A number of different models are used to describe the shear rate dependent viscosity of drilling fluids. Most, such as the Herschel-Bulkley model, have a purely empirical basis. The Quemada model, while still empirical, is based on physical principles. It is based on the notion that structural units develop in the fluid at low shear rates which are then partially broken down as the applied shear rate increases. In the current work, drilling fluid rheological data are fitted to the Herschel-Bulkley and the Quemada model. The development of the Quemada model and the calculation of each model parameter are presented. We show that the Quemada model better fits measurements over a wider range of shear rates than the Herschel-Bulkley model. We describe how to select the parameters of the Quemada model. Knowing the difficulty of obtaining a known shear rate for fluids with yield stresses, we discuss how this can affect the quality of the Quemada model fit. Furthermore, in principle, the Quemada model is not applicable in presence a non-zero yield stress. Therefore, we show how to handle the yield stress using a (very high) zero shear rate viscosity.

Rapid variations in the fluid velocity field influence pressure loss calculations. In this paper, we propose numerical methods for estimating pressure losses in a circular pipe when the flow rate oscillates. The method is described for Newtonian and two non-Newtonian rheological behaviors: the power law and the Quemada models. Also, as drilling fluids are usually thixotropic, i.e., their rheological behavior depends on the shear history, an expansion of the Quemada model is proposed to account for shear history effects. A laboratory flow-loop has been assembled and measurements conducted with a non-thixotropic aqueous solution of Carbopol and a thixotropic potassium chloride solution of xanthan gum. The measurements were analyzed and compared with estimates made with the proposed models. It is found that when applying a square wave oscillating flowrate to a non-thixotropic fluid, large surge and swab pressure spikes are generated. The same square wave signal does not produce pressure spikes when circulating a thixotropic fluid; on the contrary the acceleration and deceleration fronts are largely attenuated. When applying a triangular or sinusoidal wave form to the flowrate while circulating a non-thixotropic fluid, the peak-to-peak pressure gradient gets progressively larger when the oscillation amplitude increases or the signal period reduces, compared to the expected value when estimating the pressure losses with the steady state approximation. However, under the same conditions, when circulating a thixotropic fluid, the peak-to-peak pressure gradients are lower than those estimated with the steady state approximation.

It is nowadays well accepted that the steady state rheological behavior of drilling fluids must be modelled by at least three parameters. One of the most often used models is the yield power law, also referred as the Herschel-Bulkley model. Other models have been proposed like the one from Robertson-Stiff, while other industries have used other three-parameter models such as the one from Heinz-Casson. Some studies have been made to compare the degree of agreement between different rheological models and rheometer measurements but in most cases, already published works have only used mechanical rheometers that have a limited number of speeds and precision. For this paper, we have taken measurements with a scientific rheometer in well-controlled conditions of temperature and evaporation, and for relevant shear rates that are representative to normally encountered drilling operation conditions. Care has been made to minimize the effect of thixotropy on measurements, as the shear stress response of drilling fluids depends on its shear history. Measurements have been made at different temperatures, for various drilling fluid systems (both water and oil-based), and with variable levels of solid contents. Also, the shear rate reported by the rheometer itself, is corrected to account for the fact that the rheometer estimates the wall shear rate on the assumption that the tested fluid is Newtonian. A measure of proximity between the measurements and a rheological model is defined, thereby allowing the ranking of different rheological behavior model candidates. Based on the 469 rheograms of various drilling fluids that have been analyzed, it appears that the Heinz-Casson model describes most accurately the rheological behavior of the fluid samples, followed by the model of Carreau, Herschel-Bulkley and Robertson-Stiff, in decreasing order of fidelity.

Because of the increased importance for the drilling industry to deliver drilling automation solutions, model-based applications for the analysis and control of the drilling process, have become an attractive approach towards improved performance and increased safety. A critical characteristic for such applications is its ability to perform accurate simulations of the drilling operation in real-time, based on a detailed description of the wellbore. In a real-time context, the boundary conditions of the drilling system are seldom constant, therefore reinforcing the importance of utilizing transient models of the drilling process instead of steady state ones. Typical domains that require modelling are related to the mechanical, hydraulic and heat transfer aspects of a drilling operation. The time constants of the force-, momentum-, mass- and energy-conservation equations are sufficiently different to allow for solving each of these equations with different time discretization schemes. Yet, side effects influence the results from each other’s and therefore a time coupling shall nevertheless be accounted for. For instance, for a drilling operation conducted on a floater, the heave induced movement at the top of the string propagates along the drill-string, therefore causing a displacement that induces swab and surge pressure variations, which themselves generate counter-acting forces on the drill-string. In such conditions, both the mechanical and hydraulic frictions generate heat that changes the in situ thermal conditions and therefore the drilling fluid mass density and its rheological behavior. Consequently, heat exchange caused by the drill-string and fluid movements also influences the hydraulic response of the system. Furthermore, thermal expansion will also apply to the drill-string. In this paper, we discuss recent advances related to the coupling between transient mechanical, hydraulic and thermal models, where a key criterion is that the combined drilling model shall be capable of running in real-time on a standard computer. Incorporating these transient models is considered a necessary step towards improved accuracy of simulations, especially on floaters, where heave effects become important. We illustrate various effects by presenting and discussing several simulations results in detail.

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.

Drilling fluids are visco-elastic materials, i.e. they behave as a viscous fluid when subject to a sufficient shear stress and like an elastic solid otherwise. Both their elastic and viscous properties are time-dependent, i.e. drilling fluids are thixotropic. Because of thixotropy, it takes a finite time before the effective viscosity of a drilling fluid attains an equilibrium when the fluid is subject to a change of shear rate. This effect is visible when one changes the applied shear rate in a rheometer, as the fluid will gradually adapt to the new shearing conditions. When the velocity of a drilling fluid changes, for instance due to a change in pump flow rate, movement of the drill string, or change of flow geometry, the fluid will exhibit a time-dependent response to the new shearing conditions, requiring a certain time to reach the new equilibrium condition. Unfortunately, the time-dependence of the rheological properties of drilling fluids are usually not measured during drilling operations and therefore it is difficult to estimate how thixotropy impacts pressure losses in drilling operations. For that reason, we have systematically measured the time-dependence of the rheological properties of several samples of water-based, oil-based and micronized drilling fluids with a scientific rheometer in order to capture how drilling fluids systems respond to variations of shear rates. Based on these measurements, we propose to investigate how one existing thixotropic model manages to predict the shear stress as a function of the shear rate while accounting for the shear history and gelling conditions. Then we propose a modified model that fits better, overall, with the measurements even though there are still noticeable discrepancies, especially when switching back to low shear rates.

The automatization of the drilling process opens the opportunity to faster reactions in case of unexpected drilling conditions, therefore reducing the risk that a drilling incident escalates to a serious situation. It also allows to push the drilling performance to be as close as possible to the limits of drillability as a function of the varying drilling conditions. But to achieve high level of drilling process automation, it is necessary to have access to accurate mathematical models of the complex physical system that is composed of the drilling rig, the drill-string, the drilling fluid and the borehole itself. As the development of accurate heat transfer, mechanical and hydraulic models and their utilization in full scale drilling applications is a huge and complex task, it is tempting to recreate drilling automatization problems in a laboratory scale setup. Because of sudden variations of the downhole drilling conditions, like when transitioning from soft to hard rock or when the bit is subject to large torque variations induced by interbedded rock layers, the boundary conditions at the bit change suddenly and require quick response from the automatic top-drive and hoisting system controllers. At a small laboratory scale, the necessary reaction times are of the order of milliseconds and therefore exclude any manual intervention. It is therefore crucial that the automatic control methods utilize precise mathematical models of the physical system to accurately estimate the limits by which the drilling process can be managed under safe conditions. For that reason, a general purpose mathematical model of small-scale drilling rigs has been developed. First, the Rayleigh-Ritz method is used to determine the deflection of the drill-string and to estimate the side forces at the contact points along the drill-string and BHA (Bottom Hole Assembly). Then the dynamic response of the power transmission system is modelled for both variable frequency drive controlled tri-phase motors and for stepper motors, including friction effects at the contact points. Friction is modelled using Stribeck theory rather than the classical Coulomb laws of friction. Finally, the expected response of 3D accelerometers, that could be placed on the outside of a BHA component, is modelled to retrieve possible inclination variations and potential vibration modes such as torsional oscillations, forward or backward whirl. The generality of the model is such that it can be used for many small-scale drilling rig designs.

The efficiency of a drilling operation is to a great extent governed by how well one is able to optimize the rate of penetration (ROP) throughout each stage of the operation. ROP optimization normally involves balancing drilling speed on the one hand with acceptable wear to the drill bit on the other. The bit lifetime is largely determined by the mechanical conditions at the bit-formation rock interface and the weight on bit (WOB) and torque on bit (TOB) provide important information related to the working condition of the bit. The accuracy of WOB and TOB measurements can thus become a determining factor for the overall drilling efficiency. Due to the low bandwidth of downhole mud pulse-based telemetry systems, the WOB and the TOB are generally derived from surface measurements, i.e. from the hook load and the top-drive torque. Field experience indicates that a WOB derived from surface measurements can be of limited accuracy, such as when surface measurements suggest a negative WOB even though the ROP is positive, or when high sampling rate and high precision downhole measurements confirm a large discrepancy between the memory recorded downhole data and the estimated values based on the measurements made at the level of the hoisting equipment and the top-drive. The reason for these inconsistencies is simply that there are numerous physical processes taking place between the bit and the surface measurements that are normally not accounted for when WOB and TOB are estimated. This paper reviews and analyses the sources of these deviations and models the physical processes in order to quantify the precision for which the WOB and the TOB can be ascertained using solely surface measurements. Methods are also proposed that compensate for certain side effects by utilizing real-time torque and drag and hydraulic calculations.