Motivation

Analytical model of Temperature Profile in Homogeneous Wellbore Flow is using a combination of 

Subsurface Temperature Profile around Lateral Flow Analytical @model

and

Temperature Profile in Homogenous Stationary Pipe Flow Analytical Ramey @model

The heat flow in single-layer injector has three distinctive zones:

• Zone UP: the flow is moving from surface towards the top of flowing unit and  simulated with  RHK model
  
  
• Zone FLOW: the temperature is constant across the whole thickness of the flow unit and equal to the injecting temperature at the top of the flowing unit
  
  
• Zone BOTTOM: below flowing unit all the way to bottom hole, no flow in wellbore or in formation across this zone, only heat exchange with overlying flowing units.
  
  
  

Heat flow equation for Semispace Linear Conduction:

\frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2}

Initial Conditions

T(t=0, z) = T_G(z)

Boundary conditions

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:

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 (

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

\zeta = \frac{z}{4 a t}

The final solution for temperature  above the flowing unit is represented by RHK pipe flow solution where T_G is replaced with T_b from 

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 

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:

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:

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:

T_b(t, z) = T_G(z) + dT_{b,over}(t, z) + dT_{b,under}(t, z)

Replacing the static value of 

See also

Physics / Fluid Dynamics / Linear Fluid Flow 

[ Subsurface Temperature Profile around Lateral Flow Analytical @model ] 

[ Temperature Profile in Homogenous Stationary Pipe Flow Analytical Ramey @model ]

[ Heat Transfer Coefficient (HTC) @model ]