ABSTRACT We develop an activity model based on ice sublimation and gas diffusion inside cm-sized pebbles making-up a cometary nucleus. Our model explains cometary activity assuming no free parameters and fixing the nucleus surface temperature Ts, its gradient below the nucleus surface at thermal equilibrium, the pressure inside the porous pebbles, and the gas flux from them. We find that (i) the nucleus erosion rate and water vapour flux are independent of the nucleus refractory-to-ice ratio, which affects the dehydration rate only; (ii) water-driven dust ejection occurs in thermal quasi-equilibrium at Ts > 205 K; (iii) the smallest and largest ejected dust sizes depend on the nucleus surface temperature and its gradient at depths of few cm; and (iv) the water-driven nucleus erosion rate is independent of the water vapour flux. Regarding comet 67P/Churyumov–Gerasimenko, we find that (i) during the northern and southern polar summers, the nucleus active areas are ≈5 km2; (ii) >95 per cent of the southern pristine nucleus has a refractory-to-water-ice mass ratio >5; and (iii) the different temperature dependences of the dehydration and erosion rates explain the seasonal cycle: at perihelion, dm-sized chunks ejected by the sublimation of CO2 ices are rapidly enveloped by an insulating crust, preserving most water ice up to their fallout on the northern dust deposits; the inbound water-driven activity at low temperatures triggers a complete erosion of the fallout if its water-ice mass fraction is >0.1 per cent.