Motivation

One of the key problems in designing the pipelines and controlling the pipeline fluid transport is to predict the temperature and pressure losses during the stationary fluid transport.

Pipeline flow simulator is addressing this problem. It should account for the varying pipeline trajectory, gravity effects, fluid friction with pipeline walls and varying heat exchange with surroundings.

Definition

Given 

• space coordinates are  $//<![CDATA[ \begin{array}{l}\{ x, \, y, \, z \}\end{array} //]]>$ with  $//<![CDATA[ \begin{array}{l}z\end{array} //]]>$-ccordinate facing down to the Earth Centre
  
  
• inflow pipeline coordinates  $//<![CDATA[ \begin{array}{l}\{ x_s = 0, \, y_s = 0, \, z_s = 0 \}\end{array} //]]>$
  
  
• pipeline trajectory  $//<![CDATA[ \begin{array}{l}\{ x_w(l), \, y_w(l), \, z_w(l) \}\end{array} //]]>$, where  $//<![CDATA[ \begin{array}{l}l = \int_0^l \sqrt{dx^2 + dy^2 + dz^2} = \int_0^l \sqrt{\dot x^2 + \dot y^2 + \dot z^2} dl\end{array} //]]>$,  is pipeline length from inflow point  $//<![CDATA[ \begin{array}{l}\{ x_s = 0, \, y_s = 0, \, z_s = 0 \}\end{array} //]]>$
  
  
• pipeline cross-section area  $//<![CDATA[ \begin{array}{l}A(l)\end{array} //]]>$
  
  
• earth gravity vector  $//<![CDATA[ \begin{array}{l}{\bf g} = (0, \, 0, \, g)\end{array} //]]>$ where  $//<![CDATA[ \begin{array}{l}g = 9.81 \ \rm m/s^2\end{array} //]]>$
  
  
• inflow temperature  $//<![CDATA[ \begin{array}{l}T_s\end{array} //]]>$, inflow pressure  $//<![CDATA[ \begin{array}{l}p_s\end{array} //]]>$, inflow rate  $//<![CDATA[ \begin{array}{l}q_s\end{array} //]]>$
  
  
• PVT properties of water  $//<![CDATA[ \begin{array}{l}\rho(T, p)\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}\mu(T, p)\end{array} //]]>$
  
  
• surroundings initial temperature   $//<![CDATA[ \begin{array}{l}T_g(l)\end{array} //]]>$, thermal diffusivity  $//<![CDATA[ \begin{array}{l}a_e(l)\end{array} //]]>$, thermal conductivity  $//<![CDATA[ \begin{array}{l}\lambda_e(l)\end{array} //]]>$ of surrounding media
  
  
• heat exchange coefficient  $//<![CDATA[ \begin{array}{l}U(l)\end{array} //]]>$ based on pipeline schematics
  
  

Simulate

• along-pipe distribution of stabilized pressure  $//<![CDATA[ \begin{array}{l}p(l)\end{array} //]]>$, flow rate $//<![CDATA[ \begin{array}{l}q(l)\end{array} //]]>$ and average flow velocity  $//<![CDATA[ \begin{array}{l}u(l)\end{array} //]]>$ 
  
  
• along-pipe distribution of fluid flow temperature  $//<![CDATA[ \begin{array}{l}T(t, l)\end{array} //]]>$ after a flow period of time  $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ when the flow stabilization achieved
  
  

In water producing or water injecting wells the friction factor  can be assumed constant $//<![CDATA[ \begin{array}{l}f(l) = f_s = \rm const\end{array} //]]>$ along-hole ( see  Darcy friction factor in water producing/injecting wells ).

References