Motivation

Analytical model of Temperature The Temperature Profile in Wellbore Production Homogeneous Wellbore Flow is using a combination of Flow Analytical @model  (see Fig. 1) is a combination of the basic analytical models:

...

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

...

Image Added

Image Removed

Fig. 1.

Flow and temperature pattern for Semispace Linear Conduction model.

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

See also

...

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Wellbore Flow Model (WFM) / Wellbore Production Homogeneous Flow @modelPhysics / 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 ]