Motivation

Outputs

$\begin{array}{l}T(t, l)\end{array}$

Along-hole Temperature Profile

Inputs

$\begin{array}{l}t\end{array}$	Flowing duration
$\begin{array}{l}T_b\end{array}$
	Local Calendar Time	$\begin{array}{l}\delta T_A\end{array}$	Annual average surface temperature variation based on weather reports
$\begin{array}{l}z(l)\end{array}$	Wellbore trajectory. True Vertical Depth Sub-Sea (TVDss)	$\begin{array}{l}A_T\end{array}$	Period of annual temperature variation cycle: $\begin{array}{l}A_T = 1 \, {\rm year}\end{array}$
$\begin{array}{l}j_z\end{array}$	True vertical component of regional Earth's Heat Flux	$\begin{array}{l}\delta t_A\end{array}$	Time shift of annual highest temperature with respect to January 1
$\begin{array}{l}T_0\end{array}$	Local annual average surface temperature based on weather reports	$\begin{array}{l}\delta T_D\end{array}$	Daily average surface temperature variation based on weather reports
$\begin{array}{l}a_e\end{array}$	Local average Thermal diffusivity of the soil between Earth's surface and NTL	$\begin{array}{l}D_T\end{array}$	Period of daily temperature variation cycle: $\begin{array}{l}A_D = 1 \, {\rm day}\end{array}$
$\begin{array}{l}\lambda_r(z)\end{array}$	Subsurface Thermal Conductivity profile as function of TVDss	$\begin{array}{l}\delta t_D\end{array}$	Time shift of daily highest temperature with respect to Midnight 00:00
		$\begin{array}{l}\delta T_{\rm cut}\end{array}$	Temperature measurement threshold (usually $\begin{array}{l}\delta T_{\rm cut} = 0.01 \, {\rm °C}\end{array}$ )

where

$\begin{array}{l}l\end{array}$

wellbore trajectory Measured Depth with reference to Earth's surface ( $\begin{array}{l}l=0\end{array}$ )

Assumptions

$\begin{array}{l}q_s = {\rm const}\end{array}$	$\begin{array}{l}T_s(t) = T_s = \rm const\end{array}$	$\begin{array}{l}\theta(l) = \rm const\end{array}$	$\begin{array}{l}\rho_e = {\rm const}\end{array}$	$\begin{array}{l}\lambda_e (l) = {\rm const}\end{array}$	$\begin{array}{l}c_e (l) = {\rm const}\end{array}$

Equations

(1)	$\begin{array}{l}\displaystyle T(t, l) = T_{0e} + G_T \, z - R(t) \, G_T \, \cos \theta + \big( T_s - T_{0e} + R(t) \, G_T \, \cos \theta \big) \, e^{ - l/R(t)}\end{array}$

(2)	$\begin{array}{l}\displaystyle R(t) = \frac{q_s}{2 \pi \, a_e} \, \bigg( T_D(t) + \frac{\lambda_e}{r_f \, U} \bigg)\end{array}$

(3)	$\begin{array}{l}\displaystyle T_D(t) = \ln \big[ e^{-0.2 \, t_D} + (1.5 - 0.3719 \, e^{-t_D}) \, \sqrt{t_D} \big]\end{array}$

(4)	$\begin{array}{l}\displaystyle t_D(t) = \frac{a_e \, t}{r_w^2}\end{array}$

where

$\begin{array}{l}z_s = z(l=0)\end{array}$

TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then $\begin{array}{l}z_s = 0\end{array}$

Но для простых случаев есть аналитические оценки, которые правильно воспроизводят крупномасштабные формы температурного профиля.

Одна из популярных аналитических моделей для стационарной ( $\begin{array}{l}q_s = {\rm const}, \quad T_s(t) = T_s = \rm const\end{array}$ ) закачки в скважину с постоянным наклоном ( $\begin{array}{l}\theta(l) = \rm const\end{array}$ ), в окружении акисально-симметричного однородного пласта

( $\begin{array}{l}\rho_e = {\rm const}, \lambda_e (l) = {\rm const}, \, c_e (l) = {\rm const}\end{array}$ ) с постоянным геотермическим градиентом $\begin{array}{l}G_T(z) = \rm const\end{array}$ вдали от поверхности $\begin{array}{l}l \, \sin \theta \gg r_w\end{array}$ , дается следующей формулой (Ramey, 1962):

где

$\begin{array}{l}a_e\end{array}$	температуропроводность пород
$\begin{array}{l}\lambda_e\end{array}$	теплопроводность пород
$\begin{array}{l}с_e\end{array}$	объемная теплопроводжность пород при постоянном давлении
$\begin{array}{l}\rho_e\end{array}$	плотность пород
$\begin{array}{l}T_s\end{array}$	температура закачиваемого флюида на поверхности
$\begin{array}{l}r_f = d/2\end{array}$	радиус трубы вдоль контрой идет движение флюида
$\begin{array}{l}r_w\end{array}$	радиус скважины по долоту
$\begin{array}{l}G_T = \frac{dT_G}{dz}\end{array}$	геотермический градиент невозмущенных пород
$\begin{array}{l}q\end{array}$	дебит скважины на устье
$\begin{array}{l}\rho\end{array}$	плотность закачиваемого флюида
$\begin{array}{l}U\end{array}$	коэффициент теплопередачи между закачиваемым флюидом и породами

References

Ramey, H. J. (1962, April 1). Wellbore Heat Transmission. Society of Petroleum Engineers. doi:10.2118/96-PA

No labels

Page tree

Motivation

Outputs

Inputs

Assumptions

See Also

References

Page tree

Pipe Flow Temperature (Ramey) @model

Motivation

Outputs

Inputs

Assumptions

See Also

References