We consider the flow of a viscous incompressible fluid in a rigid homogeneous porous
medium provided with mixed boundary conditions. Since the boundary pressure can present
high variations, the permeability of the medium also depends on the pressure, so that the
model is nonlinear. A posteriori estimates allow us to omit this
dependence where the pressure does not vary too much. We perform the numerical analysis of
a spectral element discretization of the simplified model. Finally we propose a strategy
which leads to an automatic identification of the part of the domain where the simplified
model can be used without increasing significantly the error.