Motivation

One of the key tasks of in Pipe Flow Dynamics is to predict the along-hole temperature distribution during the stationary fluid transport.

In many practical cases the temperature distribution for the stationary fluid flow can be approximated by homogenous fluid flow model.

Pipeline Flow Temperature Model is addressing this problem with account of the varying pipeline trajectory, pipeline schematic and heat transfer with surroundings around pipeline.

Outputs

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

along-pipe temperature distribution and evolution in time

Inputs

$\begin{array}{l}{\bf r}(l)\end{array}$	pipeline trajectory, $\begin{array}{l}{\bf r} = {\bf r}(l) = \{ x(l), \, y(l), \, z(l) \}\end{array}$	$\begin{array}{l}\rho(T, p)\end{array}$	fluid density
$\begin{array}{l}A(l)\end{array}$	pipeline cross-section area	$\begin{array}{l}\mu(T, p)\end{array}$	fluid viscosity
$\begin{array}{l}T_0(t)\end{array}$	inflow temperature	$\begin{array}{l}T_{e0}(l)\end{array}$	initial temperature of the surroundings around the pipeline
$\begin{array}{l}p_0\end{array}$	inflow pressure	$\begin{array}{l}c_p(l)\end{array}$	specific heat capacity of the surroundings around pipeline
$\begin{array}{l}q_0\end{array}$	inflow rate	$\begin{array}{l}\lambda_e(l)\end{array}$	thermal conductivity of the surroundings around pipeline
$\begin{array}{l}U(l)\end{array}$	heat transfer coefficient based on pipeline schematic

Physical Model

Axial symmetry around the pipe

Mathematical Model

Heat transfer in wellbore due to convection and conduction along the wellbore flow

Heat transfer in rocks around the wellbore due to conduction

(1)	$\begin{array}{l}\displaystyle \rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg) - \rho \, c \, v \, \frac{dT}{dl} + \frac{2 \lambda}{\lambda_e} \cdot \frac{r_f}{r_w^2} \cdot U \cdot \left[ T_e(t, l, r_w) - T \right]\end{array}$

(2)	$\begin{array}{l}\displaystyle \rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e)\end{array}$

(3)	$\begin{array}{l}\displaystyle T(t=0, l) = T_{e0}(l)\end{array}$

(4)	$\begin{array}{l}\displaystyle T_e(t=0, l, r) = T_{e0}(l)\end{array}$

(5)	$\begin{array}{l}\displaystyle T(t, l=0) = T_0(t)\end{array}$

(6)	$\begin{array}{l}\displaystyle T_e(t, l, r \rightarrow \infty) = T_{e0}(l)\end{array}$

(7)	$\begin{array}{l}\displaystyle T (t, l=l_{max}) = T_{e0}(l_{max})\end{array}$

(8)	$\begin{array}{l}\displaystyle T_e(t, 0, r) = T_s(t)\end{array}$

(9)	$\begin{array}{l}\displaystyle T_e(t, l_{max}, r) = T_{e0}(l_{max})\end{array}$

Heat exchange between wellbore fluid and rocks around the wellbore

(10)	$\begin{array}{l}\displaystyle 2 \pi \, \lambda_e \, r_w \, \frac{\partial T_e}{\partial r} \, \bigg\|_{r=r_w} = 2 \pi \, r_f \, U \cdot \left( T_e \, \bigg\|_{r=r_w} - T \right)\end{array}$