thermodynamics - Temperature dependence for specific thermal diffusivity in the diffusion formula

I recently found this answer about the diffusion equation (nice one actually), but have one doubt about the temperature dependence of this formula.

If the "packet" of energy (terminology suggested here) is 6 degrees Celsius (i.e., temperature increase over time from 3 degrees C to 9 degrees Celsius), how much time would take to transport that temperature increase in a distance of 270 m?

I assume that it will take more time for that temperature to reach that depth (compared to a lower temperature). This "packet" of heat will be transported in material of thermal diffusivity of 10^-6 m2/s.

Any hints are appreciated, thanks in advance,

PS. This is not homework, it's part of a paper I'm reading (basically Fig. 1b), here the link: http://www.nature.com/scitable/knowledge/library/methane-hydrates-and-contemporary-climate-change-24314790