The problem here is to isolate the hole, to find a good contact where air not interfere with heat conduction,  classical methods takes temperature in one side and in the opposite side of the stone, The stone must be isolated , or you must use  a model that takes account the heat loss of the stone measured. There are calorimeters where you can measure thermal properties with small samples. Here the problem is that you must consider the heterogeneity of your stone…… in any case many measures are required to estimate  the thermal properties of the stone…..

You can take a look at the work of Kubicar and coauthors, they monitor thermal conductivity and thermal diffusivity by simple measures

http://www.tpl.fpv.ukf.sk/engl_vers/thermophys/2006/pdf/03.pdf

http://www.tpl.fpv.ukf.sk/engl_vers/thermophys/2005/papers/Kubicar_%20Bagel_Vretenar_Stofanik.pdf

No, to do that you would have to know the terrestrial heat flow, and you would have to assume that the underground tempersture was in everywhere equilibrium with the surface temperature, which is not true in these days of climate change.  There have been some attempts to measure thermal conductivity by means of down-hole heaters, but the technique is not easy.  See my book, Thermal Geophysics, pp 38-40.  In addition, during drilling the temperatue of the rocks immediately surrounding is disturbed by a combination of friction heating and water cooling, so the temperatrure field is highly disturbed.  Depending on the accuacy you need, the conductivity range for different types of rock is known, so could derive estimates from hose data.

respected sir,

thank you very much for your valuable suggestions and informations.

You've already got the right answer. Just two small comments:

- heat flux is not a rock thermal property

- I recommend reading the open-access paper "When rocks get hot: thermal properties of reservoir rocks"

http://www.slb.com/news/inside_news/2013/~/media/Files/resources/oilfield_review/ors12/aut12/2_rocks_hot.ashx

can it is possible to do in laboratory investigation sir,

U of Arizona, ret.

The original question in 2016 by Kumar was about measuring thermal properties of rocks in a drilled hole, i.e., a downhole tool.

The principles needed for such a tool are those of so-called Transient Contact methods -- of which there are many. See the review paper by Gustafsson (2014 at the 32nd International Thermal Conductivity Conference) that features the Hot Disk method and mentions many others. What is possible in principle is that a heat source injects a known energy into a rock wall at some depth for a minute or so and a thermometer records a temperature change (T(t), a few deg C measured every second or so) at a depth different by a known distance at most a few cm. From the magnitude of that change one can calculate the thermal conductivity k. The shape of that change yields an estimate of the diffusivity w.

Several challenges exist. The rock wall is very rough on the scale of a few cm. Therefore the "known energy", "known distance", and "observed temperature rise" are all in doubt. If those challenges are met, estimates of k and w to a percent or better are possible. Then C = k/w can be calculated to a percent or so, where C is the specific heat (at that temperature) per unit volume. Of high interest is that C responds to water in a fractured zone by a substantial increase relative to an intact rock wall.

Finally, there is the challenge that a deep borehole has a substantial temperature rise with depth, which tends to be a limitation to the thermometer accuracy above 100 deg C or so.

The temporary warming of the rock walls due to friction while drilling is not a problem in general. The needed downhole tool probably cannot be used near the time that drilling is done. Even if a method were accepted by the drilling company for an innovative Test While Drilling method, the duration of T(t) is at most minutes while the rock is cooling for hours after the drilling through that depth of the rock.

If the downhole tool is used while the drilling waits, a smoothing average of wall temperatures can give an estimate of the heat flux through the drilled depth.