Introduction
Fluid flow in drill and annular pipes has
received a lot of attention from oil industries in drilling operations.
Due to the constant concern with operational costs and the need for
raising production capacity higher flow rates have most frequently been
used, and consequently pressure losses in the pipe and the annular
space began to require a signifacnt amount of energy. In this work,
through MATLAB numerical simulation , pressure losses through the pipe
and the annuli have been calculated for a Newtonian and non Newtonian
fluid. Non newtonian fluid has been defined by a
rhoological model.
Problem geometry:
dint=0.14 m
dext=0..31 m
Q=0.05 m3/s
µ from 0.025 to 1.2 Pa.s
rho=1500 kg/m3
Equations for newtonian
fluid
Pressure
losses are calculated from Navier Stokes equation. We choose a small
volume portion with radius R and length dl, after discretize the
equation we finally find the pressure drop expression to solve through
MATLAB:
where
and this value depends on the flow. if laminar
if tubulent, we use Blasius relation:
Laminar flow: Pressure Profile
for pipe and annular sections
Before study the real case of the flow in the pipe which is turbulent,
we have studied Poiseuille flow (laminar flow) and solve NS equations
to find pressure profile and velocity profile:
Pressure
Profile:
Velocity profiles:
Turbulent flow: Pressure Profile
for pipe and annular sections
Pipe
Flow is turbulent with a Reynolds number over than 20 000. For this
case we use Blasius relation and solve NS equations with MATLAB. Our
step space is 10. We define a viscosity equation that increases with
depth from 0.025 Pa.s at the surface and0.6 Pa.s at 4500m.
Pressure losses= hydrostatic
pressure + friction pressure
P= 712 bar for a turbulent flow (µ =0.025, Q=0.05 and Re= 27 284)
P= 678 bar for a lamniar flow (µ =0.025, Q=0.001 and Re= 545)
> 676 bar from hydrostatic pressure
Singular Pressure Losses
 Pressure loss through bit nozzles:
Cv= the discharge coefficient ( varies with the
diameter)
TFA= total fluid area
 Other singular pressure losses are neglected.
Non newtonian fluid
Company and journal of
petroleum measurements:
Pressure profile on the
pipe with shear stress and viscosity inputs:
Rheological model:
HerschelBulckley
Today
a good compromise between the accuracy in the calculations and the
simplicity of the use is required and the best way to achieve this is
with the use of the Herschel–Bulkley rheological model
The
standard procedure of the estimations od the three HB
paramters τ,K and n; where τ, τy are the shear stress and the
yield stress respectively, K, n are the fluid consistency and fluid
behavior indices respectively and γ is the shear rate, is through
nonlinear regression of the viscometric data from concentric cylinder
geometry.
Journal of petroleum estimations of HB parameters
Comments

Input: viscosity increases with depth.

Measures results: higher the values of
apparent viscosity, less the difference between newtonian and non newtonian fluid.
 HB model
results: Difference increases with viscosity value.

Viscosity is the most important parameter for pressure losses

Pressure
losses difference between
newtonian and non newtonian model are more important in the drill than in the annular section.

In
reality, pressure tends to decrease viscosity value.