A Primer on Percolation Theory

Synopsis

Percolation theory, a key concept in statistical physics and mathematics, explores how networks behave as nodes or links are added incrementally. This phenomenon represents a geometric phase transition, where small, disconnected clusters progressively merge into larger, interconnected structures, known as spanning clusters, once a critical threshold is reached. In this talk, I will introduce some standard graph algorithms which make it possible to simulate percolation on different grid geometries. In the context of oil extraction, percolation theory helps address key challenges related to predicting the flow of fluids through porous rock formations. For example, understanding how oil moves through complex, interconnected subsurface structures, such as shale layers, allows geologists and engineers to estimate vertical permeability, manage uncertainty in reservoir models, and optimize recovery processes like water coning and gravity drainage.