Motivation
Analytical model of Temperature The Temperature Profile in Homogeneous Wellbore Flow Analytical @model is using a combination of of (see Fig. 1):
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The heat flow in single-layer injector has three distinctive zones:
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| Fig. 1. Flow and temperature pattern for Semispace Linear Conduction model. |
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| for Semispace Linear Conduction |
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\frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2} |
Initial Conditions
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T(t=0, z) = T_G(z) |
Boundary conditions
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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:
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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 |
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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
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\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
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.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
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in terms of temperature gain:| LaTeX Math Block |
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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:
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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:
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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:
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T_b(t, z) = T_G(z) + dT_{b,over}(t, z) + dT_{b,under}(t, z) |
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See also
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[ Subsurface Temperature Profile around Lateral Flow Analytical @model ]
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