The heat flow through a snow cover profoundly changes its structure and thus the thermophysical and mechanical properties of snow. Despite the great amount of empirical data collected so far, the physical processes behind these changes are poorly understood because snow sinters and recrystallizes simultaneously at temperatures typical on earth. Heat conduction in snow takes place as (i) conduction in the ice matrix and pore space, (ii) heat transport by water vapor diffusion, and (iii) latent heat release and gain due to phase changes. X-ray computed micro-tomography allows detailed representations of the 3D-microstructure of snow and its evolution during metamorphism without disturbing the snow sample. The reconstructed microstructure of snow was used to simulate the heat conduction in the ice matrix of snow and its evolution during controlled temperature gradient metamorphism. The numerical results, compared with the measured effective heat conductivites of the snow samples, suggest the relative contribution of the components of the heat flux through snow to be considerably different from what is assumed nowadays.