If I understand you correctly, you have a pipe in a cylindrical hole in a non-flowing hot rock structure with thermal conductivity k and you are passing a fluid at a known temperature, Tf, through the pipe to extract heat from the rock. If this is the case, the derivation is attached (in a much better format). You must know the convective heat coefficient of the fluid in the pipe, h and r1 and r2 represent the positions of the pipe wall and the known rock temperature, respectively. The result is that the pipe wall temperature, Tw, is

 Tw = Tf+k(T2-T1)/(k+h r1 ln (r2/r1))

and the heat flux is

q=hk(T2-Tf)/(k+h r1 ln (r2/r1))

Thank you dear Larry

First of all why did you assume that ground temperature is only function of r? What about depth(z) and time(t)? In most of papers they have used 2D conduction formula in transient mode. I mean Tground=f(r,z,t)

You can also solve a 2-D problem along the same lines as this solution. However, depending on the detail you want in the solution, the 2-D problem does not have an analytical solution. 

I know dear Larry that the problem doesn't have analytical solution but I want to know the whole scheme of solving this kind of borehole. The first step is finding ground temperature distribution. After that the matter is borehole and its wall temperature. My question is how can I calculate it?