Spatial distribution of avalanches in invasion percolation: their role in fingering

For two decades, invasion percolation (IP) has provided a simple model of ‘drainage’ where a non-wetting fluid is injected into a porous media saturated with a wetting fluid, in the limit where capillary forces dominate and viscous forces are negligible. IP produces a characteristic fingering with a fractal dimension close to that of ordinary critical percolation. Avalanches (also called ‘bursts’ or ‘Haines jumps’) have been observed. In this paper, we focus on the practical issues relating to the causes of the fingering and of the low saturations of injected fluid. We show that the saturation and the average position of the injected fluid exhibit standard fractal scaling behavior. However, the fractional flow of the injected fluid does not allow an average analysis because of the noise arising from the avalanches, even for the million site systems investigated in this paper. In studying the spatial distribution of these avalanches, we find a size cutoff depending upon the position of the avalanches; this is characteristic of the finite size of the system and signals that the systems have not achieved self-organized criticality. Furthermore, we show that the average size of these avalanches, 〈sa〉, increases with their average distance, 〈x〉, from the outlet as 〈sa〉≈〈x〉1.1. As a result, larger avalanches will tend to occur at the end of longer fingers causing preferential growth of the long fingers at the expense of the shorter fingers.