By knowing the inside hole temperature and surface temp. and area of the drill , can we calculate the thermal conductivity of that particular temperature field (while drilling in laboratory)?
Are you more interested in the temperature of the rock or the drill? Be careful not to make assumptions about one from the other. There is work that demonstrates the existence of a thermal contact resistance at the asperity level which may cause very different thermal effects on the tool or the workpiece. If you are looking at developing a thermal model of the drilling of rock it would be wise to consider the entire heat partition associated within the system. Alternatively, if you only need to know the temperature of the rock itself during cutting, this may be more achievable in the short term.