Water is the main component of interstellar ice mantles, is abundant in the solar system and is a crucial ingredient for life. The formation of this molecule in the interstellar medium cannot be explained by gas-phase chemistry only and its surface hydrogenation formation routes at low temperatures (O, O(2), O(3) channels) are still unclear and most likely incomplete. In a previous paper we discussed an unexpected zeroth-order H(2)O production behavior in O(2) ice hydrogenation experiments compared to the first-order H(2)CO and CH(3)OH production behavior found in former studies on hydrogenation of CO ice. In this paper we experimentally investigate in detail how the structure of O(2) ice leads to this rare behavior in reaction order and production yield. In our experiments H atoms are added to a thick O(2) ice under fully controlled conditions, while the changes are followed by means of reflection absorption infrared spectroscopy (RAIRS). The H-atom penetration mechanism is systematically studied by varying the temperature, thickness and structure of the O(2) ice. We conclude that the competition between reaction and diffusion of the H atoms into the O(2) ice explains the unexpected H(2)O and H(2)O(2) formation behavior. In addition, we show that the proposed O(2) hydrogenation scheme is incomplete, suggesting that additional surface reactions should be considered. Indeed, the detection of newly formed O(3) in the ice upon H-atom exposure proves that the O(2) channel is not an isolated route. Furthermore, the addition of H(2) molecules is found not to have a measurable effect on the O(2) reaction channel.