Probability current and probability density of wave packets was calculated by solving the three dimensional time-dependent Schrödinger equation for a local potential model of the scanning tunneling microscope (STM) tip - graphene system. Geometrical and electronic structure effects of the three dimensional tunneling process are identified by studying three models of increasing complexity: a jellium half space, a narrow jellium sheet, and a local one electron pseudopotential. It was found that some of the key characteristics of the STM tip - graphene tunneling process are already present at the simple jellium models. In the STM tip - jellium half space system the direction of the momentum does not change during the tunneling event, hence this setup is characterised by introducing an effective distance. For the STM tip - narrow jellium sheet system the direction of the momentum is changed from vertical to horizontal during the tunneling event. The wave packet preferentially tunnels into the bound state of the jellium sheet. For the atomistic model of the graphene sheet an anisotropic spreading of the wave packet was found for hot electrons. This may open new opportunities to build carbon based nanoelectronic devices.