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LaTeX Math Inline
body--uriencoded--T_G(t, %7B\bf r%7D)

Along-hole Geothermal Temperature Profile

LaTeX Math Inline
body--uriencoded--G_T(%7B\bf r%7D)

Geothermal Temperature Gradient

Inputs

LaTeX Math Inline
bodyt

Local Calendar Time

LaTeX Math Inline
bodyz(l)

Wellbore trajectory True Vertical Depth Sub-Sea (TVDss)

LaTeX Math Inlinebody--uriencoded--%7B

{\bf

j%7D(%7B\bf r%7D

j}(x,y, z = z_{ref})

Earth's Heat Flux at some reference depth 

LaTeX Math Inline
body--uriencoded--z = z_%7Bref%7D

LaTeX Math Inline
bodyT_s(t, x, y)

Surface temperature based on weather reports

LaTeX Math Inline
body--uriencoded--\lambda_e(%7B\bf r%7D)

Subsurface Thermal Conductivity profile as function of position vector

LaTeX Math Inline
body--uriencoded--a_%7Be%7D(%7B\bf r%7D)

Subsurface Thermal diffusivity profile as function of position vector

where

LaTeX Math Inline
bodyl

Measured Depth of wellbore trajectory with reference to Earth's surface (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
bodyz_s = z(l=0)

TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then 

LaTeX Math Inline
bodyz_s = 0


Assumptions




Equations


LaTeX Math Block
anchorT_z
alignmentleft
\rho_e \, c_e \frac{\partial T_G}{\partial t} + \nabla \left( \lambda \nabla T_G \right) = q({\bf r})



LaTeX Math Block
anchor1
alignmentleft
 T_G(t, x, y, z = z_s) = T_s(t, x, y)



LaTeX Math Block
anchor1
alignmentleft
\Big[ \lambda \nabla T_G \Big]_{z=z_{ref}} = {\bf j}(x,y, z = z_{ref}
(x,y
)



LaTeX Math Block
anchorG_T
alignmentleft
G_T({\bf r}) = \frac{j_z}{\lambda_r({\bf r})}



See Also

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Geology / Geothermal Temperature Field

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