Synonym: Geothermal Temperature Profile @model = Constant Areal Geothermal Temperature Profile @model

Motivation

In many subsurface applications which require the knowledge of subsurface temperature distributions the land area of the study is small enough to consider the subsurface heat flux $\begin{array}{l}{ \bf j}(x,y,z) = \{ j_x, \, j_y, \, j_z \}\end{array}$ and Thermal Conductivity $\begin{array}{l}\lambda_e({\bf r})\end{array}$ to be homogeneous across location area:

(1)	$\begin{array}{l}\displaystyle {\bf j}({\bf r}) ={\bf j}(x,y,z)={ \bf j}(z)\end{array}$

(2)	$\begin{array}{l}\displaystyle \lambda_e({\bf r}) =\lambda_e(z)\end{array}$

where $\begin{array}{l}z\end{array}$ is true vertical depth.

Since the heat flux is conservative (see

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) then it immediately implies that:

(3)	$\begin{array}{l}\displaystyle {\bf j}=\{ j_x = {\rm const}, \, j_y = {\rm const} , \, j_z(z) \}\end{array}$

Further admitting that a surface temperature over the study area is constant: $\begin{array}{l}T_s(x,y) = \rm const\end{array}$ one can see that lateral components of the heat flux are vanishing:

(4)	$\begin{array}{l}\displaystyle { \bf j}(x,y,z) = \{ j_x = 0, \, j_y = 0 , \, j_z(z) \}\end{array}$

Normally there are no heat sources within a subsurface volume under study other than upward Earth's Heat Flux which means that true vertical component $\begin{array}{l}j_z(z) = j_z = \rm const\end{array}$ is constant along true vertical direction. It varies across the Earth but local value is usually well known.

This simplifies the procedure of modelling a Geothermal Temperature Field $\begin{array}{l}{ \bf j}(x,y,z) = \{ 0, \, 0 , \, j_z \}\end{array}$ to modelling of a Constant Areal Geothermal Temperature Profile along a given wellbore trajectory.

Outputs

$\begin{array}{l}T_G(t, l)\end{array}$	Along-hole Geothermal Temperature Profile
$\begin{array}{l}G_T(z)\end{array}$	Geothermal Temperature Gradient
$\begin{array}{l}H_n\end{array}$	Neutral Temperature Layer (NTL)

Inputs

$\begin{array}{l}t\end{array}$	Local Calendar Time	$\begin{array}{l}\delta T_A\end{array}$	Annual average surface temperature variation based on weather reports
$\begin{array}{l}z(l)\end{array}$	Wellbore trajectory True Vertical Depth Sub-Sea (TVDss)	$\begin{array}{l}A_T\end{array}$	Period of annual temperature variation cycle: $\begin{array}{l}A_T = 1 \, {\rm year}\end{array}$
$\begin{array}{l}j_z\end{array}$	True vertical component of regional Earth's Heat Flux	$\begin{array}{l}\delta t_A\end{array}$	Time shift of annual highest temperature with respect to January 1
$\begin{array}{l}T_s\end{array}$	Local annual average surface temperature based on weather reports	$\begin{array}{l}\delta T_D\end{array}$	Daily average surface temperature variation based on weather reports
$\begin{array}{l}a_{en}\end{array}$	Local average Thermal diffusivity of the soil between Earth's surface and NTL	$\begin{array}{l}D_T\end{array}$	Period of daily temperature variation cycle: $\begin{array}{l}A_D = 1 \, {\rm day}\end{array}$
$\begin{array}{l}\lambda_e(z)\end{array}$	Subsurface Thermal Conductivity profile as function of TVDss	$\begin{array}{l}\delta t_D\end{array}$	Time shift of daily highest temperature with respect to Midnight 00:00
		$\begin{array}{l}\delta T_{\rm cut}\end{array}$	Temperature measurement threshold (usually $\begin{array}{l}\delta T_{\rm cut} = 0.01 \, {\rm °C}\end{array}$ )

where

$\begin{array}{l}l\end{array}$	Measured Depth of wellbore trajectory with reference to Earth's surface ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}z_s = z(l=0)\end{array}$	TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then $\begin{array}{l}z_s = 0\end{array}$

Assumptions

$\begin{array}{l}{ \bf j}(x,y,z) = \{ 0, \, 0 , \, j_z = {\rm const} \}\end{array}$	$\begin{array}{l}\lambda_e(x,y,z) = \lambda_e(z)\end{array}$

Equations

(5)	$\begin{array}{l}\displaystyle T_G(t, z) = T_s + \int_{z_s}^z G_T(z) dz + T_Y(t, z) + T_D(t, z)\end{array}$

(6)	$\begin{array}{l}\displaystyle G_T(z) = \frac{j_z}{\lambda_e(z)}\end{array}$

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

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

Neutral Layer

(9)	$\begin{array}{l}\displaystyle z_n = z_s + H_n\end{array}$

(10)	$\begin{array}{l}\displaystyle H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }\end{array}$

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.

GeothermalTemperatureProfile.xlsx

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