Synonym: Geothermal Temperature Profile @model = Constant Areal Geothermal Temperature Profile @model
Motivation
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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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along-hole geothermal temperature profile
| --uriencoded--%7B \bf j%7D(x,y,z) = \%7B j_x, \, j_y, \, j_z \%7D |
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and Thermal Conductivity
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True vertical component of Regional Earth's heat flux
(usually
| LaTeX Math Inline |
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| body | --uriencoded--j_z = 25 \div 55 \ mW/m%5e2 |
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)| --uriencoded--\lambda_e(%7B\bf r%7D) |
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to be homogeneous across location area:
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| {\bf j}({\bf r}) ={\bf j}(x,y,z)={ \bf j}(z) |
| | LaTeX Math Block |
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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) \} |
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 a Geothermal Temperature Field
| LaTeX Math Inline |
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| body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B 0, \, 0 , \, j_z \%7D |
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to modelling of a Constant Areal Geothermal Temperature Profile along a given wellbore trajectory.Outputs
Inputs
| 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 |
| LaTeX Math Inline |
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| body | --uriencoded--a_%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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| | LaTeX Math Inline |
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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
| LaTeX Math Inline |
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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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| | LaTeX Math Inline |
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| body | \lambda_e(x,y,z) = \lambda_e(z) |
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Equations
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| T_G(t, z) = T_s + \int_{z_s}^z G_T(z) dz + T_Y(t, z) + T_D(t, z) |
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| G_T(z) = \frac{j_z}{\lambda_e(z)} |
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| LaTeX Math Block |
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| T_Y(t,z) = \delta T_A \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_{en} \, A_T}} \, \right] \, \cos \left |
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Assumptions
Equations
| Below Neutral Temperature Layer | Above Neutral Temperature Layer |
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T_g(x,y,z) = T_{ref}(x,y) + \int_{z_{ref}}^z G_T(x,y,z) dz |
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T(t, z) = T_{srf} + \frac{j_z}{\lambda_e} (z-z_{srf}) + T_A \, \exp \bigg[ \, {(z_{srf}-z}) \sqrt{\frac{\pi}{a_e \, \delta_T}} \, \bigg] \, \cos \bigg[ \, 2 \pi \frac{t - \delta t_ |
| {min} | \delta | {srf}s -z) \sqrt {\frac{\pi}{a_ |
| e | \delta | bigg | | G | T |
G_T(z) =\frac{d T_g}{d z}= \frac{j_z}{\lambda_e} |
where
где
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TVDss of the Earth's surface
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Annual average surface temperature based on weather reports
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Annual average surface temperature variation based on weather reports
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Temperature variation cycle (usually
| LaTeX Math Inline |
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| body | --uriencoded--\delta t = 1 \, %7B\rm year%7D |
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Temperature variation cycle shift (time moment of minimal temperature with respect to astronomic midnight 0:00)
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| LaTeX Math Inline |
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| body | a_e = \frac{\lambda_e}{\rho_e \, c_e} |
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Rock specific volumetric heat capacity at constant pressure
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] |
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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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See Also
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Geology / Geothermal Temperature Field
See Also
Geology / 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.
GeothermalTemperatureProfile.xlsx
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