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

The flow of yield stress fluids in porous media presents interesting complexity due to the interplay between the non-linear rheology and the heterogeneity of the medium. A remarkable consequence is that the number of flow paths increases with the applied pressure difference and is responsible for a non-linear Darcy law. Previous studies have focused on the protocol where the pressure difference is imposed. Here we consider instead the case of imposed flow rate, $Q$. In contrast to Newtonian fluids, the two types of boundary conditions have an important influence on the flow field. Using a two-dimensional pore network model we observe a boundary layer of merging flow paths 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 paths is homogeneous and grows as $Q^\mu$. Using a mapping to the directed polymer model we identify $\delta$ with the roughness exponent of the polymer. We also characterize the statistics of non-flowing surfaces in terms of avalanches pulled at one end.