Scaling relations for interface motion through disordered media: Application to two-dimensional fluid invasion

We consider the critical transitions that occur as the force driving an interface through a random medium is increased. The total displacement of the interface, and the incremental advance after a small increase in force, diverge as the force approaches a critical depinning threshold. At the critical force there is a power-law distribution of growth sizes. General scaling relations are derived between the critical exponents associated with such transitions. These scaling relations are tested on a model system---fluid invasion of a two-dimensional porous medium. Critical exponents are determined from simulations using finite-size-scaling techniques. Two universality classes are identified: percolation and depinning. In both cases the calculated exponents obey the scaling relations.