Understanding the flow of yield stress fluids in porous media is a major challenge. In particular, experiments and extensive numerical simulations report a nonlinear Darcy law as a function of the pressure gradient. In this letter we consider a treelike porous structure for which the problem of the flow can be resolved exactly due to a mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Our results confirm the nonlinear behavior of the flow and expresses its full pressure dependence via the density of low-energy paths of DP restricted to vanishing overlap. These universal predictions are confirmed by extensive numerical simulations.

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• Flows in porous media
• Network flow
• Optimization problems
• Disordered systems
• Front propagation
• Glassy systems
• Non-Newtonian fluids
• Porous media
• Random & disordered media
• Differential equations
• Finite-element method