One common method to determine heat flow in a well is to use a Bullard plot, which graphs thermal resistance (m$^2$K W$^{-1}$) against temperature. The gradient of this line is heat flow, and the uncertainty determined from the error of linear regression.

Thermal resistance is calculated by:

$$ R = \sum_{i=0}^{n} \left( \frac{\Delta z_i}{k_i} \right) $$

which is the cumulative summation of resistors down the length of a borehole. The uncertainty on $R$ should increase as the individual errors on $k$ accumulate.

## Climate correction

The present-day temperature perturbation, $\Delta T(z, t=0)$, in a semi-infinite solid with an instantaneous change of surface temperature $\Delta T$ at time $t$ before the present is:

$$ \Delta T(z,t=0) = \Delta T \; \mathrm{erfc} \left(\frac{z}{2 \sqrt{\kappa t}} \right) $$

The effect of more than one event, $k_1, k_2, \ldots, k_n$, is found by summation - i.e. if $T(z=0,t) = T*k$ for $t*{k-1} < t < t_k$:

$$
\Delta T(z,t=0) = \sum_{k=1}^{n} T_k \left[ \mathrm{erfc}\left(\frac{z}{2 \sqrt{\kappa t*k}} \right) - \mathrm{erfc}\left(\frac{z}{2 \sqrt{\kappa t*{k-1}}} \right) \right]
$$

This formula can be used if the temperature has remained constant over a period of time in the past. For an application of this, check out our latest paper.