  We perform direct numerical simulations of the flow through a model of a
deformable porous medium. Our model is a two-dimensional hexagonal lattice,
with defects, of soft elastic cylindrical pillars, with elastic shear modulus
$G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a
viscous fluid and the solid phase is modeled as an incompressible viscoelastic
material, whose complete nonlinear structural response is considered. We
observe that the Darcy flux ($q$) is a nonlinear function -- steeper than
linear -- of the pressure-difference ($\Delta P$) across the medium.
Furthermore, the flux is larger for a softer medium (smaller $G$). We construct
a theory of this super-linear behavior by modelling the channels between the
solid cylinders as elastic channels whose walls are made of a material with a
linear constitutive relation but can undergo large deformation. Our theory
further predicts that the flow permeability is a universal function of $\Delta
P/G$, which is confirmed by the present simulations.
