@wikipedia
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
Inputs
| \delta T_A | Annual average surface temperature variation based on weather reports | Period of annual temperature variation cycle: A_T = 1 \, rm year%7D| LaTeX Math Inline |
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body | --uriencoded--%7B\%7B\bf r%7D\delta t_A | Time shift of annual highest temperature with respect to January 1| --uriencoded--z = z_%7Bref%7D |
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as function of |
T_s | Local annual average surface temperature based on weather reports | | Daily average surface temperature variation based on weather reports | a_%7Ben%7DLocal average Thermal diffusivity of the soil between Earth's surface and NTLD_T | Period of daily temperature variation cycle: | Surface temperature based on weather reports |
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A_D = 1 \, %7Brm day%7D| LaTeX Math Inline |
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body | --uriencoded--\\delta t_D | Time shift of daily highest temperature with respect to Midnight 00:00 | | LaTeX Math Inline |
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body | \delta T_%7B\rm cut%7DTemperature 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
Equations
T_G(t, {\bf r}) = T_{GS}({\bf r}) + T_Y(t, z) + T_D(t, z) |
| LaTeX Math Block |
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G_T({\bf r}) = \frac{j_z}{\lambda_r({\bf r})} |
| LaTeX Math Block |
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\nabla T_{GS} = \lambda^{-1}\rho_e \, c_e \frac{\partial T_G}{\partial t} + \nabla \left( \lambda_e \nabla T_G \right) = q({\bf r}) |
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\cdot {\bf j}rot_j | alignment | left| LaTeX Math Block |
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anchor | \nabla \times {\bf j} = 0 |
{GS}(G(t, x, y, z = z_s) = T_s |
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| LaTeX Math Block |
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T_Yz) = \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_zT_D(t,z) = \delta\Big[ \lambda_e \nabla T_ |
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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]Big]_{z=z_{ref}} = {\bf j}(x,y, z = z_{ref}) |
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| LaTeX Math Block |
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| G_T({\bf r}) = \frac{j_z({\bf r})}{\lambda_e({\bf r}) |
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| Neutral Layer |
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| LaTeX Math Block |
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z_n = z_s + H_n |
| LaTeX Math Block |
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H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut}
See Also
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Geology / Geothermal Temperature Field
[ Constant Areal Geothermal Temperature Profile @model ] [ Geothermal Temperature Gradient ]
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