The effect of palaeoclimate on heat flow data

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:

$$R=\sum _{i=0}^n\left(\frac{\mathrm{\Delta }{z}_i}{k_i}\right)$$

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

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

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

$$\mathrm{\Delta }T(z,t=0)=\sum _{k=1}^n{T}_k[\mathrm{erfc}\left(\frac{z}{2\sqrt{\kappa t_k}}\right)-\mathrm{erfc}\left(\frac{z}{2\sqrt{\kappa t_{k-1}}}\right)]$$

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.