Modelling of the Movement of a Prolate Particle in the Steady State Flow of a Non-Newtonian Fluid in an Inclined Annulus With Inner String Rotation

The determination of the slip velocity, or whether a solid particle will sediment, during its transport is of prime importance for hole cleaning evaluations during drilling operations. Yet, this task is complexified by the asymmetry of the annulus when the central pipe axis does not coincide with the borehole central line and when the inner string rotates, especially since drilling fluids typically follow a yield stress power law rheological behavior. This paper describes the modelling of the movement of a particle in such conditions yet with the following simplifications: the inner tube is eccentric but has a uniform movement, the shape of the particle is assimilated to a prolate, the change of shear rates in the fluid around the slipping particle is neglected and collisions between particles are not considered. Otherwise, gravitational effects are incorporated by accounting for the mass density difference between the particle and the surrounding fluid mixture and by considering the borehole inclination. The particle spin is also estimated as it plays an important role in the determination of the drag and lift forces. The solution to the differential equations that describe the time evolution of the position and orientation of the particle, depend largely upon the initial conditions. Therefore, an ensemble of boundary conditions is generated at a starting cross-section along the annulus and the resulting particle trajectories are estimated. It is then possible to estimate a probabilistic slip velocity for particles of the considered dimensions, far away from the entrance region. This probabilistic approach allows to define a critical transport fluid velocity as the lower limit of the bulk fluid velocity by which no particle risk to settle. Similarly, one can define a critical settling fluid velocity as the upper limit of the bulk fluid velocity where every particle will sediment regardless of the initial conditions. With the described modelling of the particle movement and its associated statistical methods, it is possible to quantitatively estimate the spatial distribution of particles in any cross-section. For those particles that get trapped between the tool-joint and the borehole, it is then possible to estimate their size reduction by grinding, resulting from the rotation of the tool-joint on the borehole wall. The grinding process impacts the particle size distribution passed a tool-joint. By applying this method iteratively up to the annulus outlet, it is possible to estimate the particle size distribution of the drill-cuttings when they arrive at the shale-shakers.