Motivation

Outputs

$\begin{array}{l}T_e(t, l)\end{array}$	along-hole geothermal temperature profile
$\begin{array}{l}G_T(z)\end{array}$	temperature gradient
$\begin{array}{l}H_N\end{array}$	neutral layer

Inputs

$\begin{array}{l}t\end{array}$	Astronomic time	$\begin{array}{l}\delta T_A\end{array}$	Annual average surface temperature variation based on weather reports
$\begin{array}{l}l\end{array}$	wellbore trajectory Measured Depth with reference to Earth's surface ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}A_T\end{array}$	Period of annual temperature variation cycle: $\begin{array}{l}A_T = 1 \, {\rm year}\end{array}$
$\begin{array}{l}z(l)\end{array}$	Wellbore trajectory. True Vertical Depth Sub-Sea (TVDss)	$\begin{array}{l}\delta t_A\end{array}$	Time moment of annual highest temperature with respect to January 1
$\begin{array}{l}j_z\end{array}$	True vertical component of regional Earth's heat flux (usually $\begin{array}{l}j_z = 25 \div 55 \ mW/m^2\end{array}$ )	$\begin{array}{l}\delta T_D\end{array}$	Daily average surface temperature variation based on weather reports
$\begin{array}{l}T_s\end{array}$	Annual average surface temperature based on weather reports	$\begin{array}{l}D_T\end{array}$	Period of daily temperature variation cycle: $\begin{array}{l}A_D = 1 \, {\rm day}\end{array}$
$\begin{array}{l}a_e\end{array}$	Average Thermal diffusivity of the soil between Earth's surface and neutral layer	$\begin{array}{l}\delta t_D\end{array}$	Time moment of daily highest temperature with respect to Midnight 00:00
$\begin{array}{l}\lambda_r(l)\end{array}$	Rock Thermal Conductivity along-hole profile	$\begin{array}{l}\delta T_{\rm cut}\end{array}$	Temperature measurement threshold (usually $\begin{array}{l}\delta T_{\rm cut} = 0.001 \, {\rm °C}\end{array}$ )

Assumptions

Equations

Neutral Layer

(1)	$\begin{array}{l}\displaystyle z_N = z_s + H_N\end{array}$

(2)	$\begin{array}{l}\displaystyle H_N = \sqrt{\frac{a_e \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }\end{array}$

Below Neutral Temperature Layer

Above Neutral Temperature Layer

$\begin{array}{l}z > z_N\end{array}$

$\begin{array}{l}z < z_N\end{array}$

(3)	$\begin{array}{l}\displaystyle T_g(l) = T_n + \int_{z_n}^z G_T(z) dz\end{array}$

(4)	$\begin{array}{l}\displaystyle T(t, z) = T_s + \frac{j_z}{\lambda_e} (z-z_s) + T_Y(t, z) + T_D(t, z)\end{array}$

(5)	$\begin{array}{l}\displaystyle G_T(z) =\frac{d T_g}{d z}= \frac{j_z}{\lambda_r}\end{array}$

(6)	$\begin{array}{l}\displaystyle T_A(t,z) = \delta T_A \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_e \, A_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_A}{A_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, A_T}} \, \right]\end{array}$

(7)	$\begin{array}{l}\displaystyle T_D(t,z) = \delta T_D \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_e \, D_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_D}{D_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, D_T}} \, \right]\end{array}$

where

$\begin{array}{l}z_s = z(l=0)\end{array}$

TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then $\begin{array}{l}z_s = 0\end{array}$

References

Kasuda, T., and Archenbach, P.R. "Earth Temperature and Thermal Diffusivity at Selected Stations in the United States", ASHRAE Transactions, Vol. 71, Part 1, 1965.

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Motivation

Outputs

Inputs

Assumptions

Equations

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Geothermal Temperature Profile @model

Motivation

Outputs

Inputs

Assumptions

Equations

See Also

References