Permeation through nanometer channels is important in the design of materials for filtration and separation techniques due to the unusual fundamental behavior arising at the molecular scale. Our molecular dynamics simulation results show that water molecules can be transported much faster than continuum theory prediction through two-dimensional capillaries formed by closely spaced graphene sheets. We attribute this fast transport to low-friction flow of a solid-like monolayer of water.We also carried out molecular dynamics simulations to investigate ions permeation through micrometer-thick laminates prepared by means of vacuum filtration of graphene oxide suspensions. Our numerical simulations confirm that small ions prefer to reside inside graphene capillaries resulting highly concentrated solutions, which named as ion sponging effect. The affinity of salts to graphene capillaries indicates an energy gain with respect to the bulk water, and this translates into a capillary-like pressure that acts on ions within a water medium.