  Context. The diffusion of volatile species on amorphous solid water ice
affects the chemistry on dust grains in the interstellar medium as well as the
trapping of gases enriching planetary atmospheres or present in cometary
material.
  Aims. The aim of the work is to provide diffusion coefficients of CH$_4$ on
amorphous solid water (ASW), and to understand how they are affected by the ASW
structure.
  Methods. Ice mixtures of H$_2$O and CH$_4$ were grown in different conditions
and the sublimation of CH$_4$ was monitored via infrared spectroscopy or via
the mass loss of a cryogenic quartz crystal microbalance. Diffusion
coefficients were obtained from the experimental data assuming the systems obey
Fick's law of diffusion. Monte Carlo simulations modeled the different
amorphous solid water ice structures investigated and were used to reproduce
and interpret the experimental results.
  Results. Diffusion coefficients of methane on amorphous solid water have been
measured to be between 10$^{-12}$ and 10$^{-13}$ cm$^2$ s$^{-1}$ for
temperatures ranging between 42 K and 60 K. We showed that diffusion can differ
by one order of magnitude depending on the morphology of amorphous solid water.
The porosity within water ice, and the network created by pore coalescence,
enhance the diffusion of species within the pores.The diffusion rates derived
experimentally cannot be used in our Monte Carlo simulations to reproduce the
measurements.
  Conclusions. We conclude that Fick's law can be used to describe diffusion at
the macroscopic scale, while Monte Carlo simulations describe the microscopic
scale where trapping of species in the ices (and their movement) is considered.
