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Motivation

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The stabilized water injection profile satisfies the assumptions of the Pressure Profile in Incompressible Stationary Quasi-Isothermal Incompressible Isoviscous Pipe Flow Pressure Profile @model.

Motivation

Inputs & Outputs

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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:

  1. Iterations
  2. Iteration
  3. Iteration ...


Outputs

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LaTeX Math Inline
bodyT_s

...

...

LaTeX Math Inline
bodyp(l)

Pressure
distribution along
distribution along the pipe
LaTeX Math Inlinebodyp_s

Intake pressure 

LaTeX Math Inline
bodyq(l)

Flowrate

 distribution along

Intake flowrate 

LaTeX Math Inlinebody

 distribution along the pipe

LaTeX Math Inline
body

q_s

u(l)

Flow velocity

distribution along

distribution along the pipe


Inputs 

...

LaTeX Math Inline
body

z(l)Pipeline trajectory TVDss

T_s

Intake temperature 

LaTeX Math Inline
body

\theta

T(l)

Pipeline trajectory inclination,
Along-pipe temperature profile 

LaTeX Math Inline
body

p_s

Intake pressure 

--uriencoded--\displaystyle \cos \theta (l) = \frac%7Bdz%7D%7Bdl%7D

LaTeX Math Inline
body

--uriencoded--%7B\bf r%7D(l)

\rho(T, p)

Fluid density 
Pipeline trajectory

LaTeX Math Inline
body

T(l)

q_s

Intake flowrate 

Along-pipe temperature profile 

LaTeX Math Inline
body\

rho

mu(T, p)

Fluid
density
viscosity 

LaTeX Math Inline
body

\mu(T, p)Fluid viscosity 

z(l)

Pipeline trajectory TVDss

LaTeX Math Inline
bodyd

Flow pipe diameter

(tubing or casing depending on where flow occurs)

LaTeX Math Inline
body\theta (l)


Pipeline trajectory inclination,

LaTeX Math Inline
body--uriencoded--\displaystyle \cos \theta (l) = \frac%7Bdz%7D%7Bdl%7D

LaTeX Math Inline
body\epsilon

Inner pipe wall roughness


Assumptions

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Stationary
fluid
flowHomogenous
fluid
flowIsothermal or Quasi-isothermal conditions
Constant cross-section pipe area LaTeX Math Inlinebody along holeIncompressible fluid  
A

Incompressible fluid  

LaTeX Math Inline
body\rho(T, p)=\rho_s = \rm const

Isoviscous  

Isoviscous flow  

LaTeX Math Inline
body\mu(T, p) = \mu_s = \rm const

Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole


Equations

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Pressure profilePressure gradient profileFluid velocityFluid rate


LaTeX Math Block
anchorPPconst
alignmentleft
p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l



LaTeX Math Block
anchorgradP
alignmentleft
\frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s



LaTeX Math Block
anchor1
alignmentleft
q(l) =q_s = \rm const



LaTeX Math Block
anchor1
alignmentleft
u(l) = u_s = \frac{q_s}{A} = \rm const


where

LaTeX Math Inline
body--uriencoded--f_s = f(%7B\rm Re%7D, \, \epsilon)

Darcy friction factor (see Darcy friction factor in water producing/injecting wells @model) at intake point

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D_s = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7D

Reynolds number at intake point

LaTeX Math Inline
body--uriencoded--A = 0.25 \, \pi \, d%5e2

flow pipe cross-section area  (tubing or casing depending on where flow occurs)


See Also

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Petroleum Industry / Upstream / Subsurface E&P Disciplines / Production Technology  / Well Flow Performance / Lift Curves (LC) / Water Injection Wellbore Profile @model

[ Pressure Profile in Incompressible Stationary Quasi-Isothermal Incompressible Isoviscous Pipe Flow Pressure Profile @model ] [ Darcy friction factor in water producing/injecting wells @model ]

[ Water Injection Wellbore Temperature Profile @model ]

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