We study the flow, through a model two-dimensional porous medium, of Newtonian fluids, power-law fluids, and viscoplastic fluids in the laminar regime and with moderate or dominant effects of the yielding term. A numerical technique able to take properly into account yielding effects in viscoplastic flows without any regularization is used to determine the detailed flow characteristics. We show that as soon as the distance between the disks forming the porous medium is sufficiently small, the velocity field and in particular the distribution function of the velocity of these different fluids in a wide range of flow regimes are similar. Moreover, the volume fraction of fluid at rest is negligible even at low flow rate. Thus the non-Newtonian character of a fluid flowing through such a complex geometry tends to be broken. We suggest that this is due to the fact that in a flow through a channel of rapidly varying cross section, the deformation, and thus the flow field, is imposed on the fluid, a situation that is encountered almost everywhere in a porous medium. These results make it possible to deduce a general expression for Darcy's law of these fluid types and estimate the parameters appearing in this expression.

• Received 13 May 2014

DOI:https://doi.org/10.1103/PhysRevE.89.063018