Motivation

Proxy model of Pressure Profile in Homogeneous Steady-State Pipe Flow @model in the form of algebraic equation for the fast computation.

Outputs

	Pressure distribution along the pipe
	Flowrate distribution along the pipe
	Flow velocity distribution along the pipe

Inputs

	Intake temperature		Along-pipe temperature profile
	Intake pressure		Fluid density
	Intake flowrate		Fluid viscosity
	Pipeline trajectory TVDss		Pipe cross-section area
	Pipeline trajectory inclination,		Inner pipe wall roughness

Assumptions

Steady-State flow	Quasi-isothermal flow

Homogenous flow	Constant cross-section pipe area along hole

Constant inclination	Constant friction along hole

Linear density
which leads to and

Equations

Pressure profile along the pipe

L = \frac{1}{2 \, G \, c^*  \rho^*}  \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}

 \cos \theta \neq 0

L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
 + \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho}

 \cos \theta = 0

where

	mass flux
	mass flowrate
	Intake volumetric flowrate
	Intake fluid density
	elevation drop along pipe trajectory
	Darcy friction factor
	Reynolds number in Pipe Flow
	dynamic viscosity as function of fluid temperature and density
	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
	gravity acceleration along pipe
	altitude drop in downwards direction (positive if descending)

See Derivation of Pressure Profile in GF-Proxy Pipe Flow @model

Alternative forms

Volumetric Flowrate in inclined pipe:

q_0^2 = \frac{2 d A^2 G}{f} \cdot \left[ 

1 + \frac{ (\rho/\rho_0)^2 -1}{1- (\rho_0/\rho)^{\frac{2}{n-1}} \cdot 
\exp \left( \frac{fL/d}{ n-1}  \right)}
\right]

Volumetric Flowrate in horizontal pipe:

q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2}{2 \ln (\rho_0/\rho) + fL/d}

where

n = \frac{f \, L^*}{d}

L^* = \frac{1}{2 \, G \, c^* \, \rho^*} = \frac{1}{2 \, G \, c_0 \, \rho_0}

\rho_0/\rho = \frac{1+c^* p_0}{1+c^* p}

with the following asymptotes:

Low compressible fluids:	High compressible fluids:

Approximations

which is equivalent to and holds true for the most of practical tube diameters, as the lowest practical values of are

q_0^2 = 
\frac{2 \, d \, A^2 \, G}{f} \cdot \left [
1 + \frac{(\rho/\rho_0)^2-1}{1- \exp (2 \, c_0 \, \rho_0 \, G \, L)}
 \right]
=
\frac{2 \, d \, A^2  \, g}{f \, L} \cdot \left [ 
\Delta Z + ((\rho/\rho_0)^2 -1) \cdot  \frac{ \Delta Z}{1 - \exp(2 \, c_0 \, \rho_0 \, g \,  \Delta Z)}
\right]

\dot m = \rho_0 \, q_0

\rho =
\rho_0 \, \exp(c_0 \, \rho_0 \, G \, L) \, \sqrt{1 - \frac{f \, q_0^2}{2 \, d \, A^2} \cdot \frac{1- \exp(-2 \, c _0 \, \rho_0 \, G \, L)}{G}} 
=\rho_0 \, \exp (с_0 \, \rho_0 \, g \, \Delta Z) \cdot \sqrt{ 1 - \frac{8}{\pi^2} \cdot   \frac{f \, L}{d^5} \cdot q_0^2 \cdot \frac{1 - \exp(- 2 \, c_0 \, \rho_0 \, g \, \Delta Z) } { g \, \Delta Z}}

p(L) = p_0 + \frac{\rho/\rho_0 -1}{c_0}

Pressure Profile in GC-proxy static fluid column @model

\rho = \rho_0 \, \exp (c_0 \, \rho_0 \, g \, \Delta Z)

p(L) =  p_0 + \frac{\exp (c_0 \, \rho_0 \, g \, \Delta Z) -1}{c_0}

References

PipeFlowSimulator.xls