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  

{\bf j}({\bf r}) ={\bf j}(x,y,z)={ \bf j}(z)

\lambda_e({\bf r}) =\lambda_e(z)

where 

Since the  heat flux is conservative (see

{\bf j}=\{ j_x = {\rm const}, \, j_y = {\rm const} , \, j_z(z) \}

Further admitting that a surface temperature over the study area is constant: 

{ \bf j}(x,y,z) = \

...

{ j_x = 0, \, j_y = 0 , \, j_z(z) \

...

}

.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 

This simplifies the procedure of modelling the a Geothermal Temperature Field 

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Equations

T_G(t, z) = T_s + \int_{z_s}^z G_T(z) dz + T_Y(t, z) + T_D(t, z)

G_T(z) = \frac{j_z}{\lambda_re(z)}

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]

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]

z_n = z_s + H_n

H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }

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Geology / Geothermal Temperature Field / Geothermal Temperature Profile

[ Geothermal Temperature Field @model ] [ Geothermal Temperature Gradient ]

[ Neutral Temperature Layer @model ]

References

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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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