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
$\begin{array}{l}{\bf r}\end{array}$	Position vector
$\begin{array}{l}z(l)\end{array}$	Wellbore trajectory True Vertical Depth Sub-Sea (TVDss)
$\begin{array}{l}{\bf j}(x,y, z = z_{ref})\end{array}$	Earth's Heat Flux at some reference depth $\begin{array}{l}z = z_{ref}\end{array}$ as function of $\begin{array}{l}(x, \, y)\end{array}$
$\begin{array}{l}q({\bf r})\end{array}$	Volumetric density of heat sources distributed throughout the subsurface rock volume
$\begin{array}{l}T_s(t, x, y)\end{array}$	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)	$\begin{array}{l}\displaystyle \rho_e \, c_e \frac{\partial T_G}{\partial t} + \nabla \left( \lambda_e \nabla T_G \right) = q({\bf r})\end{array}$

(2)	$\begin{array}{l}\displaystyle T_G(t, x, y, z = z_s) = T_s(t, x, y)\end{array}$

(3)	$\begin{array}{l}\displaystyle \Big[ \lambda_e \nabla T_G \Big]_{z=z_{ref}} = {\bf j}(x,y, z = z_{ref})\end{array}$

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

References

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Outputs

Inputs

Assumptions

Equations

See Also

References

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Geothermal Temperature Field @model

Outputs

Inputs

Assumptions

Equations

See Also

References