Numerical modelling of non-Newtonian fluids in annular space and its application to drilling operations
This thesis presents the results of investigations in two areasA) Laminar helical
flow of Herschel Bulkley fluids in annular space; and B) Cuttings transport in deviated
Description of area A: This is a theoretical study that consists of the modelling of
non-Newtonian fluid flowing through annular space. The rheology of the fluid is
represented by a three-parameter fluid model to account for a non-linear behaviour of
the rheologic curve followed by the presence of a yield stress. Two distinct
methodologies were used to study the effects of inner pipe rotation and inner pipe
eccentricity on the velocity profiles of the annular flow.
I. Laminar Helical Flow of a Herschel-Bulkley Fluid in a Concentric Annulus and its
Extension to a Narrow Eccentric Annulus: The method consists of the use of the
boundary conditions to enable the numerical integration of the motion equation. The
subsequent extension to eccentric annuli is based on division of the annulus into sectors,
where each sector is treated as an equivalent sector of a concentric annulus. Profiles of
velocity presented in 2-D and 3-D contour plots explain the effects of eccentricity, inner
pipe rotation and yield stress in many different situations. This analysis is useful to
simulate the flow field in a borehole during directional drilling and primary cementing
particularly for narrow eccentric annuli.
II. Laminar Helical Flow of a Herschel-Bulkley Fluid in an Eccentric Annulus and
The Special Case of a Pure Axial Flow: A new methodology to obtain the governing
equations is proposed here as a first step to the solution of this complicated problem. It
consists of eliminating the unknown radial and tangential pressure gradients from the
equation of motion by defining vorticity between these two components of the velocity
vector. The vorticity equation and the remaining axial component of the motion
equation, written in bipolar co-ordinates, are then made discrete using two different
finite difference approaches. Firstly, the inertial terms of both equations are made
discrete using a modified upwind scheme proposed by O. Axelsson and I. Gustafsson,
while the viscous terms are made discrete using central difference approximation.
Secondly, the moving boundary conditions are set by enforcing continuity of pressure on
the inner annular wall. The future solution of these equations will provide a very
accurate model, unavailable until now, that accounts for both effects, the eccentricity
and rotation of the drill string, to simulate the flow field of drilling mud in directional
and horizontal wells.
Description of area B: The fluid models developed are incorporated in the development
of two semi-empirical correlations to predict the critical conditions of cuttings transport
in deviated weI/bore. The numerical fluid models are modified to predict the velocity of
the fluid at the vicinity of the cutting that is at the point of being transported. An
extensive bank of data of the critical conditions of transport, emulating many different
field conditions, was used in this analysis. The experimental data was provided by an
industry sponsored project which did four years of experimental work in a simulated
testing column to generate the data.
The two semi-empirical correlations developed in this research are based on:
• a dimensional analysis of the variables involved in two distinct mechanisms of
transport experimentally observed: 1. Rolling or Sliding and 2. Suspension;
• a force balance applied to a cutting resting on the low-side wall of an inclined
annulus, under fluid dynamic conditions.
The semi-empirical correlations can be used as general criteria for evaluating and
correlating the effects of various parameters on cuttings transport, and as a guideline for
cuttings transport programme design during directional drilling.