## Summary

During drilling operations, drillstring vibrations cause many downhole dysfunctions, resulting in underperformance, equipment failure, and possibly wellbore damage. Current drillstring vibration mitigation solutions are generally located at a single site, either at the topdrive or close to the bit. However, because the sources of vibration excitation are distributed along the whole drillstring, these single-point vibration damping solutions do not succeed in removing all vibration modes. A distributed vibration damping solution is investigated, wherein damping subs are placed at intervals along the drillstring. The drillpipe rotates on bearings inside the damping subs, thereby reducing mechanical torque. The damping sub has a slightly larger diameter than the neighboring tool joints, lifting the drillstring off the borehole wall and thus eliminating most of the mechanical friction between the damping subs. Viscous friction is introduced in the form of rotary eddy current brakes within the damping subs, which cause the damping of torsional oscillations. The damping sub moves axially on spur wheels supported by bearings, thereby reducing drag forces. The decoupling of rotary and axial movement is a key element for removing a source of excitation. Simulations made with a 4*n* degree-of-freedom (DOF) transient torque and drag model confirm the expected results. The results of simulations for a use case based on a complex 3D trajectory are presented. With damping subs every 30 m and a damping coefficient of 20 N·s/rd, the topdrive torque is reduced to 40% of that of a plain drillstring, and torsional stability is obtained across the topdrive speed range of 60 to 180 rev/min. By comparison, conventional nonrotating pipe protectors (NRPPs) would have reduced the topdrive torque to 30%, but torsional stability would have not been achieved because the expected damping coefficient was less than 0.02 N·s/rd. The design of the damping sub relies on readily available technological components that can sustain very high pressures and temperatures. The damping subs are therefore compatible with high-pressure/high-temperature applications, including geothermal drilling.

## Introduction

Drilling operations are subject to many sources of nonproductive time, such as drilling fluid loss to the formation, formation fluid influx, stuck-pipe situations, formation instability, bottomhole assembly (BHA) tool failure, pipe washouts, twistoffs, and connection backoffs. Krygier et al. (2020) presents an analysis of the sources of nonproductive time in drilling operations. Poor hole cleaning is an often cited source of drilling incidents (Gulsrud et al. 2009), but drillstring vibrations are another common cause of nonproductive time. For example, drillstring vibration and more specifically high-frequency torsional oscillations (HFTO) (Bowler et al. 2016) have been identified as a common source of BHA element failure (Zhang et al. 2017). The relation between torsional stick/slip and HFTO has recently been analyzed (Hohl et al. 2020; Heinisch et al. 2020). Drillstring twistoff (Raap et al. 2011) or backoff of BHA connections (Dwars et al. 2019) can also be caused by drillstring vibrations. It has also been observed that severe tool joint wear can be caused by the forward whirl of the drillstring (Cayeux et al. 2018) while drilling hard stringers in tortuous wells. It is also suspected that drillstring vibration may be the source of wellbore instabilities (Plácido et al. 2002). Unsteady conditions at the bit tend to decrease drilling performance, either because the rock cutting process is constantly disturbed, resulting in a low rate of penetration or because the bit founder point (Dupriest and Koederitz 2005) is reduced as a result of vibration at the bit, leading to a corresponding reduction in rate of penetration.

At any curvilinear abscissa $s$ along the drillstring and relative to the initial position of the pipe in the borehole, the pipe may move axially by a distance $u$, rotate by an angle $\theta $ around the pipe centerline, move laterally by a deflection $w$ in a polar direction $\phi $, or change axial direction from its original orientation by the angles $\alpha v$ and $\alpha a$(Fig. 1). The pipe has six DOF that can be described by the displacement vector $u\u2192t=u,\theta ,w,\phi ,\alpha v,\alpha a$.

Each of those displacements can oscillate alone or in combination, but drillstring vibrations are usually characterized as axial ($u$), torsional ($\theta $), or lateral ($w,\phi $) oscillations. The sources of excitation can be varied, including bit-rock interaction (Richard et al. 2007), transition between static and kinetic friction (Aarsnes and Shor 2018; Wilson and Noynaert 2017), mass imbalance (Liu and Gao 2017), BHA stiffness and tight design (Chen et al. 2020), pipe crawling on the borehole wall (Wilson and Noynaert 2016), heave when drilling from a floating rig (Shen et al. 2017), lack of support when pulling out of the rathole in tapered borehole architecture (Bowler et al. 2016), interaction between hydraulic and mechanical forces, such as with surge pressures, and cutting particle grinding by tool joints (Cayeux et al. 2021). Furthermore, lateral vibrations can be localized on some portions of the drillstring and be observable neither at the surface nor in the BHA (Cayeux et al. 2018).

Current drillstring vibration mitigation solutions have mostly focused on damping torsional oscillations. The available solutions involve either damping torsional oscillations at the topdrive or introducing a damping element close to the bit, typically within the BHA. The first type of solution is designed to avoid the development of standing waves along the drillstring. They are either based on controlling the topdrive speed or torque to be out of phase with torsional waves propagating along the drillstring (Kyllingstad 2017) or they are based on the concept of impedance matching, in which the apparent torsional rigidity of the topdrive is matched with the one of the drillpipe such that torsional waves do not reflect back along the drillstring (Dwars 2015). These methods work relatively well for damping steady-state torsional oscillations so long as the drillstring is not too long or the drillpipe size is not too small (Shor 2016). Solutions involving a device that damps out oscillations within the BHA may be based either on passive solutions, such as the antistall tool (Selnes et al. 2009) or on active control utilizing an electromagneto-rheological fluid (Hutchinson 2013). Due to their position close to the bit, these solutions only affect excitations in the lower part of the drillstring, typically the bit-rock interaction, and only when the bit is on the bottom.

Frequency-tuned dampers have been proposed (Venugopal et al. 2016). They are typically composed of a spring combined with a dashpot that are tuned to absorb torsional vibrations at the natural frequency of the BHA. A rotary steerable system with a slow rotating housing (Jones et al. 2018) has been reported to damp HFTO as the slow rotating housing section behaves as a mass inertia/dashpot damper (Sugiura and Jones 2020). Another inertia/dashpot damper for HFTO utilizes an inertial ring rotating inside an outer housing fixed to the rotary steerable system (Wilson et al. 2022). There is a viscous fluid between the inertial ring and the housing that applies a viscous torque proportional to the difference of rotational speed of the inertial ring and its housing. Another variant consists of a sleeve kept stationary relative to the borehole, perhaps supporting the pads of a rotary steerable system and inside which the pipe body rotates. The rotation between the nonrotating sleeve and the pipe body may be a source of viscous friction with a viscous fluid filling the gap (Hohl and Mihajlovic 2021). This configuration results in increased viscous torque at the location of the damping element. Laboratory experiments and downhole recordings confirm the attenuation of HFTO based on this solution (Hohl et al. 2022). This latter solution has similarities with that described here, except that it does not address the decoupling of axial and torsional friction. These recent publications specifically focus on the damping of HFTO in the BHA and when the bit is on bottom and drilling.

A third possibility is to place one or several devices along the drillstring to damp out vibrations. Currently, such solutions have focused on deliberately introducing vibrations to assist in transmitting weight to the bit while employing a downhole motor for directional control (Barton et al. 2011; Azike-Akubue et al. 2012; Shor et al. 2015). There is also a solution that consists of placing many elements along the drillstring, namely NRPPs (Moore et al. 1996). An NRPP is a sleeve that can rotate on the pipe body. It is kept in position with collars attached to the pipe body on either side of the sleeve. Drilling fluid between the sleeve and the pipe body acts as a lubricant. The effect of NRPP is a substantial reduction in torque. However, drag forces act between the sleeve and the borehole. To reduce drag, the sleeve material can be designed with a low friction factor. In theory, the film of drilling fluid between the sleeve and the pipe body may provide viscous torque, but the effect on torsional damping will be minimal.

Because there are multiple excitation sources for drillstring vibration and several of them are not just limited to the drillstring ends but may be distributed along the entire drillstring, it seems logical that a drillstring vibration mitigation solution distributed along the drillstring might have a better effect on mitigating vibrations than solutions that are placed either at the top or bottom.

A vibration damping solution is proposed, which is distributed along the drillstring and attenuates torsional vibrations. The solution utilizes passive control and can be designed to tolerate harsh environments, such as high-pressure and high-temperature hydrocarbon production wells with temperatures reaching 200°C (Saxena et al. 2019), but also even more extreme temperatures including the one encountered during geothermal drilling, where the in-situ temperature can be in excess of 300°C (Hochstein and Sudarman 2015). Furthermore, the solution has a positive impact on energy consumption at the rigsite by reducing mechanical friction in torque and drag.

The concept has been described briefly in another publication (Ambrus et al. 2022). The latter article focused on the description of a torsional and axial transient torque and drag model modified to incorporate the effects of the proposed damping subs and an application where only a few damping subs were used. With only 5 to 10 damping subs, a very large damping coefficient is required to attenuate torsional oscillations effectively. The drawback of such a large damping coefficient is the necessity to use active control to limit its effect when the side force on the damping sub is small, as it would otherwise slip on the borehole wall. This paper details the design principles of the damping sub and the reasons behind the design choices. It focuses on the possibility of obtaining torsional stability over a wide range of topdrive speed using purely passive damping. A use case based on the regular distribution of the damping subs on every stand is analyzed, and a sensitivity study is performed on the effect of the damping subs at various operating speeds. However, the proposed damping sub design does not attempt to address the problem of attenuating HFTO.

The next two sections describe the physical principles of torsional and axial drillstring displacements and consequently explain the design principles of the proposed damping sub.

## Distributed Self-Attenuation of Torsional Vibrations

As a first approximation, only the rotational DOF (i.e., denoted $\theta $ in Fig. 1) will be considered. At any curvilinear abscissa $s$, the torque balance equation can be written (derivation in Appendix A):

where $\rho $ is the mass density of the material, $J$ is the polar moment of inertia of area, $\zeta $ is the structural damping coefficient, $\omega \tau $ is the natural frequency in the angular direction, $k\tau $ is an equivalent torsional spring constant, and $Cis,i\u2208[1,n]$ are $n$ external torques applied on the portion of pipe of length $ds$. The term $2\zeta \omega \tau k\tau \theta \u02d9p$ is the structural loss torque.

Among the various external sources of torque is the viscous friction torque ($C\upsilon $) between the drilling fluid and the pipe ( Appendix B). As most drilling fluids are non-Newtonian and more particularly shear-thinning, the viscous friction torque is not directly proportional to the angular speed. It increases rapidly at low shear rates and flattens out at high shear rates (Fig. 2b). As the structural loss and fluid viscous torques increase with pipe rotation speed, they represent sources of positive damping.

Mechanical friction is an important source of torque. When the pipe slips on the borehole wall at a relatively high speed, the resulting kinetic friction torque is approximately constant. However, when there is no slippage, the static friction torque will exactly balance all the other torques to prevent slippage. The static friction torque is greater than the kinetic friction torque. At low relative slipping speeds, the kinetic friction tends to increase as the slippage tends to zero ( Appendix C provides a more detailed explanation). Therefore, mechanical friction torque is a source of negative damping (Fig. 3a).

When cuttings are ground, the local torque on the pipe increases (Fig. 3b) due to the grinding torque ($Cg$), which tends to reduce the local rotary speed of the pipe ( Appendix D). As the rotary speed of the pipe influences whether cuttings particles being transported are on the high side or the low side of the borehole in deviated wells, this local reduction of rotary speed tends to increase the number of cuttings on the low side and therefore the number of cuttings that are ground up (Cayeux 2019). Thus, cuttings grinding has the potential for negative damping of torsional oscillations.

At the bottom of the hole, the torque on the bit is a function of the weight on bit (WOB), the rotary speed, the bit design characteristics, and the formation being drilled. The bit torque increases sharply when rotary speed increases over a narrow range close to zero, but when past a maximum bit torque, the torque decreases when the rotary speed increases further ( Appendix E). This second phase of the bit torque behavior is a source of negative damping (Fig. 3c).

At the top of the drillstring, the maximum topdrive torque capability is constant up to a certain rotary speed. Past this limit, the maximum topdrive torque ($Ctdmax$) decreases with the rotary speed (Cayeux 2018) ( Appendix F).

Here again, if the topdrive torque reaches the limit of the topdrive motor for an angular velocity larger than a threshold value, the topdrive response is responsible for a negative damping effect because the maximum available torque decreases with increasing rotary speeds (Fig. 3d).

From this overview of the terms in Eq. 1, it is apparent that there are many sources of negative damping, while the sources of positive damping are relatively weak, being limited to the drillstring material damping and the viscous friction exerted by the drilling fluid.

In deviated wells, at low topdrive speeds, a torsional wave can easily render a portion of the drillstring stationary, putting it into a static friction condition. Because more torque is required to overcome the static friction limit than is necessary with kinetic friction, stick/slip conditions will occur even with the bit off-bottom. This phenomenon has been observed on numerous occasions in downhole measurements from different wells. An initial approach to combating this effect is to reduce the mechanical friction torque at the contact points by introducing a device supported by bearings that has low mechanical friction, broadly similar to the idea of NRPP (Moore et al. 1996) (Fig. 4).

This approach can be further developed by introducing viscous friction between the sleeve and the pipe body to naturally damp down oscillations. Rotational viscous damping can be achieved by rotating a magnetic field around or inside a conductive nonmagnetic cylinder in the same way as an eddy current brake. According to Lenz’s law, a varying magnetic field induces an electromotive force in the conductor (Feynman et al. 1972):

where $\epsilon $ is the induced electromotive force and $\Phi B$ is the magnetic flux. A loop current is created in the conductor, which in turn generates a magnetic field that opposes the original magnetic field that created the loop current in first place. In a rotating configuration, this effect results in a braking torque. The typical torque response of an eddy current brake is shown in Fig. 5 (Wouterse 1991).

For low speeds, the Wouterse model can be expressed as (Wouterse 1991)

where $Fec$ is the braking force, $vt$ is the tangential velocity, $B0$ is the magnetic induction at zero velocity, $d$ is the conductor thickness, $\rho e$ is the specific resistance of the conductive material, and $A$ is a characteristic area relative to the surface of the conductor exposed to the magnetic field. In the low-speed region, braking torque is directly proportional to rotary speed and is therefore a source of positive damping. To maximize the eddy current force, it is necessary to use high-conductivity materials, such as copper or aluminum, with sufficient thickness and to maximize the surface area exposed to strong permanent magnets.

The magnetic field must be concentrated on the internal conductive cylinder to increase the generation of eddy currents while being as low as possible outside the damping element to minimize interaction with any metallic particles present in the drilling fluid or ferromagnetic components along the borehole, such as casing strings. Both requirements can be satisfied by configuring the magnets in a Halbach array and placing them into a sleeve. The cross section of the sleeve is decomposed into a polygon with $n$-edges. In each polygonal sector, a magnet is positioned with its field rotated by $2k\pi n$, where $k$ is the wave number (Soltner and Blümler 2010; Niamjan et al. 2021). If $k-1>0$, the magnetic field is concentrated inside the cylinder and very weak outside. If $k-1<0$, the magnetic field is concentrated outside the array and is almost absent inside. If $k=1$, there is no magnetic field. For $k$ = 2, the resulting field is uniform inside the sleeve (Fig. 6a). In the latter case, the magnetic field is (Skomski and Coey 1999)

where $ro$ and $ri$ are, respectively, the outer and inner radii of the magnets and $Mr$ is the ferromagnetic remanence of the magnets. Note that the magnetic field strength ($H$) is related to the magnetic flux ($B0$) through the relation $B0\mu =H$, where $\mu $ is the material permeability inside the cylinder. Therefore, the proportionality of the magnetic field to $ln\u2061rori$ also applies to the magnetic flux. When $rori>e$, where $e$ is the base of the natural logarithm, the resulting magnetic field is larger than that of the individual magnets, but one must be careful that a too strong internal magnetic field can result in demagnetizing the magnets. Fig. 6b shows a schematic view of the sleeve and its supporting pipe. Both the external part of the sleeve and the internal part of the pipe are made of a material that provides structural strength while being nonmagnetic. The internal pipe is clad with a cylinder of a highly conductive material, and the magnets are contained within the rotating sleeve.

The sleeve is supported on the pipe by tapered roller bearings which ensure low mechanical rotational friction while being capable of sustaining significant lateral and axial loads. The bearings can be sealed to avoid contamination from the surrounding drilling fluid (Fig. 7). While the seals introduce some mechanical friction, it is minimal compared to that of a standard tool joint acting on the borehole wall. The spur wheels and the possible displacement of the magnets in front of the conductive nonmagnetic cylinder will be explained in the next section.

To explore the effect of distance between the damping elements, the transfer of side forces from the borehole wall contact points to the damping element can be estimated from the deformation of the drillstring using the Rayleigh-Ritz method as first described by Walker (Walker 1986) and later reinvestigated by Sawaryn (Sawaryn 2012). Results for a drillstring composed of 5-in. drillpipe, S-135, NC50 with damping elements having a sleeve of diameter 7½-in. in a 12 ¼-in. borehole with an inclination of 90° are shown in Fig. 8. The side force is approximatively constant at every tool joint when there are no damping elements (Fig. 8a). When damping elements are positioned every 30 m, most of the side forces are supported by the damping elements and the lateral forces on the two neighboring tool joints are significantly reduced (Fig. 8b). When damping elements are spaced every 60 m, three of the five intermediate tool joints lose their contact with the borehole, while two of them get a larger side force than in the case without damping elements (Fig. 8c). Finally, when the damping elements are placed every 90 m (Fig. 8d), only two tool joints lose contact with the borehole, two tool joints have a lower lateral force than in the case without damping elements, and four tool joints have a larger side force than in the reference case of Fig. 8.

Depending on the proximity of the damping elements to each other, the side forces on the remaining tool joints are reduced. If there were damping elements at every second contact with the borehole, then no tool joints would be in contact with the borehole regardless of the inclination (Fig. 9 red curve). The calculations assume that the damping subs have no magnetic field (i.e., negligible torque between the sleeve and the pipe body of the damping sleeve). This case represents the best scenario for reducing torque but does not help with damping torsional vibrations. However, the greater the distance between the damping elements, the less the reduction of side forces contributing to mechanical friction. For a distance of 120 m (four stands of Range 3 drillpipe) between each damping element, the reduction of mechanical friction is in the range of 20–12%, depending on the inclination, while for one damping element per stand, the reduction is between 100% and 68%, depending on the inclination. This reduction in mechanical friction translates directly to a reduction in topdrive torque and, consequently, topdrive energy consumption.

The characteristics of the drillpipes and damping sleeves and drilling fluid used to calculate **Figs. 8 and 9 ** are summarized in Table 1.

Borehole diameter | 12.25 | in. |

Drilling fluid density | 1250 | kg/m^{3} |

Curvature | 0 | °/30 m |

Number of drillpipes | 50 | |

Drillpipe body outer diameter | 5 | in. |

Drillpipe body internal diameter | 4.276 | in. |

Drillpipe body length | 9 | m |

Tool joint outer diameter | 6.625 | in |

Tool joint internal diameter | 2.75 | in |

Tool joint length | 1 | m |

Damping sub body outer diameter | 5 | in. |

Damping sub body internal diameter | 4.276 | in. |

Damping sub body length | 2 | m |

Damping sub sleeve outer diameter | 7.5 | in. |

Damping sub sleeve length | 1 | m |

Damping sub tool joint outer diameter | 6.625 | m |

Damping sub tool joint internal diameter | 2.75 | in. |

Damping sub tool joint length | 1 | m |

Borehole diameter | 12.25 | in. |

Drilling fluid density | 1250 | kg/m^{3} |

Curvature | 0 | °/30 m |

Number of drillpipes | 50 | |

Drillpipe body outer diameter | 5 | in. |

Drillpipe body internal diameter | 4.276 | in. |

Drillpipe body length | 9 | m |

Tool joint outer diameter | 6.625 | in |

Tool joint internal diameter | 2.75 | in |

Tool joint length | 1 | m |

Damping sub body outer diameter | 5 | in. |

Damping sub body internal diameter | 4.276 | in. |

Damping sub body length | 2 | m |

Damping sub sleeve outer diameter | 7.5 | in. |

Damping sub sleeve length | 1 | m |

Damping sub tool joint outer diameter | 6.625 | m |

Damping sub tool joint internal diameter | 2.75 | in. |

Damping sub tool joint length | 1 | m |

Installing bearing-supported sleeves regularly along the drillstring reduces the side forces on the tool joints with a corresponding reduction in the mechanical friction. Furthermore, adding eddy current braking between the sleeve and the pipe increases viscous friction, which is useful for damping torsional oscillations. When the eddy current braking torque exceeds the maximum static frictional torque between the sleeve and the borehole, the sleeve will slip, and the resulting torque on the sleeve will equal the kinetic friction torque between the sleeve and the borehole, which will be almost constant except at very low slip velocities ( Appendix C):

where $rec$ is the internal radius of the magnets inside the sleeve. Because the left-hand side is independent of rotary speed, the right-hand side must also be constant regardless of rotary speed, meaning that $vt$ is constant. This condition is achieved by a variation of slip velocity between the sleeve and borehole such that the relative velocity between the pipe and sleeve stays constant: The faster the pipe rotates, the faster the sleeve rotates relative to the borehole **(Fig. 10).** The calculations are carried out with a magnetic field of 0.225 T, a magnet length of 0.33 m, a conductive material made of 5-mm-thick copper on a pipe diameter of 5 in. with a gap between the magnet and copper cylinder of 1 mm. When the sleeve slips on the borehole wall, the viscous damping effect disappears completely.

The static friction torque limit between the sleeve and the borehole depends on the normal force at the sleeve and the friction coefficient. In turn, the friction coefficient depends on the nature of the drilling fluid with values in the range of 0.1 to 0.25 for oil-based mud and between 0.2 and 0.35 for water-based mud. The side force on the sleeve is a function of the pipe linear weight, the drilling fluid density through the buoyancy force, the inclination, the borehole curvature, the proximity of the neighboring damping sleeves, and the tool joint diameters. When the drillstring moves axially along the borehole, the side force applied to the sleeve can vary substantially during a drilling operation because of the changes in the local inclination and borehole curvature.

Friction conditions on the sleeve can vary from small to large values, and for that reason the slipping torque limit can be very different from well to well and from drilling operation to drilling operation. On the high value side, a typical large side force on a tool joint is around 10 000 N in a curved section of wellbore. If the sleeve diameter is about 7½ in. and the drilling fluid is water-based with a kinetic friction factor reaching 0.35, then the slipping torque limit is approximately 330 N·m. With such a torque, at 180 rev/min, the power dissipated by the eddy current braking would be approximatively 6200 W, which suggests an upper limit for the eddy current braking effect.

For the low-contact friction force case, the normal force can typically be around 2500 N and the friction factor may be as low as 0.1, as when using an oil-based fluid. Then, the slipping torque limit for a 7½-in. sleeve would be approximately 24 N·m. The power dissipated by the eddy current brake at 180 rev/min would be approximately 450 W.

The resistivity of copper is about $1.71\xd710\u22128\Omega \u22c5m$. The magnetic field generated by strong permanent magnets made of neodymium (Curie temperature 310–400°C) is around 1–1.4 T, and for samarium (Curie temperature 800–1400°C) it is between 0.9 T and 1.15 T. Note that the Curie temperature is the temperature at which some materials lose their magnetic properties. Neodymium is therefore compatible with most standard oil and gas drilling applications. For high-temperature geothermal drilling operations, it may be necessary to use samarium magnets. If the conductive nonmagnetic material is 5 mm thick and attached to a 5-in. drillpipe, if there is a 1-mm clearance between the copper surface and the interior radius of the magnets, and if the magnets are 17.5-mm thick (i.e., $ri=69.5$ mm and $ro=87$ mm), then the magnetic flux is about 0.225 T (Eq. 4). In that case, the resulting eddy current braking torque at 180 rev/min is approximately 110 N·m for a magnet length of 0.2 m, which is consistent with the two bounds calculated above (24 –330 N·m). This result confirms that an eddy current braking sleeve producing braking torques compatible with downhole conditions is feasible with the readily available materials (i.e., copper, neodymium/samarium magnets, and with reasonable dimensions, e.g., 0.2 m in length on a 5-in. drillpipe).

It is legitimate to consider how the viscous damping provided by the eddy current brakes compares with that obtained with NRPPs. The damping provided by an NRPP arises from the rotation of the pipe inside the sleeve in the presence of drilling fluid in the gap between the sleeve and the pipe. The exact gap thickness is not known but can be assumed to be greater than 1 mm, because, as from drilling fluid rheometry, with a narrower gap, the rotation will not be smooth because the solid particles in the drilling fluid will obstruct the rotation. Drilling fluids are typically non-Newtonian and shear-thinning. A simple rheological model describing the viscous properties of drilling fluids is the power law model: $\tau =K\gamma \u02d9n$, where $\tau $ is the shear stress, $\gamma \u02d9$ is the shear rate, $K$ is the consistency index, and $n$ is the flow behavior index. As a crude approximation of the viscous torque induced by the drilling fluid in the gap between the pipe and the sleeve, it is possible to compute the torque of a bob rotating in a cup for a power law fluid (Bourgoyne et al. 1986):

where $Cnrpp$ is the torque generated by viscous forces between the pipe and the sleeve of the NRPP, $l$ is the length of the sleeve, $\omega $ is the angular speed, $ds$ is the sleeve internal diameter, $dp$ is the outer diameter of the pipe. Fig. 11a shows the 10th, 50th, and 90th percentiles for the probable viscous torque corresponding to a 12-in.-long NRPP clamped on a 5-in. drillpipe. The results are based on 159 rheograms of different drilling fluids measured with a scientific rheometer. **Figs. 11b and 11c ** show the probability distribution of the parameters of the power law rheological behavior for these drilling fluids. First, it is noticeable that the torque response is not directly proportional to the rotational speed. Second, the torque response depends on the viscosity of the drilling fluid. Third, the obtained viscous torque is two to three orders of magnitude lower than that obtained with the eddy current brake. NRPPs therefore have a very limited capability to damp torsional oscillations, as will be shown in the section Simulation Case Study.

## Combined Axial and Torsional Vibrations

So far, only the case of pipe rotation has been considered, where the eddy current braking force is tangential, oriented in the opposite direction to the rotation **(Fig. 12),** and its magnitude is proportional to the rotary speed. However, when the sleeve slides axially, friction between the sleeve and the borehole acts in the opposite direction to the movement between the sliding surfaces. If there is both axial and rotational motion of the sleeve relative to the borehole wall, the friction force, which is approximately constant, can be decomposed into a drag force and a friction torque. It must be recognized that the rotational kinetic friction, $\mu k,r$, is smaller than the kinetic friction, $\mu k$, since the tangential contact force must be combined vectorially with the tangential friction force: $\mu k,r=\mu k1+\mu k2$ (Kyllingstad 1993). Consequently, the rotational component of friction is proportional to: $\theta \u02d9srsu\u02d92+(1+\mu k2)\theta \u02d9s2rs2$, where $\theta \u02d9s$ is the angular velocity of the sleeve compared to the borehole and $u\u02d9$ is the axial velocity. In steady-state conditions, the rotational velocity of the sleeve relative to the borehole wall is such that it balances the eddy current torque generated by the rotation of the pipe inside the sleeve. Because the relative angular velocity between the magnets and the conductive surface on the pipe is $\theta \u02d9p-\theta \u02d9s$, where $\theta \u02d9p$ is the angular velocity of the pipe relative to the borehole, the following quartic equation can be solved in $\theta \u02d9s$:

Having determined $\theta \u02d9s$, it is then possible to calculate the tangential and axial friction forces (Fig. 12b).

Fig. 13 illustrates the effect of the axial velocity on the rotational friction component. A viscous torque, one which is proportional to drillpipe rotary speed, is available in the lower range of the drillpipe rotary speeds. On the plus side, this range increases with increasing axial velocity. However, on the negative side, there are drag forces at low drillpipe rotary speeds, meaning that in such a case, it is more difficult to transfer weight to the bit. The calculations are made with a sleeve diameter of 7½ in., a pipe diameter of 5 in., a copper conductive surface of 5-mm thickness and 0.5-m length, a magnetic flux of 0.225 T, a side force of 2500 N, and an oil-based mud friction factor of 0.2.

Using a similar method as used to derive Eq. 1, Newton’s second law of motion is applied in the axial direction ( Appendix G):

where $u$ is the axial displacement relative to unstressed conditions, $\omega a=ka\rho Ads$ is the natural frequency in the axial direction of the portion of pipe of length $ds$, and $Fis$, $i\u22081,m$ are $m$ external axial forces applied on the portion of pipe of length $ds$.

Finally, as a bottom boundary condition, there is the WOB, here denoted $Fb$.

Between the boundaries, at a contact point that is not a damping sleeve, the axial and torsional motions are therefore described by two coupled nonlinear equations:

where $Fag$ is the axial component of gravitation on a portion of pipe ( Appendix H), $Fbg$ is the buoyancy force on a portion of pipe ( Appendix H), $F\upsilon $ is the axial component of the viscous friction force ( Appendix I), $Fvp$ is the axial force engendered by viscous flow pressure drop on change of diameters (Appendix J), and $F\mu $ is the sliding friction force between the pipe and the borehole wall.

At a damping sleeve, there are two moving elements—the pipe and the sleeve. As soon as the sleeve slips on the borehole wall, for example, when there is an axial movement, their motions are described by

where $Ap$ and $As$ are, respectively, the cross-sectional area of the pipe and the sleeve, and $Jp$ and $Js$ are, respectively, the polar moment of inertia of area of the pipe and the sleeve. In Eq. 10, the fluid viscosity drag and torque are neglected on the sleeve. The cuttings grinding torque, $Cg$, applies to the sleeve and not the pipe.

The coupling through the mechanical friction can cause transfer of oscillations from axial to torsional and vice versa, and can be a source of positive or negative damping depending on the direction of movement as has been modeled and confirmed through in-situ measurements during ream-up/ream-down sequences (Cayeux et al. 2021).

It would therefore be beneficial to eliminate that coupling. One way to do this is to put spur wheels on the damping sub that roll when the sub moves axially. For a damping sub, axial friction reduces to that of the friction in the spur wheel bearings. This modification also solves the problem of weight transfer to the bit by significantly reducing the axial friction along the entire drillstring when the spacing between the damping subs is sufficiently small, typically one damping sub every 30 m. The spur wheel axles are tilted as shown in Fig. 7, and the sides of the spur wheels are rounded to increase the contact surface area with the borehole and therefore limit the additional strain applied to the wellbore wall. In this way, the configuration of the spur wheels opposes tangential slipping of the sleeve on the borehole, thereby allowing the sleeve to counteract the eddy current torque **(Fig. 14)**.

Assuming that the sleeve does not slip rotationally $(\theta \u02d9s=\theta \xa8s=0)$, the new equations of motion for the damping sub are

where $F\mu sb$ is the friction force at the level of the bearings of the spur wheels and $Cbh$ is the reaction torque between the sleeve and the borehole.

If the side force is very small, there is still a risk that the sleeve slips in the borehole. Prolonged periods of slippage may damage the spur wheels and degrade borehole quality. It is therefore desirable to introduce a mechanism to reduce eddy current braking if there is a risk that the sleeve slips in the borehole. This can be achieved by crenelating the conductive nonmagnetic cylinder attached to the pipe with a thick and a thin ring (Fig. 15). The magnet ring is attached to a carriage that translates inside the sleeve on two diametrically opposite linear rails. A ball screw controls the position of the carriage relative to the crenelated cylinder (Fig. 7a). The pitch of the ball screw results in auto-locking, so regardless of the axial force applied to the carriage, it cannot move. The carriage is only able to move when the ball screw is rotated, which prevents a change in configuration caused by axial acceleration of the damping sub.

If the magnet rings supported by the carriage are positioned across the thick conductive zone on the crenelated cylinder, the eddy current braking effect is at its maximum. When the magnet rings are positioned across the thin conductive zones, the eddy current braking effect is at its minimum. The position of the magnets relative to the crenelated conductive surface can be changed before running in hole by rotating the ball screw. Alternatively, an active controller may be used to displace the magnets such that the viscous damping is just sufficient to prevent the sleeve from slipping on the borehole wall.

## Simulation Case Study

A 4$n$ DOF transient stiff torque and drag model (Cayeux et al. 2023) has been enhanced to incorporate the effect of the damping subs. The model estimates the axial, torsional, and lateral displacements of the drillstring as a function of time. The key assumptions and limitations of the model used in the simulations are listed below:

The drillstring axis remains parallel to the wellbore at the contact points (shear deformation due to bending is assumed to be negligible).

For the lateral movement, a small displacement hypothesis is used, meaning that the model is not valid under buckling conditions.

The deformation of the string at each discretized node is assimilated to the first bending mode of a vibrating simply supported beam.

The positions of the contact points are evaluated continuously.

Shocks with the borehole wall are considered as well if the pipe rolls or slips on the wall.

At nodes where shocks occur, the lateral impact force is modeled assuming linear elastic contact with damping.

Fluid forces are included in the axial and lateral directions.

The bit-rock interaction includes a transient cutting process.

The axial movement of the damping sub takes place with negligible friction due to the presence of the spur wheels.

Rotational mechanical friction is applied to the sleeve of the damping sub, but the sleeve and pipe body are torsionally connected through a linear viscous damping force, $Fs$, corresponding to the effect of the eddy current brake:

where $kec$ is the eddy current-generated viscous damping coefficient.

For the case of a 0.2-m-long copper cylinder, 5-mm thick, on top of a 5-in. pipe body and a magnetic flux of 0.225 T, the eddy current viscous damping coefficient is $kec\u224886$ N·s/rd, and for a 0.5-m-long copper cylinder $kec\u2248214$ N·s/rd.

To illustrate the effect of the damping subs along the drillstring, consider Well A described by Cayeux et al. (Cayeux et al. 2021), a high-tortuosity horizontal well in which downhole measurements have shown frequent stick/slip events. (Fig. 16) shows the wellbore architecture, the horizontal and vertical views of the trajectory, and the drillstring and BHA dimensions used for drilling of the reservoir section of Well A. The tapered drillstring consists of 5⅞-in. drillpipe (7-in. tool joint outer diameter and 4¼-in. inner diameter), 5-in. DP (6⅝-in. tool joint outer diameter and 3¼-in. inner diameter) and 5-in. heavy-weight drillpipe (6⅝-in. tool joint outer diameter and 3-in. inner diameter). Table 2 lists the drilling fluid properties. The simulations assume damping subs with a 7½-in. sleeve outer diameter and 1-m length are placed every 30 m along the drillstring from the top of the BHA to the vertical section of the well, a total of 147 damping subs. In a first simulation, the eddy current viscous damping coefficient is set to zero, meaning that the drillpipe can rotate freely inside the supporting sleeves.

Drilling fluid density (kg/m^{3}) | 1258 |

Oil/water ratio | 75/25 |

Yield stress (Pa) | 3.35 |

Gel strength 10 seconds (Pa) | 4.79 |

Gel strength 10 minutes (Pa) | 6.22 |

Drilling fluid density (kg/m^{3}) | 1258 |

Oil/water ratio | 75/25 |

Yield stress (Pa) | 3.35 |

Gel strength 10 seconds (Pa) | 4.79 |

Gel strength 10 minutes (Pa) | 6.22 |

Fig. 17 shows an instantaneous view of several of the values calculated by the model along the drillstring. From left to right are rotary speed, axial velocity, lateral displacements in the $x^$ and $y^$ directions, radial displacement, bending moment magnitude, lateral contact angle relative to the low side of the hole, side force magnitude, and effective tension. The $x^$ and $y^$ directions are in the plane perpendicular to the borehole, and $y^$ points to the high side. The clearance between the borehole or casing and the drillstring is shown in the lateral and radial displacement tracks. The hatched pattern indicates the reduced clearances at nodes with tool joints and damping subs. A 5-m grid size was used in the discretization. The side force track shows that most of the drillstring is fully supported by the damping sleeves except above 3500 m, where there are some side forces on the tool joints.

Fig. 18 shows the time history of the simulation in Fig. 17, where the viscous damping coefficient, $kec,$ was set to zero, together with another run with the same input parameters but with $kec=5$ N·s/rd to illustrate the effect of viscous damping on the torsional oscillations. At time $t=0$, the topdrive is started and reaches 180 rev/min after a few seconds. Torsional oscillations are induced by the fast acceleration of the topdrive, but they are damped out in both cases. Thereafter, the drillstring is lowered, and at time $t\u22487seconds$, the bit starts to move downward. At time $t\u224825seconds$, the bit tags the bottom of the hole, and the WOB increases up to 8.6 kdaN while the rate of penetration reaches 50 m/h. As the WOB increases, bit torsional oscillations amplify to approximately $\xb150$ rev/min at time $t\u2248100seconds$ for the case without viscous damping ($kec=0)$). Despite a significant reduction of the topdrive torque due to the presence of the subs (around 68% compared to a drillstring configuration without subs), the absence of sufficient viscous damping along the drillstring fails to reduce drillstring oscillations. For the case with $kec=5$ N·s/rd, torsional oscillations at the bit remain at a low amplitude when increasing the WOB, and the oscillations induced by topdrive startup and tagging bottom are attenuated faster than in the case without viscous damping. In addition, the topdrive torque is about the same as in the case of no viscous damping, suggesting that the eddy current braking torque contributes only marginally to the total torque. A viscous damping coefficient of $kec=5$ N·s/rd is about 43 times smaller than what may be possible with a 0.5-m-long copper cylinder that is 5-mm thick in a 0.225 T magnetic field.

An analysis of the impact of the damping subs on surface torque reduction is presented in Fig. 19. The left-hand side of Fig. 19a shows topdrive torque as a function of surface rotary speed for the same well and conditions used in Fig. 17. The blue curve corresponds to the case where no damping subs are used (i.e., a drillstring utilizing only plain drill-pipes). The other curves are associated with different damping coefficients, ranging from zero (red) to 200 N·s/rd (black). Logically, the minimum torque is obtained when using the damping sub with a zero damping coefficient as the drillstring is mostly supported on bearings and no power is absorbed by eddy current braking. This would be roughly equivalent to using NRPPs with the topdrive torque approximately 30% of the reference value without any damping subs. The use of relatively small damping coefficients [5 N·s/rd (green) and 20 N·s/rd (cyan)] indicates marginally increased surface torques for the topdrive speed range than for the zero damping coefficient case, but still with a reduction in topdrive torque to about 40% of the reference value. However, with a large damping coefficient (200 N·s/rd), the surface torque is about 64% of the reference value at low topdrive rotary speeds and almost reaches the reference value at 180 rev/min, because the eddy current braking torque increases to that which would have been obtained by direct contact of the tool joints on the borehole. Fig. 19b shows the topdrive torque as a function of the damping coefficient for different surface rotary speeds in the same conditions as in Fig. 17. It is noticeable that for rotary speeds of 150 rev/min (cyan) and 180 rev/min (black), the surface torque decreases above a damping coefficient of 350N·s/rd and 250 N·s/rd, respectively. With large damping coefficients and high rotary speeds, the eddy current braking torque exceeds the static mechanical friction and therefore the sleeve slips on the borehole wall, at which point the torque ceases to increase and is governed by the kinetic friction, which is lower than the static friction, as shown earlier in Fig. 10.

The risk of slippage is illustrated in more detail in Fig. 20a, which shows the proportion of slipping sleeves as a function of the damping coefficient for various topdrive speeds in the same conditions as Fig. 17. Up to a damping coefficient of 50 N·s/rd, very few sleeves slip on the borehole wall until the topdrive speed reaches 180 rev/min, at which point about 12% of them slip. For damping coefficients above 300 N·s/rd, at 180 rev/min, almost all the damping sleeves slip on the borehole wall, which explains the flattening of the corresponding topdrive torque curve on Fig. 19b. More generally, above 50 N·s/rd, many damping sleeves slip on the borehole, increasing the risk of wear on the damping sleeve and formation or casing damage. It was shown earlier in Fig. 13 that even if the sleeve slips on the borehole wall, some damping effect can still be obtained as long as the axial speed stays small relative to the rotary speed.

To verify that this result remains true for a whole drillstring with many sleeves, the conditions of Fig. 17 are analyzed for torsional stability for a range of damping coefficients and topdrive rotary speeds. The drillstring is torsionally stable in the green segment of Fig. 20b. There are no visible torsional oscillations in either on- or off-bottom conditions, meaning that transients during topdrive startup or bit tagging bottom procedures are attenuated. Persistent torsional oscillations, such as stick/slip limit cycles or oscillations growing in amplitude over time, as shown in the simulation example of Fig. 18, occurring either on- or off-bottom are marked in red (i.e., torsional instability). It is noticeable that very low damping coefficients, such as 5 N·s/rd, are insufficient to totally damp out torsional oscillations across the whole spectrum of investigated topdrive speeds. The damping coefficient of NRPPs is expected to be lower than 0.02 N·s/rd (corresponding to the best case of Fig. 11) and therefore would be insufficient to obtain torsional stability at any standard topdrive speed. It is only above 20 N·s/rd that torsional stability is achieved for all topdrive speeds between 60 and 180 rev/min. Furthermore, there are torsional instabilities at very high damping coefficients (above 350 N·s/rd) and high topdrive speeds, over 150 rev/min, because most of the damping sleeves slip on the borehole wall, thus significantly reducing the effect of eddy current damping.

## Discussion

The simulation results indicate that damping sleeves can have a positive effect on topdrive torque reduction and vibration damping. Yet, it is important to choose a damping coefficient for the damping sub that is appropriate to the context of the operation. An acceptable damping coefficient can be determined by performing an analysis of torsional stability and the risk of slippage of the damping sleeves before the operation. If the damping sleeves are designed to allow the position of the magnet in front of the crenelated conductive surface to be adjusted, for instance, by rotating the ball screw (Fig. 7a), then it is possible to achieve both a large reduction in topdrive torque and attenuation of drillstring vibrations. The design of the damping sleeve remains very simple and compact. Based on the results of the simulations shown in the previous section, damping coefficients above 100 N·s/rd may be unnecessary, resulting in a permanent magnet length of about 0.2 m. Designing the adjustment mechanism to provide damping from 0% to 100% results in the length of the crenelated conductive surface being twice the length of the magnets, about 0.4 m. Allowing an additional 0.2 m on each side for the thrust bearings and spur wheels, the total length of the sleeve is about 0.8 m. Providing for the pin and box connections yields an approximate overall damping sub length of about 1.8 m. The damping sub could be manufactured as pup joints to be compatible with the weights and grades of standard drillstrings. As seen in the foregoing simulations, the additional weight of the damping sleeve has little impact on the overall loads, comprising at most 5% of the total length of the drillstring. Although the hydraulic simulations remain to be done, to a first approximation, the additional annular pressure losses should be tolerable, except for very tight margin cases, as the damping sleeve diameter will be about 10 to 15% larger than a tool joint and impacting, at most, 2% of the total length of the drillstring.

Restricting the installation to one damping sub per stand should ensure that the pipe-handling equipment will cope adequately with the additional length. An alternative would be to incorporate the damping sleeve as part of specially manufactured standard length drillpipe joints, but that would probably have drawbacks in terms of manufacturing costs and operational flexibility.

Ambrus et al. (2022) showed that vibration damping can be achieved with only two to six damping subs in the string but that their damping coefficient had to be in the range of 200 N·s/rd to be effective. However, there is a risk of slippage with such high damping coefficients (Fig. 20a), but this problem can be solved by adding active slippage control to the damping sleeve. Incorporating suitable sensors to measure pipe movement and a controller that actively positions the permanent magnets along the crenelated conductive surface could modulate the damping to prevent the sleeve from slipping on the borehole wall. The added complexity will result in the damping sub requiring electronic components and would be more expensive to manufacture and maintain. However, the added complexity is offset by the need for fewer damping subs in the string. With active damping, the reduced number of subs would limit the impact of their presence, but a greater length of the drillstring would be in direct contact with the borehole wall. Consequently, a solution based on a few damping subs will be effective for damping vibrations but only marginally reduce the topdrive torque.

## Conclusions

Drillstring vibration is the source of many costly drilling problems. The drillstring is a very slender mechanical structure that can oscillate easily. As the excitation sources are distributed along the entire drillstring, vibration mitigation solutions working at the boundaries, either at the top or bottom, only provide partial vibration attenuation.

The problem can be addressed by distributing vibration damping devices along the entire drillstring, which reduce the negative damping of mechanical friction and increase viscous damping. Damping subs, slightly larger in diameter than the drillpipe tool joints and comprising a sleeve rotating on bearings, can provide rotational viscous damping by means of an eddy current braking effect between the sleeve and the pipe body. The frictionless axial motion of the damping sub on bearing-supported spur wheels decouples axial and torsional motion, which eliminates a source of negative damping. Damping subs placed typically every stand along the deviated part of the well will effectively reduce torsional oscillations of the drillstring and reduce topdrive input torque. A design incorporating neodymium or samarium magnets can produce viscous damping coefficients as large as 214 N·s/rd with a 0.5-m-long permanent magnet assembly that allows for a compact design of the damping sub.

Simulations using a 4$n$ DOF stiff transient torque and drag model showed that distributed damping sleeves placed every 30 m would attenuate torsional drillstring vibrations in a complex well geometry based on a real case in which extreme vibrations had been experienced as indicated by recorded downhole measurements. In simulations performed with a damping coefficient as low as 20 N·s/rd, torsional stability was obtained across the topdrive speed range from 60 to 180 rev/min, while topdrive torque was reduced to 40% of that of a plain drillstring with the same operating conditions. By comparison, conventional NRPPs would have reduced the topdrive torque to 30% of the value from the plain drillstring case, but the expected damping coefficient of less than 0.02 N·s/rd would be insufficient to achieve torsional stability.

The eddy current damping sub design is based on proven mechanical components that can be manufactured to operate under high-pressure and -temperature conditions. It therefore has the potential to be a simple and robust solution for attenuating drillstring vibrations even in the harshest environments, including geothermal applications.

## Nomenclature

*A*area, [L

^{2}], m^{2}*A*_{epi}exposed cross-sectional area inside the string to the fluid pressure, [L

^{2}], m^{2}*A*_{epo}exposed cross-sectional area outside the string to the fluid pressure, [L

^{2}], m^{2}- $Ais$
inner cross-sectional area of the pipe at curvilinear abscissa

*s*, [L^{2}], m^{2} - $Aos$
outer cross-sectional area of the pipe at curvilinear abscissa

*s*, [L^{2}], m^{2} *A*_{p}cross-sectional area of the pipe, [L

^{2}], m^{2}*A*_{s}cross-sectional area of the sleeve, [L

^{2}], m^{2}*B*_{0}magnetic induction at zero velocity, [MT

^{−2}I^{−1}], T*C*_{a}Rayleigh linear damping coefficient for axial motion, [MLT

^{−1}], kg·m/s- $Cb0$
component of torque on bit that is independent of the bit rotary speed, [ML

^{2}T^{−2}], N·m *C*_{bh}reaction torque between the sleeve and the borehole, [ML

^{2}T^{−2}], N·m*C*_{e}torque related to the elasticity of the material, [ML

^{2}T^{−2}], N·m*C*_{g}grinding torque, [ML

^{2}T^{−2}], N·m- $Cis$
external torque applied on a portion of pipe length

*ds*, [ML^{2}T^{−2}], N·m *C*_{nrpp}torque generated by viscous forces between the pipe and a NRPP, [ML

^{2}T^{−2}], N·m*C*_{v}viscous friction torque between the drilling fluid and a portion of pipe of length

*ds*, [ML^{2}T^{−2}], N·m- $Ctdmax$
maximum topdrive torque, [ML

^{2}T^{−2}], N·m - $C\mu k$
kinetic mechanical friction torque, [ML

^{2}T^{−2}], N·m - $C\tau $
Rayleigh linear damping coefficient for torsional motion, [ML

^{2}T^{−1}], kg·m^{2}/s *d*conductor thickness, [L], m

*d*_{b}bit diameter, [L], m

*d*_{i}inner pipe diameter, [L], m

- $dii$
inner diameter, [L], m

*d*_{o}outer pipe diameter, [L], m

- $doi$
outer diameter, [L], m

*d*_{p}pipe outer diameter, [L], m

*d*_{s}sleeve internal diameter, [L], m

*E*Young's modulus, [ML

^{−1}T^{−2}], Pa*F*_{ag}axial component of gravitation on a portion of pipe, [MLT

^{−2}], N*F*_{b}weight on bit, [MLT

^{−2}], N*F*_{bg}buoyancy force on a portion of pipe, [MLT

^{−2}], N*F*_{ec}eddy current braking force, [MLT

^{−2}], N- $Fis$
external axial forces, [MLT

^{−2}], N *F*_{imp}impact force, [MLT

^{−2}], N- $F\u2192n$
normal force between two surfaces in contact, [MLT

^{−2}], N *F*_{s}viscous damping force between the pipe and the sleeve, [MLT

^{−2}], N*F*_{μ}friction force between the sliding surfaces, [MLT

^{−2}], N- $F\mu a$
friction force in the axial direction, [MLT

^{−2}], N - $F\u2192\mu k$
asymptotic kinetic friction force, [MLT

^{−2}], N - $F\u2192\mu s$
static friction force, [MLT

^{−2}], N - $F\mu sb$
friction force induced by the bearings of the spur wheels, [MLT

^{−2}], N *F*_{v}axial component of the viscous friction force, [MLT

^{−2}], N*F*_{vp}axial force engendered by viscous flow pressure drop on change of diameters, [MLT

^{−2}], N*g*gravitational acceleration, [LT

^{−2}], m/s^{2}*G*shear modulus, [ML

^{−1}T^{−2}], Pa- $G~$
visco-elastic shear modulus, [ML

^{−1}T^{−2}], Pa *H*magnetic field [IL

^{−1}], A/m*J*polar moment of inertia of area, [L

^{4}], m^{4}*J*_{p}polar moment of inertia of area of the pipe, [L

^{4}], m^{4}*J*_{s}polar moment of inertia of area of the sleeve, [L

^{4}], m^{4}*k*Halbach array wavenumber, dimensionless

*k*_{a}axial spring constant, [MT

^{−2}], N/m*k*_{ec}eddy current-generated viscous damping coefficient, [MLT

^{−1}], N·s/rd*k*_{τ}torsional spring constant, [ML

^{2}T^{−2}], N·m/rd*K*consistency index [ML

^{−1}T^{n−2}] Pa·s^{n}*l*length, [L], m

- $m\u02d9$
mass rate, [MT

^{−1}], kg/s *M*_{r}ferromagnetic remanence [IL

^{−1}], A/m*n*flow behavior index, dimensionless

*p*_{gi,s}hydrostatic pressure inside the pipe at curvilinear abscissa

*s*, [ML^{−1}T^{−2}], Pa*p*_{go,s}hydrostatic pressure outside the pipe at curvilinear abscissa

*s*, [ML^{−1}T^{−2}], Pa*q*exponent in the general form of the comminution law, dimensionless

*r*_{ec}internal radius of the magnets in the sleeve, [L], m

*r*_{i}inner radius of magnets in a Halbach array cylindrical configuration [L], m

*r*_{o}outer radius of magnets in a Halbach array cylindrical configuration [L], m

*r*_{s}radial distance between the axis of rotation and the sliding surfaces, [L], m

*R*electrical resistance of the topdrive rotor windings, [ML

^{2}T^{−3}I^{−2}], Ω*s*curvilinear abscissa, [L], m

*t*time, [T], seconds

- $t^s$
axial unit vector at the curvilinear abscissa, dimensionless

*u*axial displacement, [L], m

- $v\xafb$
rate of penetration, [LT

^{−1}], m/s *v*_{cs}critical Stribeck velocity, [LT

^{−1}], m/s- $vrsi$
rotational slip velocity, [LT

^{−1}], m/s - $v\u2192s$
velocity vector between two sliding surfaces, [LT

^{−1}], m/s *v*_{t}tangential velocity, [LT

^{−1}], m/s- $VE$
electrical tension in the rotor windings, [ML

^{2}T^{−3}I^{−1}], V *V*_{i}inner volume of a portion of pipe, [L

^{3}], m^{3}*V*_{o}outer volume of a portion of pipe, [L

^{3}], m^{3}*w*lateral displacement, [L], m

- $x^$
horizontal unit vector in a pipe cross section, dimensionless

*X*electrical impedance of the topdrive rotor windings, [ML

^{2}T^{−3}I^{−2}], Ω- $y^$
vertical unit vector in a pipe cross section, dimensionless

*α*stiffness-proportional damping coefficient [T], seconds

*α*_{a}azimuth of axial unit vector, [dimensionless], rd

*α*_{v}angle of axial unit vector compared to vertical, [dimensionless], rd

*β*inertia-proportional damping coefficient, [T

^{−1}], 1/s*γ*shear strain, dimensionless

*γ*_{θs}shear strain, dimensionless

- $\gamma \u02d9$
shear rate, [T

^{−1}], s^{−1} - $\Delta pva$
relative pressure inside the annulus compared to the hydrostatic pressure, [ML

^{−1}T^{−2}], Pa - $\Delta pvs$
relative pressure inside the string compared to the hydrostatic pressure, [ML

^{−1}T^{−2}], Pa *δ*characteristic length of a particle, [L], m

*ε*induced electromotive force, [ML

^{2}T^{−3}I^{−1}], V- $\u03f5$
net specific energy, [L

^{2}T^{−2}], J/kg *ε*_{a}axial strain, dimensionless

*Φ*_{B}magnetic flux, [ML

^{2}T^{−2}I^{−1}], V·s*σ*_{a}axial stress, [ML

^{−1}T^{−2}], Pa- $\u03d1$
inclination, [dimensionless], rd

*θ*angular position, [dimensionless], rd

*θ*_{p}angular position of a pipe, [dimensionless], rd

*θ*_{s}angular position of a sleeve, [dimensionless], rd

- $\theta \u02d9b$
bit angular velocity, [T

^{−1}], rd/s - $\theta \u02d9p$
angular velocity of the pipe compared to the borehole, [T

^{−1}], rd/s - $\theta \u02d9s$
angular velocity of the sleeve compared to the borehole, [T

^{−1}], rd/s - $\theta \u02d9TD$
angular velocity of the topdrive, [T

^{−1}], rd/s - $\theta \u02d9TD0$
threshold angular velocity for the topdrive constant maximum torque, [T

^{−1}], rd/s - $\eta \gamma \u02d9$
fluid viscosity at a shear rate, [ML

^{−1}T^{−2}], Pa·s - $\zeta $
structural damping coefficient, dimensionless

*ρ*mass density of the material, [ML

^{−3}], kg/m^{3}*ρ*_{i}mass density of the fluid on the inside of the portion of pipe, [ML

^{−3}], kg/m^{3}*ρ*_{o}mass density of the fluid on the outside of the portion of pipe, [ML

^{−3}], kg/m^{3}*ρ*_{e}specific resistance of the conductive material, [ML

^{3}T^{−3}I^{−2}], Ω·m*μ*permeability of the material inside the cylindrical uniform magnetic field [MLT

^{−2}I^{−2}], N/A^{2}*μ*_{k}kinetic friction factor, dimensionless

*μ*_{kb}Coulomb friction factor between the bit and the formation rock, dimensionless

*μ*_{k,r}rotational kinetic friction factor, dimensionless

*μ*_{s}static friction factor, dimensionless

*φ*polar angle of the deflection, [dimensionless], rd

*τ*shear stress, [ML

^{−1}T^{−2}], Pa*τ*_{v}shear stress exerted by the fluid at the surface of the fluid, [ML

^{−1}T^{−2}], Pa*τ*_{θs}shear stress, [ML

^{−1}T^{−2}], Pa*ω*angular speed, [T

^{−1}], rd/s*ω*_{a}natural frequency in the axial direction, [T

^{−1}], rd/s*ω*_{τ}natural frequency in the angular direction, [T

^{−1}], rd/s

## Acknowledgments

This work is part of the Center for Research-based Innovation DigiWells: Digital Well Center for Value Creation, Competitiveness and Minimum Environmental Footprint (NFR SFI project no. 309589, DigiWells.no). The center is a cooperation of NORCE Norwegian Research Centre, the University of Stavanger, the Norwegian University of Science and Technology (NTNU), and the University of Bergen and funded by the Research Council of Norway, Aker BP, ConocoPhillips, Equinor, Lundin, TotalEnergies, Vår Energi, and Wintershall-DEA.

The authors extend a special thanks to John Thorogood for very fruitful discussions and valuable inputs during the preparation of this manuscript.

### Appendix A

Supposing the material used in the drillstring obeys Hooke’s law (Landau et al. 1986):

where $\tau \theta s$ is the shear stress, $G$ is the shear modulus, and $\gamma \theta s=r\u2202\theta \u2202s$ is the shear strain at a radial distance $r$ from the center of rotation. The shear stress is related to the torque by (Landau et al. 1986)

where $Ce$ is the torque related to the elasticity of the material and $J$ is the polar moment of inertia of area. For a thin hollow cylinder, $J=\pi 32(do4\u2212di4)$, where $do$ and $di$ are, respectively, the outer and inner pipe diameters. Let us write Newton’s second law of motion for a portion of pipe of length $ds$:

where $\theta p$ is the angular position of the pipe, $\rho $ is the mass density of the material, and $k\tau =GJds$ is an equivalent torsional spring constant.

In practice, the response of drillstring elements to strain variations exhibits hysteresis, behaving like viscoelastic materials. In linear viscoelasticity theory, the shear modulus is expressed as a complex number, $G~$, for which a simple expression is

where $\zeta $ is the structural damping coefficient and $i=-1$. Typical values of $\zeta $ are between 0.03 and 0.05 for steel (Han et al. 2013). Utilizing a Rayleigh linear damping formulation, Eq. A-3 is extended to include a structural damping term:

where $c\tau $ is a linear combination of the stiffness and inertia:

with $\alpha $ being the stiffness-proportional damping coefficient and $\beta $ being the inertia-proportional damping. If we consider that structural damping is dominated by the stiffness term (i.e., $\beta \u22480$), we obtain (Fig. 2a):

Eq. A-7 can be matched to the damped harmonic oscillator equation (i.e., $x\xa8+2\zeta \omega x\u02d9+\omega 2x=0$), resulting in:

### Appendix B

Among the various external sources of torque is the viscous friction between the drilling fluid and the pipe:

where $\tau \upsilon $ is the shear stress at the wall exerted by the fluid. For a generalized Newtonian fluid, the shear stress can be expressed as

where $\eta \gamma \u02d9$ is an effective fluid viscosity at a shear rate $\gamma \u02d9$. The shear rate is the second invariant of the rate of strain tensor. The latter is a function of the fluid velocity gradient and is therefore influenced by the rotary speed of the pipe. The shear stress at the wall of the pipe increases when the shear rate increases, for example, if the rotary speed rises.

### Appendix C

The simplest form for mechanical friction is the Coulomb friction:

where $F\u2192\mu k$ is the asymptotic kinetic friction force, $\mu k$ is the kinetic friction factor, $F\u2192n$ is the normal force between the two sliding surfaces, and $v\u2192s$ is the velocity vector of the sliding surfaces. In the case of relative rotation between the sliding surfaces, the kinetic friction force generates a mechanical friction torque, $C\mu k$:

where $rs$ is the radial distance between the axis of rotation and the sliding surfaces. The kinetic friction torque is therefore proportional to the local side force at the contact point and is oriented opposite to the direction of movement.

As a first approximation, consider a pseudostatic deformation of the drillstring. In practice, the normal forces depend on the mechanical configuration, the local curvature, the inclination and diameter of the borehole, and the effect of buoyancy. The side forces at the contact points are therefore relatively constant for the time during which the drillstring remains around a given depth. For pure rotation when the direction of rotation remains constant and rotation does not stop, the kinetic mechanical torque can be considered constant; it is not a source of damping.

However, when the drillstring does not move, static friction applies. The stiction force, $F\u2192\mu s$, balances exactly all the other forces to keep the two surfaces from sliding on each other. Stiction cannot exceed a static friction force limit defined as

where $\mu s$ is the static friction factor. The static friction factor is always larger than the kinetic friction factor. For low sliding velocities, the kinetic friction force is less than the static friction and greater than the asymptotic kinetic friction. This transition from static to asymptotic kinetic friction conditions can be described by the Stribeck model (Stribeck 1902):

where $F\mu $ is the kinetic friction force and $vcs$ is the critical Stribeck velocity.

### Appendix D

Another source of external torque results from the grinding of drilled cuttings trapped between the tool joint and borehole wall. Grinding of solid particles consumes power. The general form of the comminution law is (Walker and Shaw 1954)

where $\u03f5$ is the net specific energy, $\delta $ is the characteristic dimension of the particle, $\xi $ is a constant that characterizes the material, and $q$ is an exponent that depends on the original size of the particles being ground. Grinding of particles with a pipe of outer diameter $do$ requires additional torque:

where $m\u02d9$ is the mass rate of cuttings passing under the pipe.

### Appendix E

At the bottom of the hole, the torque on bit is a function of the WOB, the rotary speed, the bit design characteristics, and the formation being drilled. Pessier and Fear (Pessier and Fear 1992) describe the relationship between bit torque and WOB using a Coulomb friction model:

where $Cb0$ is the component of the torque on bit that is independent of rotary speed, $\mu kb$ is the Coulomb friction factor between the bit and the rock, $db$ is the bit diameter, and $Fb$ is the WOB. Ritto et al. (Ritto et al. 2017) have studied several recorded downhole measurements of bit torque as a function of rotary speed and found that the following empirical relationship matches the observations fairly well:

where $Cb$ is the torque on bit and $b0$ , $b1$ , $b2$, and $b3$ are parameters of the model.

### Appendix F

At the top of the drillstring, the maximum topdrive torque capability is constant up to a certain rotary speed. Past this limit, the maximum topdrive torque ($Ctdmax$) decreases with the rotary speed (Cayeux 2018):

where $\theta \u02d9TD$ is the angular velocity of the topdrive, $\theta \u02d9TD0$ is the angular velocity threshold of the topdrive, and $VE$, $R$, and $X$ are, respectively, the electrical tension, resistance, and impedance of the rotor windings.

### Appendix G

In the axial direction, assuming that the drillstring material obeys Hooke’s law (Landau et al. 1986), the axial stress is related to the axial strain by:

where $\sigma a$ is the axial stress, $E$ is the Young’s modulus, and $\epsilon a$ is the axial strain. By integrating over a cross section, we obtain:

where $A$ is the cross-sectional area and $u$ is the axial displacement. Replacing $EAds$ by $ka$, an equivalent linear spring constant, the equation of motion in the axial direction is

As discussed for the torsional motion, there is a material damping arising from internal friction in the material and Eq. G-3 can be expanded to:

### Appendix H

Gravity acts on the drillstring and its axial component is:

where $Vo$ and $Vi$ are, respectively, the outer and inner volume of the portion of pipe, $g$ is the gravitational acceleration, and $\u03d1$ is the drillstring inclination at curvilinear abscissa $s$.

There is also a buoyancy force, $Fbg$, applied to the portion of pipe by the drilling fluid, resulting from the increase of pressure in the fluid due to the gravitational field (Cayeux et al. 2022):

where $\rho o$ and $\rho i$ are, respectively, the mass density of fluid on the outside and inside of the portion of pipe, $t^s$ is the axial unit vector at curvilinear abscissa $s$, $Aos$ and $Ais$ are, respectively, the outer and inner pipe cross-sectional area of the pipe at curvilinear abscissa $s$, and $pgo,s$ and $pgi,s$ are, respectively, the hydrostatic pressure on the outside and inside of the pipe at curvilinear abscissa $s$.

### Appendix I

As with the torsional motion, there is an axial viscous force associated with the shear stress at the wall between the drilling fluid and the pipe:

and there is an axial force engendered by the viscous pressure loss when the fluid passes around a change of pipe diameters (Cayeux et al. 2021):

where $\u2206pvs$ and $\u2206pvi$ are, respectively, the inside and outside pressure difference between the hydrodynamic and hydrostatic pressures at that location, $Aepi$ and $Aepo$ are, respectively, the area of the pipe that is exposed to the fluid pressure inside and outside of the pipe where there is a change of diameter.

## Article History

Original SPE manuscript received for review 7 December 2022. Revised manuscript received for review 7 February 2023. Paper (SPE 214675) peer approved 27 February 2023.