Effective conductivity of inertial flows through porous media

We study two-dimensional incompressible inertial flows through porous media. At core (small) scale, we prove that the constitutive, nonlinear model can be rewritten into a linear one by means of a new parameter K★ which encompasses all the inertial effects. In natural (large-scale) formations, K★ is erratically changing, and we analytically compute its counterpart, which is coined generalized effective conductivity, by the self-consistent approach (SCA). In spite of its approximate nature, the SCA leads to simple results that are in good agreement with Monte Carlo simulations.