Motivation
The Temperature Profile in Wellbore Production Homogeneous Flow Analytical @model (see Fig. 1) is a combination of the basic analytical models:
See also
...
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Wellbore Flow Model (WFM) / Wellbore Production Homogeneous Flow @model
[ Subsurface Temperature Profile around Lateral Flow Analytical @model ] [ Temperature Profile in Homogenous Stationary Pipe Flow Analytical Ramey @model ]
The heat flow in single-layer injector has three distinctive zones:
...
Heat flow equation for Semispace Linear Conduction:
LaTeX Math Block |
---|
|
\frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2} |
Initial Conditions
LaTeX Math Block |
---|
|
T(t=0, z) = T_G(z) |
Boundary conditions
LaTeX Math Block |
---|
|
T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z) |
The exact solution is given by following formula:
LaTeX Math Block |
---|
|
T(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi |
A fair approximation at late times (
) is given by expanding the integral: LaTeX Math Block |
---|
|
T(t,z) = T_f + (T_G(z) - T_f) \cdot \Bigg[ 1- \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) \Bigg] |
where
LaTeX Math Block |
---|
|
\zeta = \frac{z}{4 a t} |
The final solution for temperature above the flowing unit is represented by RHK pipe flow solution where TG is replaced with Tb from
LaTeX Math Block Reference |
---|
|
.For the intervals between two injection units the one needs to account for the SLC contribution from upper flowing unit and from lower flowing unit which can be done using the superposition.
First, let's rewrite
LaTeX Math Block Reference |
---|
|
in terms of temperature gain: LaTeX Math Block |
---|
|
dT(t, z) = T(t,z) - T_G(z)= - (T_G(z) - T_f) \cdot \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) |
Now one can write down the temperature disturbance from the overlying flowing unit A1:
LaTeX Math Block |
---|
|
dT_{b,over}(t, z) = T_{b,up}(t,z) - T_G(z)= - (T_G(z) - T_{f, A1}) \cdot \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) |
and from the underlying flowing unit A2:
LaTeX Math Block |
---|
|
dT_{b,under}(t, z) = T_{b,up}(t,z) - T_G(z)= - (T_G(z) - T_{f, A2}) \cdot \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) |
The background temperature disturbance between the flowing units will be:
LaTeX Math Block |
---|
|
T_b(t, z) = T_G(z) + dT_{b,over}(t, z) + dT_{b,under}(t, z) |
Replacing the static value of
in RHK model with dynamic value of one arrives to the final wellbore temperature model with account of heat exchange with surrounding rocks and cooling effects from flowing units (Semispace Linear Conduction).See also
Physics / Fluid Dynamics / Linear Fluid Flow
References
...
Semiplane Temperature Model.pptx
С. В. Новиков, ТЕПЛОВЫЕ СВОЙСТВА ТЕРРИГЕННЫХ КОЛЛЕКТОРОВ И НАСЫЩАЮЩИХ ФЛЮИДОВ, 2009.pdf
...