Fluid channels with inclined solid walls (e.g. cone- and slit-shaped pores) have wide and promising applications in micro- and nano-engineering and science. In this paper, we use molecular dynamics (MD) simulations to investigate the mechanisms of water infiltration (adsorption) into cone-shaped nanopores made of a hydrophobic graphene sheet. When the apex angle is relatively small, an external pressure is required to initiate infiltration and the pressure should keep increasing in order to further advance the water front inside the nanopore. By enlarging the apex angle, the pressure required for sustaining infiltration can be effectively lowered. When the apex angle is sufficiently large, under ambient condition water can spontaneously infiltrate to a certain depth of the nanopore, after which an external pressure is still required to infiltrate more water molecules. The unusual involvement of both spontaneous and pressure-assisted infiltration mechanisms in the case of blunt nanocones, as well as other unique nanofluid characteristics, is explained by the Young's relation enriched with the size effects of surface tension and contact angle in the nanoscale confinement.