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 wellbores. 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.