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Inputs

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LaTeX Math Inline
bodyT_0

Inlet temperature 

Fluid temperature at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
bodyT(l)

Along-pipe temperature profile 

LaTeX Math Inline
bodyp_0

Fluid pressure at inlet point (

LaTeX Math Inline
bodyl=0
Inlet pressure )

LaTeX Math Inline
body\rho(T, p)

Fluid density 

LaTeX Math Inline
bodyq_0

Inlet Fluid flowrate  at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
body\mu(T, p)

Dynamic fluid viscosity 

LaTeX Math Inline
bodyz(l)

Pipeline trajectory TVDss

LaTeX Math Inline
bodyA

Pipe cross-section area  
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

...

LaTeX Math Inline
body--uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D

mass flux

LaTeX Math Inline
bodyq_0 = q(l=0)

fluid flow rate at pipe intake

Fluid flowrate at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
body\rho_0 = \rho(T_0, p_0)

fluid density at intake temperature and pressure

Fluid density at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
bodyс(p)

Fluid Compressibility

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

Darcy friction factor

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bj_m \cdot d%7D%7B\mu(T, p)%7D

Reynolds number in Pipe Flow

LaTeX Math Inline
body--uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D

characteristicCharacteristic linear dimension of the pipe

(or exactly a pipe diameter in case of a circular pipe)

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LaTeX Math Inline
body--uriencoded--f_0 = f(%7B\rm Re%7D_0, \, \epsilon)

Darcy friction factor

 at

 at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D_0= \frac%7Bj_m \, d%7D%7B\mu_0%7D

Reynolds number

 at

 at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
body\mu_0 = \mu(T_0, p_0)

Dynamic Fluid Viscosity at inlet point (

LaTeX Math Inline
bodyl=0
)


Expand
titleDerivation


Panel
borderColorwheat
bgColormintcream
borderWidth7

Incompressible fluid 

LaTeX Math Inline
body\rho(T, p) = \rho_s 0 = \rm const
 means that compressibility vanishes 
LaTeX Math Inline
bodyc(p) = 0
 and fluid velocity is going to be constant along the pipeline trajectory 
LaTeX Math Inline
body--uriencoded--u(l) = u_s 0 = \frac%7Bq_s%7D%7BA%7D 0%7D%7BA%7D = \rm const
.

For the constant viscosity 

LaTeX Math Inline
body\mu(T, p) = \mu_s 0 = \rm const
 along the pipeline trajectory the Reynolds number 
LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D(l) = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7Bfrac%7Bj_m%5e2 \, d%7D%7B\mu_s%7D 0%7D = \rm const
 and Darcy friction factor 
LaTeX Math Inline
body--uriencoded--f(l) = f(%7B\rm Re%7D, \, \epsilon) = f_s 0 = \rm const
 are going to be constant along the pipeline trajectory.

Equation 

LaTeX Math Block Reference
anchorPP
 becomes:

LaTeX Math Block
anchorPP
alignmentleft
\frac{dp}{dl} = \rho_s0 \, g \, \frac{dz}{dl}  - \frac{\rhoj_sm^2 \, qf_s^2 0}{2 A^2 \, \rho_0 \, d} f_s

which leads to 

LaTeX Math Block Reference
anchorgradP
 after substituting 
LaTeX Math Inline
body--uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D
  and can be explicitly integrated leading to 
LaTeX Math Block Reference
anchorPPconst
.


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In many practical applications the water in water producing wells or water injecting wells can be considered as incompressible and friction factor  can be assumed constant

LaTeX Math Inline
body f(l) = f_s 0 = \rm const
 along-hole ( see  Darcy friction factor in water producing/injecting wells ).

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