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

Darcy law for yield stress fluid