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SynonymGeothermal 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  

LaTeX Math Inline
body--uriencoded--%7B \bf j%7D(x,y,z) = \%7B j_x, \, j_y, \, j_z \%7D
 to be independent on areal location:   and Thermal Conductivity 
LaTeX Math Inline
body--uriencoded--
%7B
\
bf j%7D(x,y,z) =
lambda_e(%7B\bf
j%7D(z) and vanishing lateral components: 
LaTeX Math Inline
body--uriencoded--%7B \bf j%7D(x,y,z) = \%7B j_x = 0, \, j_y = 0 , \, j_z(z) \%7D
.

Normally there are no heat sources within a subsurface volume under study other than Earth's Heat Flux which means that true vertical component 

LaTeX Math Inline
bodyj_z(z) = j_z = \rm const
 is constant along true vertical direction.

r%7D)
 to be homogeneous across location area: 

LaTeX Math Block
anchor1
alignmentleft
{\bf j}({\bf r}) ={\bf j}(x,y,z)={ \bf j}(z)
LaTeX Math Block
anchor1
alignmentleft
\lambda_e({\bf r}) =\lambda_e(z)

where 

LaTeX Math Inline
bodyz
 is true vertical depth.

Since the  heat flux is conservative (see

LaTeX Math Block Reference
anchorrot_j
pageGeothermal Temperature Field @model
) then it immediately implies that:

LaTeX Math Block
anchor1
alignmentleft
{\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: 

LaTeX Math Inline
bodyT_s(x,y) = \rm const
 one can see that lateral components of the heat flux are vanishing: 

LaTeX Math Block
anchorWS78A
alignmentleft
{ \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
bodyj_z(z) = j_z = \rm const
 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
body
This simplifies the procedure of modelling the Geothermal Temperature Field 
LaTeX Math Inline
body--uriencoded--%7B \bf j%7D(x,y,z) = \%7B 0, \, 0 , \, j_z \%7D
 along  to modelling of a Constant Areal Geothermal Temperature Profile along a given wellbore trajectory.

...

LaTeX Math Inline
bodyT_G(t, l)

LaTeX Math Inline
bodyG_T(z)

Geothermal Temperature Gradient

LaTeX Math Inline
bodyH_n

Neutral Temperature Layer (NTL)

...

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_0s

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
body--uriencoded--a_e%7Ben%7D

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_re(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 = 0.01 \, %7B\rm °C%7D

where

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

LaTeX Math Inline

bodyl

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

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
bodyz_s = z(l=0)

TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then 

LaTeX Math Inline
body

l

z_s = 0

)


Assumptions

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

...

LaTeX Math Inline

Neutral Layer LaTeX Math Block
anchorz_N
alignmentleft
z_n = z_s + H_n

body\lambda_e(x,y,z) = \lambda_e(z)


Equations

z < z_n_g_GlT_s + \int_{z_s}^z G_T(z) dz 0 + \frac{j_z}{\lambda_e} (z-z_s) + T_Y(t, z) + T_D(t, z)
LaTeX Math Block
anchorHT_Nz
alignmentleft
H_nT_G(t, z) = \sqrt{\frac{a_e \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }
Below Neutral Temperature LayerAbove Neutral Temperature Layer

LaTeX Math Inline
bodyz > z_n

LaTeX Math Inline
body
T_s + \int_{z_s}^z G_T(z) dz + T_Y(t, z) + T_D(t, z)



LaTeX Math Block
anchorG_T
alignmentleft
G_T
(
z) = 
\frac{j_z}{\lambda_e(z)}
LaTeX Math Block
anchorT_z
alignmentleft
T_Y(t,
z) = \delta T_
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_Y(t,z) = \delta T_A \, \exp \left[ \, {(z_s-z}) \sqrt
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_
e
{en} \, A_T}} \, \right]
LaTeX Math Block
anchorT_z
alignmentleft
T_D(t,z) = \delta T_D
\, \cos \left[
 \, 
2
\
pi
exp \
frac{t -
left[ \
delta
, 
t_A}
{
A_T} +
(z_s
-z}) \sqrt
{\frac{\pi}{a_
e
{en} \, 
A
D_T}} \, \right]
LaTeX Math Block
anchorT_z
alignmentleft
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
{en} \, 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]

where

Neutral Layer
LaTeX Math Block
anchorz_N
alignmentleft
z_n = z_s + H_n
LaTeX Math Block
anchorH_N
alignmentleft
H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }

LaTeX Math Inline
bodyz_s = z(l=0)

TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then  LaTeX Math Inlinebodyz_s = 0


See Also

...

Geology / Geothermal Temperature Field / Geothermal Temperature Profile

Geothermal Temperature Field @model ] [ Geothermal Temperature Gradient ]

Neutral Temperature Layer @model ]

References

...

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.

...