A method is reported to fabricate silicon-glass nanofluidic chips with non-uniform channel depths in the range 20-500 nm and micrometer resolution in width. The process is based on grayscale laser lithography to structure photoresist in 2.5 dimensions in a single step, followed by a reactive ion etching to transfer the resist depth profile into silicon. It can be easily integrated in a complete process flow chart. The method is used to fabricate a network of interconnected slits of non-uniform depth, a geometry mimicking a nanoporous medium. The network is then used to perform a pressure step controlled drainage experiment, i.e. the immiscible displacement of a wetting fluid (liquid water) by a non-wetting one (nitrogen). The drainage patterns are analyzed by comparison with simulations based on the invasion percolation algorithm. The results indicate that slow drainage in the considered nanofluidic system well corresponds to the classical capillary fingering regime.