scale was found to be unaffected by the length of the external hull, providing the standard literature values of fractal dimension for the problematic short-wide systems.