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Motivation

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One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary fluid transport.

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Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.

Outputs

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LaTeX Math Inline

body	p(l)

Pressure distribution along the pipe

LaTeX Math Inline

body	q(l)

Flowrate distribution along the pipe

LaTeX Math Inline

body	u(l)

Flow velocity distribution along the pipe

Inputs

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LaTeX Math Inline

body	T_s0

Intake temperature

LaTeX Math Inline

body	T(l)	\rho_0

Fluid densityAlong-pipe temperature profile

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body	p_s0

Intake pressure

LaTeX Math Inline

body	\

rho(T, p)

mu_0

Fluid viscosityFluid density

LaTeX Math Inline

body	q_s0

Intake flowrate

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body

\mu(T, p)

Pipe cross-section area Fluid viscosity

LaTeX Math Inline

body	z(l)

Pipeline trajectory TVDss

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body

\epsilon

Inner pipe wall roughness Pipe cross-section area

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body	\theta (l)

Pipeline trajectory inclination,

LaTeX Math Inline

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

Assumptions

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Steady-State flow

Isothermal

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body

\epsilonInner pipe wall roughness

Assumptions

--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0

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body	T(t, l)=T_0 = \rm const

Homogenous flow

Constant cross-section pipe area

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body	A

along hole

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body	--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)

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body	A(l) = A = \rm const

Incompressible fluid

Stationary flow Homogenous flow Isothermal or Quasi-isothermal conditionsIncompressible fluid

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body	\rho(T, p)=\rho_

0 = \rm const

Isoviscous flow

→

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body	\mu(T,

\rho) =\mu_

0 = \rm const

Constant cross-section pipe area

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body	A

along hole

Equations

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Pressure profile	Pressure gradient profile

Fluid velocityFluid rate

LaTeX Math Block

anchor	PPconst
alignment	left

p(l) = p_

0 + \rho_

0 \, g \, \Delta z(l) - \frac{\rho_

0 \, q_

s^2

0^2 }{2 A^2 d} \, f_

0 \, l = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{8}{\pi^2} \frac{f_0 \, l}{d^5} \rho_0 \, q_0^2

LaTeX Math Block

anchor	gradP
alignment	left

\frac{dp}{dl} = \rho_

0 \, g \cos \theta(l) - \frac{\rho_

0 \, q_

s^2

0^2 }{2 A^2 d} \, f_

Mass Flux

Mass Flowrate

LaTeX Math Block

anchor

MassFlux
alignment	left

q(l) =q_s = \rm const

j_m = \rho_0 \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_0}

LaTeX Math Block

anchor	MassFlowrate
alignment	left

\dot m = j_m \cdot A = \rho_0 \cdot A \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_s}

Volumetric Flowrate

Intake Fluid velocity

LaTeX Math Block

anchor

PPconst
alignment	left

u(l) = u_s = \frac{q_s}{A} = \rm const

where

q_0 = \dot m / \rho_0 = A \cdot \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{  g \, \Delta z(l) + (p_0 - p)/ \rho_0 }

LaTeX Math Block

anchor	PPconst
alignment	left

u_0 = j_m/ \rho_0 =q_0 / A = \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{  g \, \Delta z(l) + (p_0 - p)/ \rho_s }

where

LaTeX Math Inline

body	j_m

Intake mass flux

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body	\dot m

mass flowrate

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body	u_0 = u(l=0)

Intake Fluid velocity

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body	\Delta z(l) = z(l)-z(0)

elevation drop along pipe trajectory

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body

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

Darcy friction factor at intake point

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body	--uriencoded--\displaystyle %7B\rm Re%7D_0 = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s 0 q_s%7D%7B0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D_0%7D

Reynolds number at intake point

LaTeX Math Inline

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

characteristic linear dimension of the pipe

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

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Expand

title	Derivation

Panel

borderColor	wheat
bgColor	mintcream
borderWidth	7

Incompressible fluid

LaTeX Math Inline

body	\rho(T, p) = \rho_s 0 = \rm const

means that compressibility vanishes

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body	c(p) = 0

and fluid velocity is going to be constant along the pipeline trajectory

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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

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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

anchor	PP

becomes:

LaTeX Math Block

anchor	PP
alignment	left

\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

anchor	gradP

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

anchor	PPconst

The first term in the right side of

LaTeX Math Block Reference

anchor	gradP

defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:

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

LaTeX Math Inline