When considering transport phenomena in thin porous media, the questions arise as to whether thin porous media must be considered as a distinct class of porous media and when a porous medium can be defined as thin. A general answer to these questions is not obvious since the answer can depend on the particular transport phenomenon considered. Hence here we consider a specific example motivated by the study of two phase flows in some porous layers of proton exchange membrane fuel cells (PEMFC). This example is the quasi-static water invasion in a porous layer with multiple independent injection points at the inlet. We study the statistics of breakthrough points and show that that the behaviour of sufficiently thin porous media is distinct from the one of thicker porous media. For example, in a two-dimensional hydrophobic system, the average number of breakthrough points varies as /L/^-1 for a sufficiently thick porous medium, where /L/ is the thickness of the porous layer. The evolution is different for a thin a porous medium and slower than the trivial scaling /L/^-1 . This leads to identify a “critical” thickness, below which the system can be considered as thin. The effect of trapping and the influence of wettability will be also addressed.