Nanoporous graphene has emerged as an excellent material for desalination and water purification. Holey graphene (HG) is a special form of nanoporous graphene, where multilayers of nanoporous graphene get arranged in spatially separated stacks. In this paper, we employ molecular dynamics simulations to unravel the dynamics of a water drop in presence of an applied force F in such holey graphene architecture, which is characterized by the presence of either hydrophilic functionalization (HIF) or hydrophobic functionalization (HOF) of the edges of the holes. For realistic values of F, the consideration of water drop makes the capillary effects important, which in turn interplays with the wettability of the surface functionalization to ensure that the HG with the HOF causes both an enhanced flux and an enhanced permeated water volume. We relate these phenomena to the augmented water-hydrophilic-edge attraction that arrests the dewetting of water from the graphene stacks and slows the speed of water flowing past the graphene edges. Finally, we discover a time interval when a quasi-steady flux of water comes out of the HG for either types of functionalization and therefore attempts a Darcy’s law-like description of the water flux only to witness a capillarity-induced breakdown of Darcy’s law with the flux being proportional to F^α, where α_HOF > α_HIF > 1.