n7     Flow model for a vertical wellbore mud drilling fluid


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:



and this value depends on the flow. if laminar  


if tubulent, we use Blasius relation:     blasius

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 N-S equations to find pressure profile and velocity profile:

Pressure Profile:

Poiseuille1         poiseuille2             

Velocity profiles:

velocity1                 velocicty2

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 N-S 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.
peffect               feffect

                                                            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

  Cv= the discharge coefficient ( varies with the diameter) 

  TFA= total fluid area


    Non newtonian fluid

Company and journal of petroleum measurements:

shearstress                       viscosity

Pressure profile on the pipe with shear stress and viscosity inputs:

geo1                Art1

Rheological model: Herschel-Bulckley

 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 H-B 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 non-linear regression of the viscometric data from concentric cylinder geometry.

Pressure losses for different sections:

Company estimations of H-B parameters:

input parameters: H-B parameters K,n and τ.
Viscosity variation with depth from 0.025 to 0.6.

geocompar               geocompar

geocompar2         geo2CI

Journal of petroleum estimations of H-B parameters

artcomp1                                art1zoom

art2                                   art2zoom