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Outputs

LaTeX Math Inline
bodyT_e(t, l)

LaTeX Math Inline
bodyG_T(z)

Geothermal Temperature Gradient

LaTeX Math Inline
bodyH_Nn

Neutral Temperature Layer (NTL)

Inputs

LaTeX Math Inline
bodyt

Local Calendar Time

LaTeX Math Inline
body\delta T_A

Annual average surface temperature variation based on weather reports

LaTeX Math Inline
bodyz(l)

LaTeX Math Inline
bodyA_T

Period of annual temperature variation cycle:

LaTeX Math Inline
body--uriencoded--A_T = 1 \, %7B\rm year%7D

LaTeX Math Inline
bodyj_z

True vertical component of regional Earth's Heat Flux

LaTeX Math Inline
body\delta t_A

Time shift of annual highest temperature with respect to January 1

LaTeX Math Inline
bodyT_

s

0

Local annual average surface temperature based on weather reports

LaTeX Math Inline
body\delta T_D

Daily average surface temperature variation based on weather reports

LaTeX Math Inline
bodya_e

Local average Thermal diffusivity of the soil between Earth's surface and NTL

LaTeX Math Inline
bodyD_T

Period of daily temperature variation cycle:

LaTeX Math Inline
body--uriencoded--A_D = 1 \, %7B\rm day%7D

LaTeX Math Inline
body\lambda_r(z)

Subsurface Thermal Conductivity profile as function of TVDss

LaTeX Math Inline
body\delta t_D

Time shift of daily highest temperature with respect to Midnight 00:00



LaTeX Math Inline
body--uriencoded--\delta T_%7B\rm cut%7D

Temperature measurement threshold (usually

LaTeX Math Inline
body--uriencoded--\delta T_%7B\rm cut%7D = 0.

001

01 \, %7B\rm °C%7D

where

LaTeX Math Inline
bodyl

wellbore trajectory Measured Depth with reference to Earth's surface (

LaTeX Math Inline
bodyl=0
)


Assumptions

LaTeX Math Inline
body--uriencoded--%7B \bf j%7D(x,y,z) = \%7B 0, \, 0 , \, j_z = %7B\rm const%7D \%7D

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Neutral Layer


LaTeX Math Block
anchorz_N
alignmentleft
z_Nn = z_s + H_Nn



LaTeX Math Block
anchorH_N
alignmentleft
H_Nn = \sqrt{\frac{a_e \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }


Below Neutral Below Neutral Temperature LayerAbove Above Neutral Temperature Layer

LaTeX Math Inline
bodyz > z_Nn

LaTeX Math Inline
bodyz < z_Nn


LaTeX Math Block
anchorT_g
alignmentleft
T_g(l) = T_s + \int_{z_s}^z G_T(z) dz



LaTeX Math Block
anchorT_z
alignmentleft
T(t, z) = T_s0 + \frac{j_z}{\lambda_e} (z-z_s) + T_Y(t, z) + T_D(t, z)



LaTeX Math Block
anchorG_T
alignmentleft
G_T(z) =\frac{d T_g}{d z}= \frac{j_z}{\lambda_r}



LaTeX Math Block
anchorT_z
alignmentleft
T_A(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]




LaTeX Math Block
anchorT_z
alignmentleft
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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