Motivation

For the stabilized flow the wellbore pressure profile is constant and wellbore temperature profile is changing very slowly.

This allows solving the pressure-temperature problem iteratively:

Outputs

$\begin{array}{l}p(l)\end{array}$	Pressure distribution along the pipe
$\begin{array}{l}q(l)\end{array}$	Flowrate distribution along the pipe
$\begin{array}{l}u(l)\end{array}$	Flow velocity distribution along the pipe

Inputs

$\begin{array}{l}T_s\end{array}$	Intake temperature	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_s\end{array}$	Intake pressure	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_s\end{array}$	Intake flowrate	$\begin{array}{l}\mu(T, p)\end{array}$	Fluid viscosity
$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss	$\begin{array}{l}d\end{array}$	Flow pipe diameter (tubing or casing depending on where flow occurs)
$\begin{array}{l}\theta (l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$	$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness

Stationary flow	Homogenous flow	Isothermal or Quasi-isothermal conditions
Incompressible fluid $\begin{array}{l}\rho(T, p)=\rho_s = \rm const\end{array}$	Isoviscous flow $\begin{array}{l}\mu(T, p) = \mu_s = \rm const\end{array}$	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole

Pressure profile

Pressure gradient profile

Fluid velocity

Fluid rate

(1)	$\begin{array}{l}\displaystyle p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l\end{array}$

(2)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s\end{array}$

(3)	$\begin{array}{l}\displaystyle q(l) =q_s = \rm const\end{array}$

(4)	$\begin{array}{l}\displaystyle u(l) = u_s = \frac{q_s}{A} = \rm const\end{array}$

where

$\begin{array}{l}f_s = f({\rm Re}, \, \epsilon)\end{array}$	Darcy friction factor (see Darcy friction factor in water producing/injecting wells @model) at intake point
$\begin{array}{l}\displaystyle {\rm Re}_s = \frac{4 \rho_s q_s}{\pi d} \frac{1}{\mu_s}\end{array}$	Reynolds number at intake point
$\begin{array}{l}A = 0.25 \, \pi \, d^2\end{array}$	flow pipe cross-section area (tubing or casing depending on where flow occurs)