  Using the gradient discretisation method (GDM), we provide a complete and
unified numerical analysis for non-linear variational inequalities (VIs) based
on Leray--Lions operators and subject to non-homogeneous Dirichlet and
Signorini boundary conditions. This analysis is proved to be easily extended to
the obstacle and Bulkley models, which can be formulated as non-linear VIs. It
also enables us to establish convergence results for many conforming and
nonconforming numerical schemes included in the GDM, and not previously studied
for these models. Our theoretical results are applied to the hybrid mimetic
mixed method (HMM), a family of schemes that fit into the GDM. Numerical
results are provided for HMM on the seepage model, and demonstrate that, even
on distorted meshes, this method provides accurate results.
