  We study numerically local models for the mechanical contact between two
solids with rough surfaces. When the solids softly touch either through
adhesion or by a small normal load $L$, contact only forms at isolated patches
and fluids can pass through the interface. When the load surpasses a threshold
value, $L_c$, adjacent patches coalesce at a critical constriction, i.e., near
points where the interfacial separation between the undeformed surfaces forms a
saddle point. This process is continuous without adhesion and the interfacial
separation near percolation is fully defined by scaling factors and the sign of
$L_c-L$. The scaling factors lead to a Reynolds flow resistance which diverges
as $(L_c-L)^\beta$ with $\beta = 3.45$. Contact merging and destruction near
saddle points becomes discontinuous when either short-range adhesion or
specific short-range repulsion are added to the hard-wall repulsion. These
results imply that coalescence and break-up of contact patches can contribute
to Coulomb friction and contact aging.
