Nonlinear Dyn (2019) 98:2631–2650https://doi.org/10.1007/s11071-019-05075-6

ORIGINAL PAPER

Modelling and dynamic analysis of an anti-stall tool in adrilling system including spatial friction

Roeland Wildemans · Arviandy Aribowo ·Emmanuel Detournay · Nathan van de Wouw

Received: 1 February 2019 / Accepted: 15 June 2019 / Published online: 27 June 2019© The Author(s) 2019

Abstract This paper investigates the effects of adown-hole anti-stall tool (AST) in deviated wellson the drilling performance of a rotary drilling sys-tem. Deviated wells typically induce frictional con-tact between the drill-string and the borehole, whichaffects the drill-string dynamics. In order to study theinfluence of such frictional effects on the effective-ness of the AST in improving the rate-of-penetrationand drilling efficiency, a model-based approach is pro-posed. A dynamic model with coupled axial and tor-sional dynamics of a drilling system including thedown-hole tool in an inclined well is constructed. Fur-thermore, the frictional contact between the drill-stringand the borehole is modelled by a set-valued spatialCoulomb friction law affecting both the axial and tor-sional dynamics. These dynamics are described bystate-dependent delay differential inclusions. Numer-ical analysis of this model shows that the rate-of-penetration and drilling efficiency increases by inclu-sion of the AST, both in the case with and without spa-tialCoulomb friction. Furthermore, a parametric designstudy of the AST in different inclined drilling scenariosis performed. This study reveals a design for the AST,

R. Wildemans (B) · A. G. Aribowo · N. van de WouwDepartment of Mechanical Engineering, EindhovenUniversity of Technology, Eindhoven, The Netherlandse-mail: r.wildemans@tue.nl

E. Detournay · N. van de WouwDepartment of Civil, Environment and Geo Engineering,University of Minnesota, Minneapolis, USA

which gives optimal drilling efficiency, robustly over abroad range of inclined drilling scenarios.

Keywords Drill-string dynamics · Set-valued forcelaws · Spatial Coulomb friction · State-dependentdelay differential inclusions

1 Introduction

Drilling is used for the exploration of oil, gas, min-erals and increasingly for geothermal energy. Currentdrilling operations are challenging as often complex,deviated wells need to be drilled, for oil and gas explo-ration, while, for geothermal applications, drilling intohard rock is required. Improving the efficiency of thesedrilling operations will significantly reduce the costs.Particularly in geothermal drilling operations, where30–50%of the total development costs are fromdrilling[4,6,14]. Therefore, the development of new technolo-gies to improve drilling performance is key to increasethe economic feasibility of geothermal drilling opera-tions.

Rotary drilling with polycrystalline diamond com-pact (PDC) bits is widely accepted as most efficientfor exploration and production drilling operations.Figure 1 depicts the major components of a rotarydrilling system: rig, drill-string, bottom hole assembly(BHA), including stabilizers and down-hole tools, anddrill-bit. In a rotary drilling system, a key indicator ofits efficiency is the rate-of-penetration (ROP), which is

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Fig. 1 Schematic overview of a drilling system in an inclinedwell. (Adapted from Marck [26])

the speed at which the bit is drilling into the sub-surfaceformation. In order to optimize theROP, suitable valuesof the hook-load, which translates into an axial force onthe bit referred to as weight-on-bit (WOB), and rota-tion speed (RPM) imposed at the surface (rig) mustbe set by drilling operators. For this ROP optimiza-tion purpose, the Bourgoyne and Young ROP model iswidely used in industry [8,30]. This model takes intoaccount different aspects, such as formation strength,WOB, angular velocity and bit wear, which influencethe ROP. Based on the optimal operational parameters,the ROP is then optimized by using various controllers[7]. However, the dynamics of the drill-string, the bit–rock interaction and the (frictional) contact along thedrill-string in a deviated well are not taken into accountin this modelling approach. A different approach forROP improvement is the selection of drill-string com-ponents, such as drill-bits or down-hole tools located atthe BHA of the drill-string [4,5,15]. Down-hole toolscan be either active [5,15] or passive [4,31,37]. Activetools typically provide axial excitation during drilling,which can alter the effect of axial friction and con-sequently improve the drilling efficiency. The currentpaper aims to model and analyse the effect of a passive

down-hole tool on the drilling efficiency in presence offriction between the drill-string and boreholewall. Thiswork is motivated by field data that show evidence thatthe AST can increase the drilling efficiency in terms ofROP, also in deviated wells [31,33].

In particular, this paper focuses on themodelling andanalysis of the couplingof axial and torsional vibrationsin drill-string dynamics, and studies how a down-holetool, called the anti-stall tool (AST) [31], affects thedrilling performance (in terms of ROP) in a deviatedborehole. Previous works, e.g., [9,10,32] have shownthat a rate-independent bit–rock interactionmodel [11],including both cutting and frictional contact processes,is essential in the coupling between the axial and tor-sional dynamics. In the current paper, we also pursuesuch type of modelling approach, further motivated bythe fact that the AST also operates by coupling theaxial and torsional dynamics [31,37]. The dynamics ofdrill-string systems, including such rate-independentbit–rock interaction model, has been described by avariety of dynamical models. In [9,10,17,19,20,23–25,28,29,32] lumped-parameter models for the axial-torsional drill-string dynamics have been proposed,while in [2,3,13,16] both finite-element based and dis-tributed models have been developed. These modelshave been employed to study instabilities and axial andtorsional vibrations of drill-string systems, and recentlyto investigate the effect of the AST on the ROP [37].However, the effect of friction between the drill-stringand the borehole due to deviated well scenarios has notyet been taken into account.

The modelling of the frictional contact along thedrill-string has been considered extensively in the scopeof so-called torque and drag models [1,21,34]. Themagnitude of the frictional forces mainly depends onthe normal force acting between the drill-string andthe borehole wall. Hence, in highly deviated wells theeffects of this friction indeed becomes more promi-nent, because the drill-string is resting under its ownweight on the borehole wall. However, in these modelsno vibrational dynamics and down-hole tooling havebeen taken into account, while it has been shown thatthe functioning of the AST is intrinsically related to thedrill-string dynamics [37].

Thiswork builds on and extends the developments in[37] by modelling and analysis of the (frictional) con-tact between the BHA and the borehole wall, therebybroadening its applicability to deviated well scenarios.The main contributions of this paper are as follows:

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Modelling and dynamic analysis of an anti-stall tool 2633

– Firstly, a benchmark model of the drill-stringdynamics (without AST) and a drill-string modelincluding AST, both with spatial friction betweenthe borehole and drill-string, are developed.Herein,both unilateral contact and frictional characteris-tics of the bit–rock interaction and the spatial fric-tion between the borehole and drill-string have beenmodelled by set-valued force laws. This allows fora unified treatment of these nonlinear model fea-tures in a time-stepping-based simulation tool forthe resulting delay differential inclusion model;

– Secondly, a model-based analysis of the effect offrictional contact between BHA and borehole wallon the drilling performance (in terms of ROP anddrilling efficiency) is performed;

– Finally, a parametric performance study on theASTdesign is performed leading to an optimal design tomaintain a high drilling efficiency, which is robustfor a wide range of deviated wells.

The outline of this paper is as follows. In Sect. 2, thedynamic models of a drill-string system without andwith AST for drilling in a deviated well are derived. InSect. 3, a dynamic analysis is performedwith a focus onthe effect of the spatial frictional contact on drilling per-formance (ROP and drilling efficiency; for both caseswithout and with AST). Subsequently, a parametricdesign study on the AST design is presented in Sect. 4.Finally, conclusions are drawn in Sect. 5.

2 Dynamic modelling

In this section, two models of a drill-string includingspatial friction between the BHA and the borehole arepresented. In Sect. 2.1, two drill-string models, withand without AST, are introduced, enabling the com-parative analysis of systems with and without the AST.In Sect. 2.2, the bit–rock interactionmodel is discussed.In Sect. 2.3, the model that describes the frictional con-tact between the borehole and the BHA is presented.Finally, in Sect. 2.4 the two dynamic models, with andwithout AST, are expressed in dimensionless perturba-tion coordinates around their nominal solutions in orderto identify a minimum set of parameters characterizingthe dynamics.

A total overview of a drilling system in a deviatedwell is depicted in Fig. 1. A typical drilling systemis operated from the rig located at the surface, wherethe top-drive equipment sets the angular speed and

adjusts the weight by regulating the hook-load. Theseoperational conditions are transmitted via slender drill-pipes and a BHA to the drill-bit. The BHA is specifi-cally designed to fulfil a particular drilling objective(s)based on the sub-surface geological conditions and canbe composed by several down-hole components, suchas stabilizer(s), rotary steerable system (RSS), loggingtools, mud motor(s), etc. Due to the larger diameter ofthe stabilizers compared to the rest of the BHA com-ponents, the stabilizers are in contact with the bore-hole wall, which consequently results in additionalfriction affecting the drill-string dynamics. The phys-ical aspects to be considered in the modelling are theboundary conditions at the rig, the drill-string dynam-ics including the AST, the bit–rock interaction as thedown-hole boundary conditions and the frictional con-tact between the stabilizers and the borehole.

2.1 Drill-string dynamics

In this section, the dynamic models of the drill-stringsystems are presented. First the benchmark model,excluding AST, is discussed. Thereafter, the modelincluding AST is discussed.

2.1.1 Benchmark drill-string dynamics

In Fig. 2, the lumped-parameter benchmarkmodel (i.e.,without AST) is schematically depicted. The axial andtorsional dynamics are described by this model, whichconsists of two degrees-of-freedom (DOFs), namelythe axial displacement of the bit Ub and the angulardisplacement of the bit Φb.

At the rig, the boundary conditions are given by animposed constant angular velocity Ω0 and a constantupward force H0, the so-called hook-load. The totalmass and inertia of the drill-string including the BHAare lumped in the discrete mass M and inertia I . Thetorsional stiffness of the drill-pipes is modelled as atorsional spring with stiffness Cp. The viscous frictionalong the drill-string and BHA in axial and angulardirections are characterized by the parameters D andDΦ , respectively. The parameters λTa and λTt are asso-ciated with the spatial Coulomb friction between thestabilizers and the borehole in axial and torsional direc-tion, respectively, which is discussed in more detail inSect. 2.3. The weight acting on the bit is denoted byWand the torque acting on the bit is denoted by T .

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Fig. 2 Schematic representation of the benchmark model

The bit–rock interaction model, which is discussedin more detail in Sect. 2.2, relates the weight-on-bit(WOB) W and the torque-on-bit (TOB) T to the axialand angular motion of the bit. This bit–rock law con-siders two independent processes, namely a pure cut-ting process and a frictional contact. Hence, the forceand torque can be decomposed in a cutting and fric-tional component, denoted by the superscripts c and f ,respectively, i.e., for the total WOB W = Wc + W f ,and for the total TOB T = T c + T f . The force andtorque contributions associated with the wearflat willfrom now on be denoted as follows

W f = −λba , T f = −λbt . (1)

By using a Lagrangian approach, the equations ofmotion (EOMs) for this model are obtained. In general,these can be written in the following form:

Mq − h(t,q, q) = Wλ, (2)

where q represent the column with generalized coordi-nates. M is the mass matrix and the column h(t,q, q)

contains all generalized forces except the friction forces(both due to frictional contacts at the bit and betweenthe borehole and the stabilizers at the BHA). The vec-tor λ contains the generalized forces associated withthe set-valued force laws, characterizing both due tofrictional contacts at the bit and between the boreholeand the stabilizers at the BHA, see Sects. 2.2 and 2.3,and the matrix W contains the associated generalizedforce directions. In case of the benchmark model with

Fig. 3 The working principle of the anti-stall tool (AST): anincrease in torque (M2) will cause a contraction (S) to off-loadthe weight from the cutters (F2) [4]

the generalized coordinates q = [Ub Φb]T, this resultsin the following matrices and columns in (2):

M =[M 00 I

]

h(t,q, q) =[ − DUb − H0 + Ws − Wc

− DΦΦb + Cp(Ω0t − Φb) − T c

](3)

W =[1 0 1 00 1 0 R

]

λ = [λba λbt λTa λTt

]Twhere Wc and T c satisfying (6), (7) and (8) and λ sat-isfying (9), (12) and (25) and Ws represents the sub-merged weight of the drill-string. A detailed derivationof the equations of motion can be found in [38].

2.1.2 Drill-string dynamics including AST

TheAST is designed to influence the coupling betweenthe axial and torsional displacement. The AST consistsof two tool bodies connected by a helical spline and anaxial internal spring, as viewed in Fig. 3. According toSelnes et al. [33], the working principle of the tool is

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Modelling and dynamic analysis of an anti-stall tool 2635

Fig. 4 Schematic representation of the drill-stringmodel includ-ing AST

that a torsional load with sufficient magnitude to over-come the compressed spring will make the upper toolbody, with the internal helical spline, rotate on the mat-ing lower body. When the upper and lower part screwtogether in this manner, the tool telescopically con-tracts. Consequently, this results in an adjustment of theaxial and torsional loading acting on the bit. Hence, thetool prevents dynamic forces from reaching destructivelevels.

In Fig. 4, the lumped-parameter model includingAST is schematically depicted. The AST separates thedrill-string in two parts, where the coordinates U andΦ are related to the displacement of the top part andUb

and Φb to displacement of the part below the tool. Themass and inertia of the drill-string including the partof the BHA above the tool are lumped into a discretemassMa and inertia Ia , while themass and inertia of thepart of the BHA below the tool are lumped into a dis-crete mass Mb and inertia Ib. The torsional stiffness ofthe drill-pipes and the viscous friction components areidentical to the benchmark model. However, the axialviscous friction only acts on the part above the tool.The viscous friction in the angular direction is mod-

elled by two dampers characterized by DΦ and DΦb .The spatial friction acts both on the DOFs above thetool and on the DOF at the bit, and the distribution ofthe friction between the two parts of the drill-string isdenoted by Δ ∈ [0, 1], as introduced in (22) and (23).The radius of the stabilizer below the tool is assumed tobe the same as above the tool. Let us now introduce themodel of the AST, which introduces an additional axialspring Kb and damper Db, see Fig. 4. Furthermore, thehelical spline in the tool introduces a kinematic con-straint, which is characterized by the lead p, lead angleβ and the radius rspline of the helical spline, and can bewritten as

U −Ub = p

2πrspline(Φrspline − Φbrspline)

= p

2π(Φ − Φb) =: α(Φ − Φb).

(4)

HereinU (Φ) andUb (Φb) represent the axial (angular)positions above the tool and at the bit, respectively (seeFig. 4). The lead is given by p = 2πrspline tan β.

The generalized coordinates of the model includingAST are given by qc = [U Φ Ub Φb]T. However, dueto the kinematic, holonomic constraint of the AST, thismodel can be alternatively formulated in terms of threeindependent generalized coordinatesq = [U Ub Φb]T.This coordinate transformation is discussed in detail in“Appendix A”. Using a Lagrangian approach for sys-tems with constraints and after eliminating the DOFΦ,the obtained EOMs can be written in the general formof (2), with the following matrices and columns:

M =

⎡⎢⎢⎣Ma + 1

α2 Ia − 1α2 Ia

1α Ia

− 1α2 Ia

1α2 Ia + Mb − 1

α Ia1α Ia − 1

α Ia Ia + Ib

⎤⎥⎥⎦

h(t,q, q) =

⎡⎢⎢⎣

− Kb(U −Ub) − 1αCpY − 1

α DΦ Y − DU

Kb(U −Ub) + 1αCpY + 1

α DΦ Y

−CpY − DΦ Y − DΦb Φb

− Db(U − Ub) + 1αCpΩ0t + Ws − H0

+ Db(U − Ub) − Wc − 1αCpΩ0t + Wbs

− T c + CpΩ0t

⎤⎥⎥⎦ (5)

W =⎡⎢⎣0 0 1 R

α 0 0

1 0 0 − Rα 1 0

0 1 0 R 0 R

⎤⎥⎦

λ = [λba λbt λTa λTt λTba λTbt

]Twith the auxiliary variables Y = Φb + 1

α(U −Ub) and

Y = Φb + 1α(U − Ub) in the expression for h(t,q, q),

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whileWc and T c satisfy (6), (7) and (8), λ satisfies (9),(12) and (25). The parameters Ws and Wbs denote thesubmerged weights of the drill-string parts above andbelow the tool, respectively. A detailed derivation ofthe equations of motion can be found in [38].

2.2 Bit–rock interaction model

In this paper, the rate-independent bit–rock interac-tion law as introduced in [11,12] is adopted, whichrelates the WOB and the TOB to the axial and angularmotions of the bit. The bit–rock interaction involvestwo independent processes: a pure cutting process tak-ing place at the front of the cutters and a frictionalcontact between the rock and the so-called wearflatunderneath the cutters. According to Detournay et al.[11,12], the cutting contributions for a bit consistingof n identical and symmetrically distributed blades ofcutters around the axis of revolution and a bit radius ofa, can be written as

Wc = naεζdn, T c = 1

2na2εdn, (6)

where ε is the intrinsic specific energy related to therock strength and ζ is related to the orientation of thecutting face. The cutting force and torque are propor-tional to the depth-of-cut (DOC) dn produced by a sin-gle blade, which is in general not constant. The DOCdepends on the axial position of the bit and the rocksurface generated by the previous blade according to

dn = Ub(t) −Ub(t − tn(t)), (7)

whereUb(t) is the axial bit position and t denotes time,see Fig. 5. Furthermore, tn(t) is the time required forthe bit to rotate by an angle of 2π/n, which is the anglebetween two successive blades. This time-dependentdelay tn(t) (actually state-dependent) is characterizedby the implicit equation:∫ t

t−tn(t)

dΦb(s)

dsds = Φb(t) − Φb

(t − tn(t)) = 2π

n, (8)

where Φb(t) denotes the angular position of the bit attime t .

In the contributions associated with the wearflatas introduced in (1), the wearflat reaction force λbais essentially discontinuous in terms of the bit axialvelocity. When the bit moves downwards, the contact

Fig. 5 Bottom hole profile between two successive blades [9]

between the wearflat and the rock is fully mobilized.However, when the bit moves upwards, the contactis lost and consequently the reaction force vanishes.Hence, the wearflat reaction force can be described ina set-valued force law by the following inclusion:

λba ∈ −naσ ln1 + Sign(Ub)

2, (9)

where σ is the maximum contact stress and ln is thewearflat length per blade. The axial velocity of the bitis denoted by Ub and the set-valued sign-function in(9) is defined as

Sign(y) =

⎧⎪⎨⎪⎩1, y > 0,

[ − 1, 1], y = 0,

− 1, y < 0.

(10)

As a consequence of the set-valued nature of the law in(9), the admissible values of the wearflat reaction forceform a convex set Ca given by

Ca = {λba | − naσ ln ≤ λba ≤ 0}. (11)

The force acting on thewearflat also induces a frictionaltorque λbt . Since the friction always acts in oppositedirection compared to the bit rotational velocity, thisfrictional torque is discontinuous with respect to therotational velocity and can be modelled by the follow-ing inclusion:

λbt ∈ 1

2aμξλbaSign(Φb). (12)

Herein, μ is a rate-independent friction coefficient andξ characterizes the orientation and spatial distribution

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Modelling and dynamic analysis of an anti-stall tool 2637

of the frictional contact of the surfaces along the bitblade(s). The angular velocity of the bit is denoted byΦb. The admissible values of the frictional torque formsa convex setCt , which depends on thewearflat (normal)reaction force λba . This set is given by

Ct (λba ) ={λbt |

1

2aμξλba ≤ λbt ≤ −1

2aμξλba

}.

(13)

The set-valued force laws for reaction force λba andfrictional torque λbt can be formulated by using normalcones of the convex sets (11) and (13), respectively[18,22]:

− Ub ∈ NCa (λba ), (14)

− Φb ∈ NCt (λbt ). (15)

From convex analysis, these inclusions are equiv-alent to implicit proximal point formulations [22].Hence, these formulations transform the associatedinclusions into nonlinear implicit equations, which areultimately used in the (time-stepping-based) numeri-cal solver which is developed in this work. These readas

λba = proxCa(λba − r1Ub), (16)

λbt = proxCt(λbt − r2Φb), (17)

for r1, r2 > 0 arbitrary, positive constants.

Remark During a torsional slip phase, the cutting edgeis in full contact with the rock. However, during tor-sional stick, this contact is not necessarily fully mobi-lized,which results in an unknowndistribution betweentorque associated with cutting and friction in this case.To include torsional stick in the model, it is assumedthat during torsional stick the contact between the cut-ting edge of the bit blade and the rock remains fullymobilized. Therefore, this assumption has conditionedthat the cutting component of the model in (6) is onlyvalid under the conditions of a nonnegative angularmotion of the bit (Φb ≥ 0) and with nonnegative DOC(dn ≥ 0). Furthermore, a negative DOC is associatedwith bouncing of the bit, which indicates total lossof contact between the bit and the rock. Hence, bit-bouncing is not analysed in this work.

2.3 Spatial Coulomb friction model

In drilling operations in inclined wells, the drill-stringrests on the boreholewall with its ownweight, resultingin additional frictional contact between the drill-stringand borehole. This contact is mainly generated by aspecific BHA component, the stabilizers, due to theirlarger diameter compared to the rest of the BHA com-ponents (see Fig. 6). The frictional contact between thedrill-string and borehole has been modelled by torqueand drag models [21,34]. In these models, the fric-tional contact forces depend on the normal force andthe frictional coefficient between contact surfaces. Ina drilling operation, the drill-string rotates and trans-lates in axial direction. Hence, the sliding velocity ofthe contact point has two components, both in axial andtangential direction. Due to this spatial contact, the spa-tial Coulomb’s friction law involves a two-dimensionalforce λT = [λTa λTt ]T. In Fig. 6 a schematic repre-sentation of the forces acting on the BHA is depicted.Figure 6 also depicts the AST located in between thetop and bottom stabilizer of the BHA. In this case, thenormal force FN is distributed over the two stabilizers.In the benchmark model (excluding AST), all frictionforce acts on a single stabilizer, because in the bench-mark model both stabilizers related to the same DOF.Furthermore, homogeneous rock formations are con-sidered in this study, such that the spatial friction law

Fig. 6 The BHA resting on its own weight in an inclinedwell with the axial (λTa , λTba ) and tangential (λTt , λTbt , point-ing out the plane) components of the spatial friction actingon the stabilizers and the distributed normal force FN withdistribution Δ

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is assumed to be isotropic and its force reservoir is rep-resented by a disc. The admissible values of the spatialCoulomb friction force for a single frictional contactpoint (i.e., as in the benchmark model) are given by theconvex set CT :

CT ={λT ∈ R

2 | ‖λT ‖ ≤ μwFN

}, (18)

whereμw is the friction coefficient and FN denotes thenormal force between the stabilizer in the BHA and theborehole wall.

Low-order lumped-parameter models for the drill-string dynamics are used in this study, such that onlythe gravitational effect (represented by the drill-stringweight) is considered to contribute to the normal force.Thus, the possible force contributions due to the cur-vature of the borehole are neglected. Furthermore, itis assumed that the stabilizers are always in contactwith the borehole wall. However, when the geometricalstructure of the borehole is perfectly vertical (Θ = 0◦),the spatial friction force vanishes. As the normal forcepresumptively depends on the buoyed weight of theBHA and the inclination of the borehole structure Θ ,as depicted in Fig. 6, this normal force is defined as

FN = BFMBH Ag sinΘ, (19)

where MBHA denotes the total mass of the BHA, gthe gravitational acceleration and BF is the buoyancyfactor given by

BF = ρ − ρm

ρ. (20)

Herein, ρ is the density of the BHA and ρm is the den-sity of the mud in which the BHA is submerged belowthe surface. Hence, the magnitude of the normal forceincreases as the borehole inclination (Θ) increases.

In the benchmarkmodel without the tool, the slidingvelocity γ T at the frictional contact between boreholeand BHA is given by

γ T =[Ub

ΦbR

], (21)

where the first component is associated with the axialand the second with the tangential velocity component.In this definition, R is the outer radius of the stabilizers.

In the drill-string model with the tool, the BHA isseparated in two parts, such that the spatial friction canact partly above and partly below the tool as shown inFig. 6. Essentially, the normal force FN is distributedbetween the two locations, namely above and belowthe AST. As a consequence, the force reservoir CT can

be segregated into two smaller isotropic reservoirs. Inorder to enable the analysis of the cases where all thefriction acts only above or below the tool, a linear distri-bution parameter Δ ∈ [0 1] is introduced. The admis-sible friction force reservoirs associated to the frictionforces above and below the tool are, respectively, givenby the following convex sets:

CT ={λT ∈ R

2 | ‖λT ‖ ≤ ΔμwFN

}, (22)

CTb ={λTb ∈ R

2 | ‖λTb‖ ≤ (1 − Δ)μwFN

}. (23)

The index b denotes the contributions, which arelumped at the bit. Note that for a straightforward com-parison between the model with and without the tool,the sum of maximal allowable friction forces are equal.When Δ = 1 holds, the spatial friction only acts abovethe AST; when Δ = 0 holds, all spatial friction actsbelow the tool. The corresponding sliding velocitiesare given by

γ T =[U

ΦR

]and γ Tb =

[Ub

ΦbR

]. (24)

The relation between the sliding velocity and the spa-tial friction force can be expressed by the followinginclusion, using the normal cone formulation of theset-valued spatial Coulomb friction law [22]

− γ T ∈ NCT (λT ). (25)

This inclusion can equivalently bewritten as an implicitproximal point formulation:

λT = proxCT(λT − rγ T ), (26)

where r > 0 is an arbitrary positive constant.

2.4 Dimensionless perturbation models

The EOMs, given by (2), are scaled in order to reducethe number of parameters. Furthermore, the dynamicsare expressed around its nominal solution (reflected bya constant angular velocity and ROP) by introducingperturbation coordinates. Following [32], a timescalet∗ and a characteristic length L∗ are introduced, whichare defined by

t∗ =√

ItotCp

and L∗ = 2Cp

εa2, (27)

with the total inertia in the benchmark model Itot = Iand in the model including AST Itot = Ia + Ib. Since

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Modelling and dynamic analysis of an anti-stall tool 2639

the total inertia is equal in both models, the timescale isthe same in bothmodels. By using these scaling param-eters, the following dimensionless perturbation coordi-nates are introduced

u(τ ) = U −U0

L∗, ub(τ ) = Ub −Ub0

L∗,

φb(τ ) = Φb − Φb0,

(28)

which are functions of the dimensionless time

τ = t

t∗. (29)

These coordinates represent the dimensionless axialand torsional perturbations with respect to its nomi-nal responses, where the coordinates denoted with sub-script b are associated with the bit. Explicit expressionsfor the nominal displacementsU0,Ub0 andΦb0 in bothmodels are given in the subsequent sections.

The generalized forces associated with set-valuedforce laws are scaled by a characteristic cutting forcecorresponding to a DOC equal to the characteristiclength L∗. This results in the following dimensionlessperturbation forces and torque associated to the set-valued force laws introduced in Sects. 2.2 and 2.3:

λba = a

2ζCp(λba − λba0), λbt = 1

Cp(λbt − λbt0),

λT = a

2ζCp(λT − λT0).

(30)

Note that λT is a column containing the dimensionlessaxial and tangential components of the spatial Coulombfriction. Furthermore, in the model including the tool,the spatial Coulomb friction acting above and belowthe tool both satisfy the same dimensionless perturbedform as above for λT . However, the values of each asso-ciated friction forces can be different as these are scaledby the parameter Δ in (22) and (23). Furthermore, thenominal values of the frictional contact component inthe bit–rock interaction law are λba0 = −naσ ln andλbt0 = − 1

2na2μξσ ln . Expressions for λT0 for both

models are given in the subsequent sections.The dimensionless form of the time delay, depth-of-

cut and axial and torsional nominal velocities are givenby

τn = tnt∗

, δ = d

L∗,

v0 = V0t∗L∗

, ω0 = Ω0t∗. (31)

The nominal axial velocity V0 (in original coordinates)in both models is given by

V0 = 1

D

(− H0 + Msg − aεζd0 + λba0 + λTa0

). (32)

The nominal DOC d0 and the axial component of thenominal spatial friction λTa0 are both functions of thenominal velocity V0 and given by

d0 = V0tn0 (33)

λTa0 = − V0Ω0R

√√√√√ μ2wF2

N

1 +(

V0Ω0R

)2 , (34)

with tn0 = 2π/(nΩ0). Substitution of these expres-sions results in a fourth-order polynomial in V0, whichis monotone for positive values of V0. Hence, (32)exhibits a unique solution for normal drilling opera-tions (reflected by a positive nominal axial velocity V0).

Moreover, the dimensionless nominal time delay isdefined as τn0 = tn0/t∗ = 2π/(nω0). The dimension-less depth-of-cut (δ) can be expressed in terms of aperturbation δ from the nominal depth-of-cut per rev-olution (δ0 = 2πv0/ω0):

δ = δ + δ0. (35)

The dimensionless perturbed DOC δ is given by

δ = n (ub(τ ) − ub(τ − τn)) + nv0τn . (36)

Herein, τn is the dimensionless time delay τn = τn +τn0, where τn is its perturbation from the nominal timedelay τn0. This time delay is obtained with the implicitdelay equation, given in (8), which reads in a dimen-sionless formulation:

φb(τ ) − φb(τ − τn) + ω0τn = 0. (37)

2.4.1 Benchmark drill-string model

The column with the dimensionless perturbation coor-dinates in the benchmark model is given by z =[ub φb]T. In the case of a nominal drilling operation,there are no vibrations; thus, the axial and torsionalvelocities are constant and positive. Due to the con-stant velocities, the accelerations are equal to zero. Bysubstitution of the constant velocities and zero acceler-ations in the dynamic models, expressions for the nom-inal values of the displacements, velocities and forcesare obtained. The nominal values of the axial and angu-lar bit displacement, Ub0 and Φb0 (see (28)), are givenby

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2640 R. Wildemans et al.

Ub0 = V0t, (38)

Φb0 = Ω0t + 1

Cp

(−DΦΩ0 − 1

2na2εd0

+ λbt0 + RλTt0

). (39)

The nominal values of the spatial Coulomb friction(λT0 = [λTa0 λTt0 ]T) are given by

λTa0 = − V0Ω0R

√√√√√ μ2wF2

N

1 +(

V0Ω0R

)2 , (40)

λTt0 = −√√√√√ μ2

wF2N

1 +(

V0Ω0R

)2 . (41)

The scaling and introduction of the perturbation coor-dinates leads to the dimensionless EOMs in generalform:

M z′′ − H (τ, z, z′) = W λ. (42)

Then, the corresponding matrices and columns in thebenchmark are given by

M =[1 00 1

]H (τ, z, z′) =

[− γ u′

b − ψδ

− γφφ′b − φb − δ

]

W =[ψ 0 ψ 00 1 0 χ

]λ = [

λba λbt λTa λTt

]T(43)

2.4.2 Drill-string model including AST

The column with the dimensionless perturbation coor-dinates in the model including AST is given by z =[u ub φb]T. The nominal values of the axial and angu-lar displacements, U0, Ub0 and Φb0, are given by

U0 = V0t+ 1

αKb

(1

2na2εdn0+αnaεζdn0 + DΦbΩ0

− αMbsg − αλba0 − λbt0 − αλTba0 − RλTbt0

),

(44)

Ub0 = V0t, (45)

Φb0 = 1

αKb

(− naεζdn0 + Mbsg + 1

αDΦΩ0

+ λba0 + λTba0

)

+(

1

Cp+ 1

α2Kb

)(−1

2na2εdn0 − (DΦ + DΦb )Ω0

+ λbt0 + RλTbt0

)+ R

CpλTt0 + Ω0t. (46)

The nominal values of the spatial Coulomb fric-tion located above and below the AST (λT0 =[λTa0 λTt0 λTba0 λTbt0 ]T) are given by

λTa0 = − V0Ω0R

√√√√√ Δ2μ2wF2

N

1 +(

V0Ω0R

)2 , (47)

λTt0 = −√√√√√ Δ2μ2

wF2N

1 +(

V0Ω0R

)2 , (48)

λTba0 = − V0Ω0R

√√√√√ (1 − Δ)2μ2wF2

N

1 +(

V0Ω0R

)2 , (49)

λTbt0 = −√√√√√ (1 − Δ)2μ2

wF2N

1 +(

V0Ω0R

)2 . (50)

Then, the scaled EOMs in dimensionless perturbationcoordinates for themodel includingAST can bewrittenin the general formof (42). The correspondingmatricesand columns are given by

M =⎡⎣m∗ + κι − κι νι

− κι −m∗ + κι + 1 − νικνι − κ

νι 1

⎤⎦

H (τ, z, z′) =⎡⎣ − η2b(u − ub) − γ u′ − γb(u′ − u′

b) − νφbνφb + κ(u − ub) + νγφ1φ

′b + κγφ1 (u

′ − u′b)− φb − κ

ν(u − ub) − γφ1φ

′b

− κ(u − ub) − νγφ1φ′b − κγφ1 (u

′ − u′b)

+ η2b(u − ub) + γb(u′ − u′b) − ψδ

− γφ1κν(u′ − u′

b) − γφ2φ′b − δ

⎤⎦

W =⎡⎣0 0 ψ νχ 0 0

ψ 0 0 − νχ ψ 00 1 0 χ 0 χ

⎤⎦

λ = [λba λbt λTa λTt λTba λTbt

]T.

(51)

The definitions of the characterizing dimensionlessparameters in (43) and (51) are given in Table 1, alongwith their values used in Sect. 3. The models presentedin this section will now be used to analyse their dynam-ics, in particular to study the effect of the AST andspatial frictional on the drilling performance.

3 Drilling performance analysis

In this section, the effect of spatial Coulomb frictionon the drilling performance is investigated. The drilling

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Modelling and dynamic analysis of an anti-stall tool 2641

Table 1 Characteristicsystem parameters

Parameter Symbol Value

Characteristic length L∗ = 2Cp

εa27.13 × 10−4

Characteristic time t∗ =√

ItotCp

0.45

Mass ratio m∗ = MaMtot

0.92

Inertia ratio ι = IaItot

0.86

Scaled axial damping γ = DMtot

√ItotCp

6.43 × 10−3

Torsional damping above tool γφ1 = DΦ√Itot Cp

2.05 × 10−4

Torsional damping below tool γφ2 = DΦb√Itot Cp

7.39 × 10−6

Drill-string design ψ = Itot aεζMtotCp

37.5

Arm Coulomb friction force χ = 2ζ Ra 0.96

Wearflat friction λ = a2 σ l2ζCp

16.15

Drill-bit design β = μξζ 0.81

Inertia mass ratio κ = Itotα2Mtot

0.56

Scaled lead of tool ν = ακL∗ 50.64

Scaled axial stiffness of tool ηb =√

Kb ItotMtotCp

2.06

Scaled axial damping of tool γb = DbMtot

√ItotCp

0.45

performance is characterized by the drilling efficiencyand ROP. This paper focuses on the effect of frictionon the drilling performance under different operationalconditions, namely the prescribed angular speed andthe hook-load at the surface. The characterizing param-eters are introduced in Sect. 3.1. Next, in Sect. 3.2 thedrilling performance of the benchmark model is inves-tigated. In Sect. 3.3, the drilling performance of themodel including AST is investigated and these resultsare compared to the benchmarkmodel in order to inves-tigate the effectiveness of the AST.

3.1 Drilling performance variables

From stability analyses of the benchmark model inthe absence of spatial Coulomb friction, it is observedthat the nominal solution is typically unstable forrealistic operating conditions [10,32,36]. As a conse-quence, solutions diverge away from the unstable nom-inal response and result in a time-varying steady-stateresponse from the nonlinear dynamics, where the non-linearities are related to the set-valued nonlinearities inthe bit–rock interaction law, the set-valued spatial fric-

tion law and the state-dependent delay effect. The drill-string system exhibits both axial and torsional vibra-tions, where the torsional vibrations typically evolveover a significantly slower timescale compared to theaxial vibrations. Since the torsional vibrations typicallyconverge to a steady-state torsional limit cycle, in thissection, the performance characterization variables areaveraged over a torsional limit cycle.

The performance of a drilling operation is mainlycharacterized by the drilling efficiency [27]. Thedrilling efficiency reflects howmuch of the total torqueprovided to the bit is used for cutting.

Remark Note that the torque provided at the bit is ingeneral not equal to the torque applied at the surfacedue to frictional losses along the drill-string.

In line with previous studies [32,37], this efficiencyis defined as the ratio between the energy devoted tothe cutting process and the total energy dissipated at thebit (i.e., by cutting and frictional forces). The averagedrilling efficiency η is given by

η = 〈T c〉〈T c〉 + 〈T f 〉 . (52)

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2642 R. Wildemans et al.

Herein T c represents the cutting torque and T f denotesthe frictional torque at the bit. The brackets 〈.〉 denotethe average over a torsional limit cycle. However, theaveraged frictional torque at the bit, 〈T f 〉, is not directlyobtained from the numerical simulation, since the fric-tional torque at the wearflat, T f , acts on the same DOFas the tangential component of the set-valued Coulombfriction force below the AST, λTbt (see (3) and Fig. 2for the benchmark model and (5) and Fig. 4 for themodel including AST). Besides, both torques are gov-erned by a similar set-valued force law, see (15) and(25). As a consequence, it is not possible to distinguishbetween the wearflat torque and the set-valued fric-tional torque below the AST in numerical simulationswith the model.

Let us now explain how we obtain an accurate mea-sure for the frictional losses acting at the bit in orderto assess the efficiency in (52). According to (12), thefrictional torque at the bit is proportional to the wearflatreaction force W f with a factor 1/2aμξ . From themodel including the tool, as depicted in Fig. 4, the aver-age of the sum of the wearflat force and the axial set-valued friction force below the AST, 〈W f +λTba 〉, candirectly be computed. Since the averaged axial velocityis much smaller compared to the averaged tangentialvelocity (i.e., 〈Ub〉/〈RΦb〉 = O(10−4 − 10−2)), thefrictional contact basically only produces a frictionaltorque and thus the axial component λTba is negligi-ble. Hence, the average wearflat force 〈W f 〉 is approx-imated by 〈W f + λTba 〉 and can be used to calculatethe average frictional torque at the bit 〈T f 〉.

A higher drilling efficiency will result in more effi-cient drilling and consequently in saving drilling costs.Moreover, a higher drilling efficiency, as defined in(52), implies less frictional dissipation at the bit, whichis generally favourable from a bit wear perspective (i.e.,longer bit life-time or maintaining bit sharpness).

A control parameter in both models is the hook-load H0 at the surface (an upward force). It can bededuced that an increase in the hook-load is causinga decrease in the total weight applied to the bit, andthis consequently will decrease the ROP. However, thetotal weight applied on the bit is not only defined by thehook-load, but also by the gravitational forces acting onthe submerged drill-string. Therefore, instead of vary-ing the hook-load, the total nominal weight applied onthe bit W0 is varied as a control parameter in the sim-ulations. The total weight applied on the bit is definedas

W0 = Ws − H0, (53)

with Ws the submerged weight of the drill-string.

Remark In general, the WOB depends on the incli-nation of the well, since the submerged weight of adrill-string decreases when the inclination increases.However, it is beyond the scope of this paper to studyhowall individual force components varywith the incli-nation. Therefore, it is assumed that the hook-load isadjusted when the inclination changes such that theWOB remains constant.

3.2 Drilling performance of the benchmark model

The performance analysis pursued in this sectionfocuses on the axial bit velocity, because this ultimatelydetermines the ROP. The dynamic models as presentedin Sect. 2 are simulated with a time-stepping-basednumerical simulator. The structure of the numericalsimulator is based on [35].

Time-domain responses of the axial bit velocity ofthe benchmark model with and without spatial frictionbetween the BHA and the borehole wall are depicted inFig. 7. In both simulations, the same boundary condi-tions are applied (the total weight applied on the bitW0 = 171 kN and the angular velocity at the top-drive Ω0 = 80 RPM). The initial conditions in bothsimulations are chosen close to the desired nominaloperating conditions (ub(0) = φb(0) = 1 × 10−4 andub(0) = φb(0) = 0), such that the initial perturbationsare small with respect to the nominal solution. Fig-ure 7a, b shows the axial bit velocity without and withspatial friction, respectively. Both cases show unstabletransient behaviour where the oscillations grow untilthe bit experiences an axial (and a torsional) stick-sliplimit cycle. Furthermore, the transient phase in the casewith friction is longer, i.e., it requiresmore time to reachthe axial (and torsional) limit cycle. This implies thatthe friction has a stabilizing effect on the drill-stringdynamics, which reduces the growth rate of the axial(and torsional) vibrations. The spatial Coulomb fric-tion does not qualitatively change the drill-string sys-tem response of the benchmark model for this specificset of operation conditions.

Next, the effect of the spatial friction is investi-gated for a broad range of operation conditions. Therange of the nominal WOB (W0) corresponds with therange of hook-load forces of H0 = 370 − 440 kN.

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Modelling and dynamic analysis of an anti-stall tool 2643

Fig. 7 a The axial bitvelocity for the benchmarkmodel without frictionbetween the BHA andborehole wall (Θ = 0) andb with Coulomb friction(Θ = 90◦) (W0 = 171 kNand Ω0 = 80 RPM)

(a) (b)

This range is chosen such that lower values result inbit-bouncing and higher values in a negative nominalaxial velocity (see (32) for the relation between H0 andV0). The range of angular velocities corresponds withΩ0 = 30 − 150 RPM. The drilling efficiency η andaveraged ROP for this range of operation conditionsare depicted in Figs. 8, 9 and 10 for different inclinedscenarioswithΘ = 0◦,Θ = 45◦ andΘ = 90◦, respec-tively.

In Figs. 8a, b, 9a, b and 10a, b, it can be observedthat the ROP increases with increasing nominal WOBand prescribed angular velocity. An increase in W0,under a prescribed constant angular velocity, results inan increased ratio between the cutting and frictionalforces. Also with faster rotation (higher values of Ω0)more volume of rock is cut in a given time window.Furthermore, the spatial Coulomb friction has a smalleffect on the ROP, implying that the portion of the totalforce used for the cutting process is not affected sig-nificantly by the spatial friction between the BHA andborehole.

The drilling efficiency for different operation con-ditions is shown in Figs. 8c, d, 9c, d and 10c, d. Anincrease in the nominalWOB results in a higher drillingefficiency. This indicates that for a higher W0 moreenergy is used for the cutting process, which is inline with the results in Figs. 8a, 9a and 10a. However,an increasing prescribed angular velocity results in adecreasing drilling efficiency, which indicates that lessenergy is used for cutting. This implies a decrease inDOC. Even with the decrease in DOC, an increasedROP is still maintained. This consequently happenssince with a higher angular velocity more volume ofrock is removed by cutting in a given amount of time.From these results, it can be concluded that the spatialCoulomb friction mainly acts in tangential direction.

This is a direct consequence of the large angular veloc-ity compared to the axial velocity of the drill-string,which results in a sliding velocity (between stabilizerand borehole) with a relative small axial componentcompared to the tangential component. Consequently,this is reflected by the ratio between the axial and tan-gential components of the spatial Coulomb friction(λT = λTa/λTt ), which is of O(10−4 − 10−2). Thisobservation indicates that the spatial Coulomb frictionbasically only produces a torsional friction and conse-quently hardly influences the axial motion of the bit, asreflected in the ROP observations.

3.3 Drilling performance of the model including AST

In this section, we analyse the drilling performance ofthe system with AST in the presence of spatial frictionbetween borehole and stabilizers.

Steady-state time-domain responses of the axial bitvelocity for the drill-string model including the ASTwith andwithout spatial friction are depicted in Fig. 11.These responses are obtained with the same boundaryconditions as used in benchmark model (W0 = 171 kNand Ω0 = 80 RPM). The initial conditions are cho-sen close to the nominal operating conditions (u(0) =ub(0) = φb(0) = 1 × 10−4 and u(0) = ub(0) =φb(0) = 0). In the absence of spatial Coulomb friction,the axial vibrations have a larger amplitude compared tothe cases with spatial Coulomb friction. Furthermore,the axial bit velocity exhibits stick-slip transitions incase with and without spatial friction. Comparing theresponse of the model including AST with response ofthe benchmark shows a significant difference betweenthe axial responses. In particular, the amplitude of theaxial bit velocity (Ub) increases by including the AST

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2644 R. Wildemans et al.

Fig. 8 Drillingperformance without andwith AST in a vertical wellwith inclination angleΘ = 0◦. Therate-of-penetration asfunction of a applied WOBW0 and b rotational velocityΩ0 and drilling efficiency asa function of c appliedWOB W0 and d rotationalvelocity Ω0 (W0 = 184 kNand Ω0 = 80 RPM, unlessparameter is varied)

(a) (b)

(c) (d)

up to two times the amplitude of the axial bit velocityobtained with the benchmark model.

The drilling performance for a range of operationalconditions is depicted in Figs. 8, 9 and 10 for differ-ent inclined scenarios with Θ = 0◦, Θ = 45◦ andΘ = 90◦, respectively. The trends are comparable withthe benchmark model, since the increase of the nomi-nal WOB and prescribed angular velocity result in anincreasing ROP, while the drilling efficiency increaseswith W0 and decreases with increasing Ω0. These fig-ures also indicate that the spatial friction hardly affectsthe ROP. However, Fig. 10 shows that it makes a dif-ference if the spatial Coulomb friction fully acts below(Δ = 0) or above (Δ = 1) the AST. In the simulations,it is observed that the axial vibrations above the tool(U ) decreases when the spatial friction acts fully abovethe tool (Δ = 1). This indicates that smaller vibrationsabove the AST have a positive effect on the effective-ness of the tool, since it results in a slight improvementof ROP. When the friction fully acts below the tool(Δ = 0), the ROP is slightly lower compared to thecase without spatial friction. Furthermore, the influ-

ence of the location where the spatial friction acts isalso observed in the drilling efficiency, which is lowerin the case when all spatial friction acts below the tool.Hence, it can be concluded that the effect of the spatialfriction on the axial vibrations, which are related to theimproved drilling performance, depends on the loca-tion where the spatial friction acts. For various oper-ational conditions (W0 and Ω0), the drilling perfor-mance is higher for the case where all the additionalfriction acts above the tool. This insight reveals that itis more beneficial in practice to place the AST closerto the bit, such that the friction acts mainly above thetool.

A comparison between the ROP and drilling effi-ciency obtained with the benchmark model and withthe model including AST shows that incorporating theAST significantly improves the ROP and the drillingefficiency for a broad range of spatial friction levels.For example, in the case of a prescribed angular veloc-ity Ω0 = 80 RPM and a nominal WOB of W0 = 192kN, the benchmark model results in the absence of spa-tial friction in a ROP of 〈Ub〉 = 3.94× 10−3 m/s and a

123

Modelling and dynamic analysis of an anti-stall tool 2645

Fig. 9 Drillingperformance without andwith AST in an inclinedwell with inclination angleΘ = 45◦. Therate-of-penetration asfunction of a applied WOBW0 and b rotational velocityΩ0 and drilling efficiency asa function of c appliedWOB W0 and d rotationalvelocity Ω0 (W0 = 184 kNand Ω0 = 80 RPM, unlessparameter is varied)

1.7 1.8 1.9 2 2.1 2.2 2.3

105

2

4

6

8

10

1210-3

(a)

40 60 80 100 120 140

2

4

6

8

10

1210-3

(b)

1.7 1.8 1.9 2 2.1 2.2 2.3

105

0

0.1

0.2

0.3

0.4

0.5

0.6

(c)

40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

(d)

drilling efficiency η = 0.28, see Fig. 8. Under the sameoperational condition, a ROP of 〈Ub〉 = 6.08 × 10−3

m/s and a drilling efficiency η = 0.37 is obtainedwith the model including AST. In this specific case,an increase of more than 50% in ROP and more than30% in drilling efficiency is achieved by incorporatingthe AST. In the presence of spatial friction, the increasein ROP and drilling efficiency are comparable.

3.4 Discussion

From the performance analysis, it is concluded that thespatial friction hardly affects the ROP, since the axialcomponent of the friction is relatively small comparedto the tangential component. Furthermore, simulationresults have revealed that incorporating the AST in thedrill-string results in an improved drilling efficiencyand ROP for a broad range of deviated wells. In casewhen the spatial friction acts fully above the AST, aslight improvement of drilling performance is observedcompared to the case without spatial friction and whenall friction acts below the tool.

4 Parametric design study AST

Based on the analysis performed in the previous sec-tion, incorporating the AST can provide a solution toimprove the drilling efficiency, also in inclined drillingscenarios with increased frictional contact. From thispoint of view, the question arises how to find the opti-mal tool settings that provide the highest drilling effi-ciency. A parametric design study is performed in orderto investigate the optimal tool design and to understandwhether the optimality of this design is influenced bythe frictional contacts between the BHA and the bore-hole wall (i.e., whether optimal tool settings can befound that robustly optimize performance for a broadrange of deviated wells).

In practical field cases, the tool is placed in the bot-tom part of the BHA [33], which results in more massof the BHA above the tool than below the tool. Sincethe largest contribution to the spatial friction comesfrom the heaviest part of the BHA, all simulations inthis section are performed under the assumption thatall spatial Coulomb friction forces fully act above thetool (Δ = 1).

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2646 R. Wildemans et al.

Fig. 10 Drillingperformance without andwith AST in an inclinedwell with inclination angleΘ = 90◦. Therate-of-penetration asfunction of a applied WOBW0 and b rotational velocityΩ0 and drilling efficiency asa function of c appliedWOB W0 and d rotationalvelocity Ω0 (W0 = 184 kNand Ω0 = 80 RPM, unlessparameter is varied)

1.7 1.8 1.9 2 2.1 2.2 2.3

105

2

4

6

8

10

1210-3

(a)

40 60 80 100 120 140

2

4

6

8

10

1210-3

(b)

1.7 1.8 1.9 2 2.1 2.2 2.3

105

0

0.1

0.2

0.3

0.4

0.5

0.6

(c)

40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

(d)

140 142 144 146 148 1500

0.01

0.02

0.03

0.04

0.05

(a)

140 142 144 146 148 1500

0.01

0.02

0.03

0.04

0.05

(b)

140 142 144 146 148 1500

0.01

0.02

0.03

0.04

0.05

(c)

Fig. 11 a The steady-state axial bit velocity for the model including ASTwithout friction between the BHA and borehole wall (Θ = 0),b with Coulomb friction (Θ = 90◦) acting fully above the tool Δ = 1 and c with Coulomb friction (Θ = 90◦) acting fully below thetool Δ = 0 (W0 = 171 kN and Ω0 = 80 RPM)

The tool design is mainly reflected by two param-eters, namely the lead of the helical spline β andthe spring stiffness Kb. In the current design of theAST, a lead angle of β = 45◦ and a spring stiffnessKb = 1522.5 kN/m are used. The investigated rangeof the lead angle is in between β = 10 − 70◦ and thespring stiffness range is in between 50− 11133 kN/m.Higher values of spring stiffness results in bit-bouncing

in absence of spatial friction. Two different operationalscenarios are investigated, namely with a low angularvelocity, where Ω0 = 50 RPM, and a high angularvelocity, where Ω0 = 120 RPM. By considering thesetwooperation scenarios, it can be investigated if the tooldesign is robust for different operational conditions. Inall simulations W0 = 171 kN.

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Modelling and dynamic analysis of an anti-stall tool 2647

Fig. 12 Drilling efficiency for different tool settings at a Θ = 0, b Θ = 45◦ and c Θ = 90◦ with Ω0 = 50 RPM and W0 = 201 kN.The highest drilling efficiency is denoted by the red dots. (Color figure online)

Fig. 13 Drilling efficiency for different tool settings at a Θ = 0, b Θ = 45◦ and c Θ = 90◦ with Ω0 = 120 RPM and W0 = 201 kN.The highest drilling efficiency is denoted by the red dots. (Color figure online)

From the resulting parametric study, it is observedthat a higher spring stiffness results in a higher fre-quency of the vibrations induced by the tool contrac-tion. Furthermore, a smaller value of the leadwill resultin less contraction at a certain torsional displacement.

In Fig. 12, the drilling efficiency is plotted againstthe lead angle and the spring stiffness for different wellinclination angles, and for the low angular velocity case(Ω0 = 50 RPM). The shape of the surfaces charac-terizing the drilling efficiency have similar trends fordifferent values of Θ , indicating that the influence ofthe spring stiffness and the lead on the drilling effi-ciency is comparable under various levels of spatialCoulomb friction. Figure 12 shows that for various spa-tial Coulomb frictions, the optimal value for the leadangle is around 30◦. Since in the current designs a leadangle of 45◦ is used, between 2 and 7% in drilling effi-ciency can be gained by changing the lead angle to 30◦in this specific case with the above-mentioned opera-tional conditions. Furthermore, it is observed that forlower values of the spring stiffness, the tool providesa higher drilling efficiency. However, the influence of

the spring stiffness is relatively small compared to theinfluence of the lead angle.

In Fig. 13, the drilling efficiency for the high angularvelocity case (Ω0 = 120 RPM) is depicted for variousvalues of the lead angle and spring stiffness and fordifferent values of Θ . The shapes in these figures areslightly different compared to the low angular velocitycase. The dependency of the drilling efficiency η on thespring stiffness Kb reveals a less clear trend comparedto the low angular velocity case. The optimal lead anglein this case is also aroundβ = 30◦, which is equal to thelow angular velocity case. However, it is observed thatin the high angular velocity case the drilling efficiencyis less sensitive for an increase in lead angle comparedto the low angular velocity case.

Based on these results, the general conclusion is thatthe optimal values of the tool design are robust for dif-ferent friction levels and a range of angular velocitiesimposed at the surface (rig). Hence, an optimal designfor the AST that gives optimal drilling efficiency overa broad range of inclined scenarios is feasible.

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2648 R. Wildemans et al.

5 Conclusions

In this paper, the effect of a passive down-hole anti-stalltool on the drilling performance of rotary drilling sys-tems has been investigated for deviated well scenarios.A model including the coupled axial-torsional drill-string dynamics, the bit–rock interaction, the tool andthe frictional effects of the stabilizers, due to boreholeinclination, has been developed. A set-valued mod-elling approach for all contact and frictional effectshas been pursued leading to a model in terms of adelay differential inclusion for which a time-steppingmethod is employed for simulation purposes. Numer-ical analysis results revealed that the down-hole toolsignificantly improves drilling efficiency and ROP fora broad range of deviated wells. Moreover, based on aparametric design study it is concluded that an optimaltool design, in terms of drilling efficiency, can be foundthat is robust for a large range of borehole inclinationsand operational conditions.

Acknowledgements This study is supported by the Indone-sian Endowment Fund for Education (LPDP) of the Republic ofIndonesia (Grant No. PRJ-4653/LPDP.3/2016).

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrest-ricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

A Coordinate transformation

The drill-string dynamics including AST are expressedin a set of dependent coordinatesqc = [U, Φ, Ub, Φb]Tcombined with the following holonomic constraint:

hAST = α (Φ − Φb) − (U −Ub) = 0. (54)

The dynamics of the constrained system can also begiven in variational form:

(δqc)T(Mcqc − hc(t,qc, qc) − Wcλ

) = 0, (55)

which holds for all the virtual displacements δqc thatsatisfy the variational holonomic constraint given by

WTcδq

c = 0, (56)

where WTc is the constraint Jacobian:

WTc = ∂hAST

∂qc=

⎡⎢⎢⎣

− 1α

1−α

⎤⎥⎥⎦ . (57)

Expressions for Mc and hc(t,qc, qc) can be foundin [38]. The set of independent coordinates q =[U Ub Φb]T is introduced, such that it uniquely deter-mines qc = qc(q). The relation between the set ofdependent and independent coordinates is explicitlygiven by a constant transformation matrix T. Hence,the following relations hold:

qc = Tq, qc = Tq, qc = Tq, (58)

with

T =

⎡⎢⎢⎢⎢⎣

1 0 01α

− 1α

1

0 1 0

0 0 1

⎤⎥⎥⎥⎥⎦ . (59)

Moreover, the virtual displacement in terms of the inde-pendent coordinates can be written as follows:

δqc = Tδq. (60)

Substitution of Eqs. (58) and (60) into (55) results inthe following equation

δqT(TTMcTq − TThc(t,Tq,Tq)

−TTWcλ)

= 0, ∀δq. (61)

Since the generalized coordinatesq are independent, allvirtual displacements δq are admissible and thereforethe dynamics in independent coordinates can bewrittenas follows:

TTMcTq − TThc(t,Tq,Tq) = TTWcλ. (62)

The new system matrices, explicitly given in (5), arenow given by

M := TTMcT,

h(t,q, q) := TThc(t,Tq, Tq), W := TTWc. (63)

123

Modelling and dynamic analysis of an anti-stall tool 2649

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