  Predicting the flow of non-Newtonian fluids in porous structure is still a
challenging issue due to the interplay betwen the microscopic disorder and the
non-linear rheology. In this letter, we study the case of an yield stress fluid
in a two-dimensional structure. Thanks to a performant optimization algorithm,
we show that the system undergoes a continuous phase transition in the behavior
of the flow controlled by the applied pressure drop. In analogy with the
studies of the plastic depinning of vortex lattices in high-$T_c$
superconductors we characterize the nonlinearity of the flow curve and relate
it to the change in the geometry of the open channels. In particular, close to
the transition, an universal scale free distribution of the channel length is
observed and explained theoretically via a mapping to the KPZ equation.
