Introduction to Invasion Percolationg
The invasion-percolation model was introduced by Wilkenson and Willemsen [1] in order to study the injection of an invading fluid (water) to drive the defending fluid (petroleum) to production wells. The description of the two dimensional square model on a grid of sites connected by bonds is:
- • The invasion-percolation cluster grows from a central site where fluid is injected.
- • The bonds connecting this site to adjacent sites are assigned random numbers in the range 0 to 1.
- • The weakest bond (smallest number) opens and fluid flows to the adjacent site.
- • This process continues and a site is added at each time step.
The growing cluster is illustrated below for 1000, 2000, and 3000 time steps.
Here is a video showing the typical growth of an invasion percolation cluster.
Cluster growth occurs in bursts, the failure of a strong bond allows fluid to enter a region where there are weaker bonds. An example of burst structure for a cluster with 50,000 sites is shown below. The four largest bursts are shown in color, smaller bursts are shown in black.
Bursts satisfy power-law (Gutenberg-Richter) number (Nb)-area (Ab) scaling.
[1] Wilkinson, D. & Willemsen, J. F. (1983). Journal of Physics A, 16(14), 3365–3376.