Water self-diffusion is investigated by molecular dynamics (MD) simulation through armchair single walled carbon nanotubes (SWCNTs) implanted in 1-palmytoil-2-oleoyl-sn-glycero-3-Phosphatidylcholine (POPC) membrane patches. Four systems are investigated, each containing one of (5,5), (6,6), (8,8) and (11,11) CNTs with diameters of 6.89, 8.20, 11.04 and 15.02 Å and of varying lengths, oriented normal to the membrane. The CHARMM 27 potential has been employed, in conjunction with TIP3P water, with particle-mesh Ewald electrostatics. Equilibrium and non-equilibrium MD simulations are performed in the respective absence and presence of static electric and time-varying electromagnetic fields, applied both along the axis normal to the membrane and in the plane parallel to the membrane. Single-file diffusion is observed in the (5,5) and (6,6) cases, compared to classical diffusion in the larger pores. From an analysis of the molecular dipole moment distributions, time correlation functions, the number of water molecules present in the CNTs, and the hydrogen bonding characteristics of water inside the CNTs and at their mouth, we qualify the observed non-monotonic trends in the water molecule self-diffusion fluxes.