We measured the pressure drop vs flow rate during the flow, in a wide range of velocities, of well controlled yield stress fluids through confined packings of glass beads of different sizes. A detailed analysis of the data makes it possible to extract a general expression for the pressure drop vs flow rate curve through a porous medium as a function of the flow rate and the characteristics of the system. This general law has a form similar to the Herschel-Bulkley model describing the rheological behavior of such fluids in simple shear, i.e. it expresses as the sum of a critical (yielding) pressure drop and a flow rate dependent term. This law involves the rheological parameters of the fluid, one characteristic length of the medium, and two coefficients which only depend on the structure of the porous medium. The first coefficient is related to the path of maximum width throughout the porous medium while the second coefficient reflects the pore size distribution. The values of these coefficients were determined in the case of a granular packing.

From NMR measurements we show that the velocity field of a yield stress fluid flowing through a disordered well-connected porous medium is very close to that for a Newtonian fluid. In particular, it is shown that no arrested regions exist even at very low velocities, for which the solid regime is expected to be dominant. This suggests that these results obtained for strongly nonlinear fluid can be extrapolated to any nonlinear fluid. We deduce a generalized form of Darcy's law for such materials and provide insight into the physical origin of the coefficients involved in this expression, which are shown to be moments of the second invariant of the strain rate tensor.

We show that frustrated creep flows of yield stress fluids give rise to a boundary layer, which takes the form of a liquid region of uniform significant thickness separating two solid regions. In this boundary layer the shear rate is approximately constant for a given flow rate and the layer thickness extremely slowly varies with the flow rate.