Motivation

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

Outputs

$\begin{array}{l}p(l)\end{array}$	Pressure distribution along the pipe
$\begin{array}{l}q(l)\end{array}$	Flowrate distribution along the pipe
$\begin{array}{l}u(l)\end{array}$	Flow velocity distribution along the pipe

Inputs

$\begin{array}{l}T_0\end{array}$	Intake temperature	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_0\end{array}$	Intake pressure	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_0\end{array}$	Intake flowrate	$\begin{array}{l}\mu(T, p)\end{array}$	Fluid viscosity
$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss	$\begin{array}{l}A\end{array}$	Pipe cross-section area
$\begin{array}{l}\theta(l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$	$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness

Assumptions

Steady-State flow	Quasi-isothermal flow
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = 0\end{array}$	$\begin{array}{l}\displaystyle \frac{\partial T}{\partial t} =0 \rightarrow T(t,l) = T(l)\end{array}$
Homogenous flow	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)\end{array}$	$\begin{array}{l}A(l) = A = \rm const\end{array}$
Constant inclination	Constant friction along hole
$\begin{array}{l}\displaystyle \theta(l) = \theta = {\rm const} \rightarrow \cos \theta = \frac{dz}{dl} = {\rm const}\end{array}$	$\begin{array}{l}f(l) = f = \rm const\end{array}$
Linear density
$\begin{array}{l}\rho = \rho^* \cdot ( 1 + c^* \cdot p)\end{array}$ which leads to $\begin{array}{l}\displaystyle c(p) = \frac{c^}{1 + c^ \cdot p }\end{array}$ and $\begin{array}{l}c^* \, \rho^* = c_0 \, \rho_0\end{array}$

Equations

Pressure profile along the pipe

(1)	$\begin{array}{l}\displaystyle L = \frac{1}{2 \, G \, c^* \rho^*} \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F} -\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}\end{array}$

(2)	$\begin{array}{l}\displaystyle \cos \theta \neq 0\end{array}$

(3)	$\begin{array}{l}\displaystyle L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2) + \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho}\end{array}$

(4)	$\begin{array}{l}\displaystyle \cos \theta = 0\end{array}$

where

$\begin{array}{l}\displaystyle j_m = \frac{ \dot m }{ A}\end{array}$	mass flux
$\begin{array}{l}\displaystyle \dot m = \frac{dm }{ dt}\end{array}$	mass flowrate
$\begin{array}{l}\displaystyle q_0 = \frac{dV_0}{dt} = \frac{ \dot m }{ \rho_0}\end{array}$	Intake volumetric flowrate
$\begin{array}{l}\rho_0 = \rho(T_0, p_0)\end{array}$	Intake fluid density
$\begin{array}{l}\Delta z(l) = z(l)-z(0)\end{array}$	elevation drop along pipe trajectory
$\begin{array}{l}f = f({\rm Re}(T,\rho), \, \epsilon) = \rm const\end{array}$	Darcy friction factor
$\begin{array}{l}\displaystyle {\rm Re}(T,\rho) =\frac{j_m \cdot d}{\mu(T,\rho)}\end{array}$	Reynolds number in Pipe Flow
$\begin{array}{l}\mu(T,\rho)\end{array}$	dynamic viscosity as function of fluid temperature $\begin{array}{l}T\end{array}$ and density $\begin{array}{l}\rho\end{array}$
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}} = \rm const\end{array}$	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
$\begin{array}{l}G = g \, \cos \theta = \Delta Z/L = \rm const\end{array}$	gravity acceleration along pipe
$\begin{array}{l}\Delta Z = Z_{out} - Z_{in}\end{array}$	altitude drop in downwards direction (positive if descending)
$\begin{array}{l}F = j_m^2 \cdot f/(2d) = F(l) = \rm const\end{array}$

Derivation

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

Alternative forms

Volumetric Flowrate in inclined pipe: $\begin{array}{l}\cos \theta \neq 0\end{array}$

(5)	$\begin{array}{l}\displaystyle q_0^2 = \frac{2 d A^2 G}{f} \cdot \left[ 1 + \frac{ (\rho/\rho_0)^2 -1}{1- (\rho_0/\rho)^{\frac{2}{n-1}} \cdot \exp \left( \frac{fL/d}{ n-1} \right)} \right]\end{array}$

Volumetric Flowrate in horizontal pipe: $\begin{array}{l}\cos \theta = 0\end{array}$

(6)	$\begin{array}{l}\displaystyle q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2}{2 \ln (\rho_0/\rho) + fL/d}\end{array}$

where

(7)	$\begin{array}{l}\displaystyle n = \frac{f \, L^*}{d}\end{array}$

(8)	$\begin{array}{l}\displaystyle L^* = \frac{1}{2 \, G \, c^* \, \rho^*} = \frac{1}{2 \, G \, c_0 \, \rho_0}\end{array}$

(9)	$\begin{array}{l}\displaystyle \rho_0/\rho = \frac{1+c^* p_0}{1+c^* p}\end{array}$

with the following asymptotes:

Low compressible fluids: $\begin{array}{l}c^* p \ll 1, \, \, c^* p_0 \ll 1\end{array}$	High compressible fluids: $\begin{array}{l}c^* p \gg 1, \, \, c^* p_0 \gg 1\end{array}$
$\begin{array}{l}\displaystyle \rho_0/\rho = c^* \cdot (p_0-p)\end{array}$	$\begin{array}{l}\displaystyle \rho_0/\rho = p_0/p\end{array}$

Approximations

$\begin{array}{l}n \geq 1\end{array}$ which is equivalent to $\begin{array}{l}L^* \geq d\end{array}$ and holds true for the most of practical tube diameters, as the lowest practical values of $\begin{array}{l}L^* \geq d\end{array}$ are $\begin{array}{l}L^* \geq 7,000 \, {\rm m}\end{array}$

(10)

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

(11)	$\begin{array}{l}\displaystyle \dot m = \rho_0 \, q_0\end{array}$

(12)

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

(13)	$\begin{array}{l}\displaystyle p(L) = p_0 + \frac{\rho/\rho_0 -1}{c_0}\end{array}$

Pressure Profile in GC-proxy static fluid column @model

(14)	$\begin{array}{l}\displaystyle \rho = \rho_0 \, \exp (c_0 \, \rho_0 \, g \, \Delta Z)\end{array}$

(15)	$\begin{array}{l}\displaystyle p(L) = p_0 + \frac{\exp (c_0 \, \rho_0 \, g \, \Delta Z) -1}{c_0}\end{array}$

Page tree

Motivation

Inputs

Assumptions

Equations

Alternative forms

Approximations

References

Page tree

Pressure Profile in GFC-Proxy Pipe Flow @model

Motivation

Inputs

Assumptions

Equations

Alternative forms

Approximations

References