  A theoretical and computational study analysing the initiation of
yield-stress fluids percolation in porous media is presented. Yield-stress
fluid flows through porous media are complicated due to the non-linear
rheological behaviour of this type of fluids, rendering the conventional Darcy
type approach invalid. A critical pressure gradient must be exceeded to
commence the flow of a yield-stress fluid in a porous medium. As the first step
in generalising the Darcy law for yield-stress fluids, a universal scale based
on the variational formulation of the energy equation is derived for the
critical pressure gradient which reduces to purely geometrical feature of the
porous media. The presented scaling is then validated by both exhaustive
numerical simulations (using an adaptive finite element approach based on the
augmented Lagrangian method), and also the previously published data. The
considered porous media is constructed by randomised obstacles with various
topologies; namely, square, circular and alternatively polygonal obstacles
which are mimicked based on Voronoi tessellation of circular cases. Moreover,
computations for the bi-dispersed obstacle cases are performed which further
demonstrate the validity of the proposed universal scaling.
