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

Temperature distribution along the wellbore trajectory

	Intake temperature		Pipeline trajectory TVDss
	Intake pressure		Pipeline trajectory inclination,
	Intake flowrate		Flow pipe diameter (tubing or casing depending on where flow occurs)
	Fluid density		Inner pipe wall roughness
	Fluid viscosity

Stationary fluid flow	Isothermal or Quasi-isothermal conditions	Constant cross-section pipe area along hole
Incompressible fluid	Isoviscous

The water injection wellbore temperature profile can be split into the following components:

Upward vertical heat conduction from Earth's Centre towards Earth's surface leading to a static geothermal profile
Upward & Downward vertical heat conduction from reservoir with non-geothermal temperature (invaded by injection water)
Heat exchange between wellbore fluid and surrounding rocks above and below the invaded reservoir
The temperature in water invaded reservoir stays constant from top to bottom

T(l) = Ts + ...

where