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
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body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B j_x, \, j_y, \, j_z \%7D |
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to be independent on areal location: and Thermal Conductivity %7B bf j%7D(x,y,z) = j%7D(z. Further admitting that lateral inhomogeneity with the study area is not high the Thermal Conductivity is going to be a function of true vertical depth only LaTeX Math Inline |
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body | \lambda_rto be homogeneous across location area: LaTeX Math Block |
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| {\bf j}({\bf r}) ={\bf j}(x,y,z)= |
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| \lambda_e({\bf r}) =\lambda_e(z) |
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where
is true vertical depth.Since the heat flux is conservative (see LaTeX Math Block Reference |
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anchor | rot_j |
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page | Geothermal Temperature Field @model |
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) then it immediately implies that:
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{\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:
one can see that lateral components of the heat flux are vanishing: LaTeX Math Block |
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{ \bf j}(x,y,z) = \{ j_x = 0, \, j_y = 0 , \, j_z(z) \} |
which leads to vanishing lateral components of the heat flux:
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body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B j_x = 0, \, j_y = 0 , \, j_z(z) \%7D |
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.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 LaTeX Math Inline |
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body | j_z(z) = j_z = \rm const |
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is constant along
true vertical direction. It varies across the
Earth but local value is usually well known.This simplifies the procedure of modelling the a Geothermal Temperature Field
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body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B 0, \, 0 , \, j_z \%7D |
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along to modelling of a Constant Areal Geothermal Temperature Profile along a given
wellbore trajectory.
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| Local Calendar Time | | Annual average surface temperature variation based on weather reports |
| | | Period of annual temperature variation cycle: LaTeX Math Inline |
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body | --uriencoded--A_T = 1 \, %7B\rm year%7D |
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| True vertical component of regional Earth's Heat Flux | | Time shift of annual highest temperature with respect to January 1 |
| Local annual average surface temperature based on weather reports | | Daily average surface temperature variation based on weather reports |
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body | --uriencoded--a_e%7Ben%7D |
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| Local average Thermal diffusivity of the soil between Earth's surface and NTL | | Period of daily temperature variation cycle: LaTeX Math Inline |
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body | --uriencoded--A_D = 1 \, %7B\rm day%7D |
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| Subsurface Thermal Conductivity profile as function of TVDss | | Time shift of daily highest temperature with respect to Midnight 00:00 |
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body | --uriencoded--\delta T_%7B\rm cut%7D |
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| Temperature measurement threshold (usually LaTeX Math Inline |
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body | --uriencoded--\delta T_%7B\rm cut%7D = 0.01 \, %7B\rm °C%7D |
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where
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| TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then |
Assumptions
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body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B 0, \, 0 , \, j_z = %7B\rm const%7D \%7D |
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body | \lambda_re(x,y,z) = \lambda_re(z) |
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Equations
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body | z < z_n LaTeX Math Block |
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| z_nT_G(t, z) = zT_s + H_n |
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| H_n = \sqrt{\frac{a_e \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} } |
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Below Neutral Temperature Layer | Above Neutral Temperature Layer |
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\int_{z_s}^z G_T(z) dz + T_Y(t, z) + T_D(t, z) |
| | _g | _G | l | T_s + \int_{z_s}^z G_T(z) dz |
| | 0 + \frac{j_z}{\lambda_r(z)} (z-z_s) + T_Y(t, z) + T_D(t, z) LaTeX Math Block |
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G_T(z) =\frac{d T_g}{d z}= \frac{j_z}{\lambda_r} |
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T_Y(t,z) = \delta T_A \, \exp \left[ \, {(z_s-z}) \sqrtA \, \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_ |
| e | LaTeX Math Block |
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| T_D(t,z) = \delta T_D |
| \, \cos \left[ | 2 | pi | frac{t - | delta | t_A} | A_T} + | | | e | A | LaTeX Math Block |
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T_D(t,z) = \delta T_D \, \exp \left[ \, {(z_s-z}) \sqrt \, \cos \left[ \, 2 \pi \frac{t - \delta t_D}{D_T} + (z_s -z) \sqrt {\frac{\pi}{a_ |
| e | \, \cos \left[ \, 2 \pi \frac{t - \delta t_D}{D_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, D_T}} \, \right]
where
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Neutral Layer |
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LaTeX Math Block |
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| z_n = z_s + H_n |
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| H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} } |
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| TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then LaTeX Math Inline |
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body | z_s = 0
See Also
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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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