For the typical small-diameter boreholes both arguments in the exponential integral will be also small. Thus, the integral allows approximation Ei(—x) = ln x + 0.577 22. The latter number is the Euler gamma constant. Final expression for Td will be
The heat release Q is proportional to the diameter of borehole and the temperature difference between the drilling fluid and surrounding rock. The typical values range in the interval 10 to 20W/m. In a shallow borehole the temperature of the drilling fluid is close to that on the surface. Additional heating caused by the friction of the drilling bit may slightly increase its value; however, because the temperature in borehole increases with depth, the value of Q may somewhat decrease with depth in case of other conditions being constant.
Figure 19 shows the re-establishment of the undisturbed temperature field (having existed prior to the drilling) calculated for different values of the heat released during the drilling process. The magnitude of the borehole temperature disturbance due to drilling decreases almost linearly during the period compared with the drilling time. At t = th the disturbance falls to approximately 12% of its initial value. For times t > th the attenuation significantly slows down. The above calculations were performed under assumption of continuous drilling; however, suggested method can be further developed for the case of drilling regime interrupted by several breaks (Stulc, 1995). At a certain stage the above approximation may not be valid (Drury, 1984); this happen when the arguments of the exponential integrals (Eq. (2)) are not small enough due to low circulation time and/or large-diameter boreholes.
As shown below, a reliable climate reconstruction using borehole temperature-depth data can be substantially improved by the knowledge of the thermophysical properties
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