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

Motivation

In some specific subsurface applications which require the knowledge of subsurface temperature distributions the assumption of the Constant Areal Geothermal Temperature Profile is not valid and the problem requires a proper 3D modelling solution.

Outputs

$\begin{array}{l}T_G(t, {\bf r})\end{array}$	Along-hole Geothermal Temperature Profile
$\begin{array}{l}G_T({\bf r})\end{array}$	Geothermal Temperature Gradient

Inputs

$\begin{array}{l}t\end{array}$	Local Calendar Time
$\begin{array}{l}z(l)\end{array}$	Wellbore trajectory True Vertical Depth Sub-Sea (TVDss)
$\begin{array}{l}{\bf j}({\bf r})\end{array}$	Earth's Heat Flux
$\begin{array}{l}T_s(t, x, y)\end{array}$	Surface temperature based on weather reports
$\begin{array}{l}\lambda_e({\bf r})\end{array}$	Subsurface Thermal Conductivity profile as function of position vector
$\begin{array}{l}a_{e}({\bf r})\end{array}$	Subsurface Thermal diffusivity profile as function of position vector

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

Equations

(1)	$\begin{array}{l}\displaystyle T_G(t, {\bf r}) = T_{GS}({\bf r}) + T_Y(t, z) + T_D(t, z)\end{array}$

(2)	$\begin{array}{l}\displaystyle G_T({\bf r}) = \frac{j_z}{\lambda_r({\bf r})}\end{array}$

(3)	$\begin{array}{l}\displaystyle \nabla T_{GS} = \lambda^{-1}({\bf r}) \cdot {\bf j}\end{array}$

(4)	$\begin{array}{l}\displaystyle \nabla \times {\bf j} = 0\end{array}$

(5)	$\begin{array}{l}\displaystyle T_{GS}(x, y, z = z_s) = T_s\end{array}$

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

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

Neutral Layer

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

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

Page tree

Outputs

Inputs

Assumptions

Equations

See Also

References