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The flow of yield stress fluids in porous media presents interestingcomplexity due to the interplay between the non-linear rheology and theheterogeneity of the medium. A remarkable consequence is that the number offlow paths increases with the applied pressure difference and is responsiblefor a non-linear Darcy law. Previous studies have focused on the protocol wherethe pressure difference is imposed. Here we consider instead the case ofimposed flow rate, $Q$. In contrast to Newtonian fluids, the two types ofboundary conditions have an important influence on the flow field. Using atwo-dimensional pore network model we observe a boundary layer of merging flowpaths of size $\ell(Q) \sim Q^{-\mu/\delta}$ where $\mu = 0.42 \pm 0.02$ and$\delta \simeq 0.63 \pm 0.05$. Beyond this layer the density of the flow pathsis homogeneous and grows as $Q^\mu$. Using a mapping to the directed polymermodel we identify $\delta$ with the roughness exponent of the polymer. We alsocharacterize the statistics of non-flowing surfaces in terms of avalanchespulled at one end.

Influence of the imposed flow rate boundary condition on the flow of Bingham fluid in porous media