Prior to large earthquakes (e.g. 1995 Kobe earthquake, Japan), an increase in the atmospheric radon concentration is observed, and this increase in the rate follows a power-law of the time-to-earthquake (time-to-failure). This phenomenon corresponds to the increase in the radon migration in crust and the exhalation into atmosphere. An irreversible thermodynamic model including time-scale invariance clarifies that the increases in the pressure of the advecting radon and permeability (hydraulic conductivity) in the crustal rocks are caused by the temporal changes in the power-law of the crustal strain (or cumulative Benioff strain), which is associated with damage evolution such as microcracking or changing porosity. As the result, the radon flux and the atmospheric radon concentration can show a temporal power-law increase. The concentration of atmospheric radon can be used as a proxy for the seismic precursory processes associated with crustal dynamics.