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Motivation

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


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

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Stationary fluid flowHomogenous fluid flowIsothermal or Quasi-isothermal conditions

Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole

Incompressible fluid  

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

Isoviscous  

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




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)

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

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

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

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