Invasion percolation models fluid displacement in porous media — e.g., water pushing oil out of rock. Each cell has a random "capillary pressure" (throat size). The invading fluid always enters the neighboring cell with the lowest resistance (widest throat), growing the invasion cluster greedily one cell at a time.
Unlike ordinary percolation (random bulk filling), invasion percolation self-organizes to the critical threshold p_c without any tuning — a form of self-organized criticality. Trapping occurs when the defending fluid is surrounded and cannot escape; trapped clusters are shown in orange. The fractal dimension of the invasion front is D_f ≈ 1.82 (2D).