Motivation

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

In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.

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

Outputs

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

Inputs

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

Assumptions

Stationary flow

Homogenous flow

Isothermal or Quasi-isothermal conditions

Constant cross-section pipe area along hole

Equations

\left(\rho(p) -  j_m^2 \cdot c(p)   \right)  \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l)  - \frac{ j_m^2 }{2 d} \cdot  f({\rm Re}, \, \epsilon)

p(l=0) = p_0

u(l) = \frac{\rho_0 \cdot q_0}{\rho(T(l), p(l))) \cdot A}

q(l) = \frac{\rho_0 \cdot q_0}{\rho(T(l),p(l))}

where

	mass flux
	fluid flow rate at pipe intake
	fluid density at intake temperature and pressure
	Fluid Compressibility
	Darcy friction factor
	Reynolds number in Pipe Flow
	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)

See Derivation of Pressure Profile in Steady-State Homogeneous Pipe Flow @model.

Approximations

Incompressible pipe flow with constant viscosity

Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model

Pressure profile

Pressure gradient profile

Fluid velocity

Fluid rate

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

\frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{j_m^2  f_0}{2 \, \rho_0 \, d}

q(l) =q_0 = \rm const

u(l) = u_0 = \frac{q_0}{A} = \rm const

where

	Darcy friction factor at inlet point
	Reynolds number at inlet point
	Dynamic Fluid Viscosity at inlet point

Incompressible fluid means that compressibility vanishes and fluid velocity is going to be constant along the pipeline trajectory .

For the constant viscosity along the pipeline trajectory the Reynolds number and Darcy friction factor are going to be constant along the pipeline trajectory.

Equation becomes:

\frac{dp}{dl} = \rho_s \, g \, \frac{dz}{dl}  - \frac{\rho_s \, q_s^2 }{2 A^2 d} f_s

which leads to after substituting and can be explicitly integrated leading to .

The first term in the right side of defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:

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 along-hole ( see Darcy friction factor in water producing/injecting wells ).