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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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| 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: | LaTeX Math Inline |
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| body | --uriencoded--%7B \bf j%7D(x,y,z) =%7B \bf j%7D(z) |
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. Further admitting that lateral inhomogeneity with the study area is not high the Thermal Conductivity is going to be a function of depth only Thermal Conductivity which leads to vanishing lateral components of the heat flux and vanishing lateral components: | LaTeX Math Inline |
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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
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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 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 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 |
| 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
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(x,y,z) = \lambda(z) |
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Equations
| Neutral Layer |
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| z_n = z_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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| T_G(l) = T_s + \int_{z_s}^z G_T(z) dz |
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| T(t, z) = T_0 + \frac{j_z}{\lambda_e} (z-z_s) + T_Y(t, z) + T_D(t, z) |
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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}) \sqrt{\frac{\pi}{a_e \, A_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_A}{A_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, A_T}} \, \right] |
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| T_D(t,z) = \delta T_D \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_e \, D_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_D}{D_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, D_T}} \, \right] |
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