3D Invasion Percolation: Fractal Geometry and Self-Organised Criticality

Basic explanation

Imagine oil displacing water in porous rock. At each step the invading fluid does not occupy sites at random: it takes the easiest available route - the unoccupied neighbour of the wet region with the lowest capillary resistance. Repeating that local choice grows a branching cluster with trapped “fjords” of defending fluid rather than a solid ball.

The written report studies this process - invasion percolation - on three-dimensional lattices. The browser demo below is deliberately two-dimensional so the same growth rule is visible step by step; it illustrates the algorithm, not the 3D measurements.

Research question

Standard site percolation occupies sites independently with a fixed probability $p$. That model is fundamental, but it is a poor description of dynamic drainage: real invasion follows paths of least resistance and has no externally tuned occupation probability.

The individual project therefore asks two linked questions:

1. What fractal dimension $d_f$ do invasion clusters attain in three dimensions?
2. Does the process exhibit self-organised criticality - approaching the ordinary percolation threshold $p_c$ through its own growth rule rather than by setting $p = p_c$ by hand?

The work extends a preceding group report on classical percolation definitions and one-dimensional cases; the individual report focuses on the invasion algorithm, optimisation, and 3D analysis.

Computational method

Each site $i$ on an $L \times L \times L$ lattice receives a uniform random resistance $r_i \sim U(0,1)$. Growth proceeds by a burst dynamic:

1. Occupy a central seed.
2. Maintain the interface: empty neighbours of occupied sites.
3. Invade the interface site with globally minimal resistance and expand the interface.

A naïve scan of the interface each step costs $O(N)$ work and $O(N^2)$ overall - impractical for $N > 10^5$. I stored the boundary in a min-heap so the weakest site is always at the root. Insertions and extractions then cost $O(\log N)$, giving roughly $O(N \log N)$ total time and making side lengths such as $L = 100$ routine.

Fractal scaling

Invasion clusters are fractal because the fluid seeks rare low-resistance corridors. Mass $M$ and radius of gyration $R_g$ are compared through a power-law fit whose exponent is the fractal dimension $d_f$.

Simulations on large lattices ($L = 400$ in 2D, $L = 150$ in 3D), repeated across realisations, produced

• $d_f \approx 1.88 \pm 0.02$ in two dimensions, close to the literature value $\approx 1.89$ for invasion percolation with trapping;
• $d_f \approx 2.52 \pm 0.03$ in three dimensions, denser because the extra dimension lets the fluid bypass obstacles.

Self-organised criticality

In ordinary percolation, fractal spanning structures appear only near a tuned $p_c$. In invasion percolation the acceptance profile of invaded resistances is nearly uniform up to a sharp cutoff near the known three-dimensional site threshold $p_c \approx 0.31$. The algorithm never sets an occupation probability equal to $p_c$; the critical scale emerges from always invading the weakest boundary site.

What the demo shows

The interactive lattice uses the same weakest-boundary rule on a smaller 2D grid, with a frontier highlighted and an invaded-resistance histogram. Geometry and the reference threshold differ from the 3D report, so treat the demo as a visual explanation of the data structure and growth rule, not a reproduction of the measured $d_f$.