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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   { \bf j}(x,y,z) = \{ j_x, \, j_y, \, j_z \}  and Thermal Conductivity  \lambda_e({\bf r}) to be homogeneous across location area: 

(1) {\bf j}({\bf r}) ={\bf j}(x,y,z)={ \bf j}(z)
(2) \lambda_e({\bf r}) =\lambda_e(z)

where  z is true vertical depth.

Since the  heat flux is conservative (see

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) then it immediately implies that:

(3) {\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:  T_s(x,y) = \rm const one can see that lateral components of the heat flux are vanishing: 

(4) { \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  j_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  { \bf j}(x,y,z) = \{ 0, \, 0 , \, j_z \} to modelling of a Constant Areal Geothermal Temperature Profile along a given wellbore trajectory.

Outputs

T_G(t, l)

G_T(z)

Geothermal Temperature Gradient

H_n

Neutral Temperature Layer (NTL)

Inputs

t

Local Calendar Time

\delta T_A

Annual average surface temperature variation based on weather reports

z(l)

A_T

Period of annual temperature variation cycle: A_T = 1 \, {\rm year}

j_z

True vertical component of regional Earth's Heat Flux

\delta t_A

Time shift of annual highest temperature with respect to January 1

T_s

Local annual average surface temperature based on weather reports

\delta T_D

Daily average surface temperature variation based on weather reports

a_{en}

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

D_T

Period of daily temperature variation cycle: A_D = 1 \, {\rm day}

\lambda_e(z)

Subsurface Thermal Conductivity profile as function of TVDss

\delta t_D

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



\delta T_{\rm cut}

Temperature measurement threshold (usually \delta T_{\rm cut} = 0.01 \, {\rm °C}

where

l

Measured Depth of wellbore trajectory with reference to Earth's surface ( l=0)

z_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  z_s = 0


Assumptions

{ \bf j}(x,y,z) = \{ 0, \, 0 , \, j_z = {\rm const} \}

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


Equations

(5) T_G(t, z) = T_s + \int_{z_s}^z G_T(z) dz + T_Y(t, z) + T_D(t, z)
(6) G_T(z) = \frac{j_z}{\lambda_e(z)}
(7) T_Y(t,z) = \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]
(8) 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]
Neutral Layer
(9) z_n = z_s + H_n
(10) H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }

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.

GeothermalTemperatureProfile.xlsx













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