The phenomenon of granular capillary rising under vertical vibration provides a novel technical route for hoisting, transporting and collecting granular materials. However, there are still obvious deficiencies in the existing studies of the granular capillary rising behavior, especially the intensive investigation on the effects of gravitational acceleration, horizontal vibration component and particle size distribution are still lacking. To address these problems, the discrete element method is used to numerically simulate the granular capillary rising phenomenon under different operating conditions. The final capillary rising height and average capillary rising velocity of the granular matter are computed and analyzed based on the numerical simulations. The results show that the granular capillarity can also occur under low gravity conditions, and that the final capillary rising height and the average capillary rising velocity first increase and then decrease with the gravitational acceleration. It is also found that the final capillary rising height is insensitive to the variation of horizontal vibrational component, whereas the average capillary rising velocity increases with the augmentation of horizontal vibrational component. Compared with the mono-sized particles, the particles with the same mean size but having a Gaussian size distribution exhibit a maximal capillary rising height at a larger critical tube diameter. Meanwhile, the average capillary rising velocity of the particles having a Gaussian size distribution is faster in the tube diameter range where the granular capillary dynamics for both size distributions is dominated by the jamming effect.