Advanced Drilling SPE Book Chapter 3.2: Drilling

Advanced Drilling SPE Book Chapter 3.2: Drilling vibrations Authors June 2008

Chapter 1 Drillstring Vibrations 1.1

Introduction

The drillstring vibrations are extremely complex due to the random nature of a multitude of factors (eg bit-formation interaction, drillstring-wellbore interaction, hydraulics etc). Drillstring vibrations involve a collection of simultaneous vibration phenomena that render the analysis quite challenging. Three primary modes of vibration are present while drilling: axial, torsional, and lateral. Related to these are phenomena including bit-bounce, stick-slip, and whirling, respectively. Drillstring vibrations can be induced by external excitations such as bitformation [45]. In these cases, the tuning of the excitation source to a natural frequency of the drilling assembly or its components can yield destructive motions. Self-excited vibrations are also present downhole [49]. Vibrations can also be caused by the flow in the drillstring annulus [93]. The dynamic behavior of the drillstring can be either transient or steady-state. Drillstring vibrations affect directly the drilling performance because drillstring assembly components experience premature wear and damage [47, 79, 81] and the rate of penetration (ROP) decreases as part of the drilling energy needed to cut the rock is wasted in vibrations [32, 80, 130]. Further, vibrations can cause interference with measurement-while-drilling (MWD) tools [69]. Finally, vibrations often induce well-bore instabilities that can worsen the condition of the well and reduce the directional control and the overall

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shape of the well-bore [45]. Drill collars and adjacent drill pipes have been recognized over a period of decades as the components subjected to the most harmful vibrations. Accordingly, the bottom-hole assembly (BHA) not only influences the overall dynamic response of the assembly, but it is also the location of most failures [35, 51]. Therefore, vibration mitigation requires understanding the dynamic behavior of the BHA. However, downhole vibrations can be a valuable source of information that provides insight into bit wear, formation properties, and drillstring – wellbore interactions. It has also been suggested that they can be used as a potential seismic source [10, 100]. Finally, drillstring vibrations have been considered as a means to enhance drilling effectiveness by increasing the available power at the bit [36].

1.2 1.2.1

Axial Vibrations Preliminary Remarks

Axial vibrations of a drillstring involve motions of its components along its longitudinal axis. Drill strings are subject to both static and dynamic axial loadings. The classical buckling theory provides the maximum static weight-on-bit (WOB) that the assembly can sustain without buckling [78, 87]. It yields a certain operational static axial constraint corresponding to the applied WOB that fulfills the appropriate criteria (safety factor, rate-of-penetration requirements, etc.). On the other hand, dynamic axial loads on the drilling assembly originate primarily from bit–formation interactions. They give rise to timedependent fluctuations of the weight applied to the bit, and are rather erratic. Historically, the drillstring axial modes of vibration have been observed in the field, together with rotational modes, before their lateral counterparts [49], because they can travel from the bottom of the well to the surface, whereas lateral vibrations are usually trapped below the neutral point [56]. Drillstring axial dynamic behavior was the object of pioneer theoretical in2

vestigations of downhole vibrations in the early 1960s [3]). Severe axial vibrations develop quite often when drilling with roller-cone bits, owing to their type of interaction with the formation. Specifically, the multi-lobes pattern generated by tricone bits at the bottom of the well is a major source of axial excitations for vertical or near vertical wells in which the drillstring – bore-hole interactions are limited, and the effective damping is reduced [114]. In the most severe cases, the axial vibrations can be observed at the surface as they may induce the bouncing of the kelly and the whipping of draw-works cables [33, 34]. Axial vibrations can be detrimental [95] or beneficial [36] to drilling, as they affect the WOB and consequently the ROP. Axial vibrations can be especially harmful to drilling, if they have large amplitude oscillations as in the case of the tuning of the drillstring natural frequencies with excitations of frequency of three cycles per bit revolution [34]. Further, the well-bore does not directly restrain the axial displacement of drill collars, resulting in large amplitude oscillations. This can make the bit start bouncing off the formation, rendering the rock-breakage process erratic, and thereby reducing the overall ROP. Axial vibrations also have indirect consequences due to downhole coupling mechanisms that result, for instance, in significant lateral displacements [45, 113]. Existing literature reviews on this subject include [20, 46, 96, 110, 118, 119].

1.2.2

The Bit-Bounce Phenomenon

An important feature of axial vibrations is the temporary lift-off of the drillbit from the formation, known as the bit-bounce phenomenon. Analytical investigations of bit-bouncing have been reported in [95, 118]. Downhole MWD tools have enabled the detection of wide and frequent WOB fluctuations that are sometimes not noticeable at the rig surface. In extreme cases, the axial load of the drill-bit vanishes rapidly and more or less periodically [30, 40, 126, 131], together with the torque-on-bit (TOB) [7]. These instances correspond to a lift-off of the drill-bit, and the process is accordingly called bit-bounce.

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Two primary causes are considered for the drill-bit lift-off. The first is the irregularities in the formation surface, sometimes resulting from drilling with a tricone bit [110], which can generate a three-lobe pattern downhole. A second postulated source is the frequency tuning of mud pressure with the axial natural frequencies of the drilling assembly [30]. Bit-bouncing has many consequences including low ROP, excessive fatigue of the downhole drillstring components, and eventually well damage [92].

1.2.3

Modeling Perspective

The effects of the axial vibrations on drilling performance have led to the analytical investigation of drillstring axial vibrations since the early 1960s [8, 9, 11, 95] when the first measurements-while-drilling data recording devices became available. The similarities between the propagation of axial and torsional waves in drilling assemblies have encouraged their joint examination to determine, for instance, the natural frequencies of drilling assemblies [3]. Although the wave equation can be used for some analytical solutions, the increasing availability of computers has fostered the use of numerical methods to study the axial vibration modes [37, 114]. The influence of downhole equipment such as shock subs, which are designed to absorb some of the axial energy, has also been investigated [67]. The insight gained through practical and theoretical examinations has allowed identifying the bottom-hole assembly as a primary vibration source governing most of the assembly dynamic behavior [33, 35]. For instance, a prevalent source of axial vibrations is bit kinematics that can become quite adverse as the bit bounces off the rock. In 1963, Paslay and Bogy [95] undertook the first analytical investigation of bit-bouncing. Since then, downhole MWD tools have allowed detection of wide and frequent WOB fluctuations that are often not noticeable at the rig surface.

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1.2.4

Continuous Model

This section discusses the mathematical representation of the axial behavior of the drillstring. The equation governing the undamped axial motion ξ(x, t) of a linear elastic bar, is a second-order partial differential equation (PDE) [3, 11, 28, 35]) called the undamped classical wave equation ∂ 2 ξ(x, t) 1 ∂ 2 ξ(x, t) = 2 , (1.1) ∂x2 c ∂t2 whose general solution involves the superposition of terms of the form ³ ωn ωn ´ ξn (x, t) = An sin x + Bn cos x (Cn sin ωn t + Dn cos ωn t) , c c

n = 1, 2, . . . ,

(1.2) where An , Bn , Cn , Dn are constants, and ωn x/c are dimensionless parameters [3]. The constants An , Bn , and Cn , Dn are determined by imposing the boundary and initial conditions. The axial wave velocity, c, can be expressed in terms of the Young’s modulus, E, and the density, ρ, of the material as c2 =

E . ρ

(1.3)

The equation of motion for the axial vibrations of a drillstring of crosssection As , accounting for damping and subjected to an external forcing function can be described by the second-order hyperbolic equation [15, 23, 110] µ ¶ ∂ 2ξ ∂ξ ∂ 2ξ ∂ξ ∂ξ ρ 2 + ca − E 2 + ρg = ga x, t, ξ, , , (1.4) ∂t ∂t ∂x ∂x ∂t where ca is a damping factor, g is the acceleration due to gravity, and ga is the external axial force per unit mass applied on the drillstring. In many cases, such as in the presence of nonlinearities or arbitrary forcing functions in time and space, a closed-form solution is difficult, or even impossible, to obtain. In these cases, alternative procedures based on numerical techniques can prove useful including finite differences [4, 121], boundary elements [12, 17, 18] and finite elements [5, 62, 83, 102, 104]. 5

1.2.5

Bit-Bounce Modeling

The bouncing of the bit correspond to the intermittent lift of the drilling assembly off the formation. This phenomenon relates primarily to tricone bits as those tend to create a pattern on the surface of the rock that may result in large amplitude longitudinal vibrations of the BHA. Spanos et al. [118] presented a model that considers the coupling of axial and torsional vibrations of the BHA submitted to an excitation originating from the rock surface. This representation relied on a sinusoidal angular variation of the elevation of the surface without radial variation. Further, a quarter-sine radial variation established the continuity of the surface in its center; that is, ( S(r, φ) =

³ S0 sin

r π ∆rb 2

´ sin(3φ), 0 ≤ r ≤ ∆rb and 0 ≤ φ ≤ 2π

So sin(3φ),

∆rb ≤ r ≤ rb ,

(1.5)

where rb is the radius of the bore-hole, ∆rb is a smaller radius than rb , and r and φ are the radial and angular coordinates, respectively. Then, the axial model was combined with a torsional one, in an effort to capture the coupling of these two vibration modes during a lift-off period of the bit. Finally, formulating conditions for lift-off of the bit and for resuming contact permitted to conduct numerical analysis of the phenomenon. The special aspects of the solution are as follows. Condition for Lift-Off. When the drill bit moves in contact with the formation at a certain time, its axial displacement after the time step, due to free vibration, can be calculated from the governing equation of motion by setting the excitation equal to zero. If this displacement is above the corresponding value of the profile elevation, then the drill bit is no more in contact with the formation. Condition for Resuming Contact. When the drill bit is not in contact with the formation at a certain time and the displacement calculated from the free vibration equation for the next time is below the corresponding value of the profile elevation, then the drill-bit will be intercepted by the profile. 6

To calculate the intermediate value of the time at which the drill bit comes in contact with the formation, an interpolation scheme such as in the NewtonRaphson method can be applied. Criterion of Bit Motion. When the drill bit is in contact with the formation, its rotational motion is governed by the motive torque from stiffness of the drillstring above and the resisting TOB from the formation. The drill bit accelerates when the first is greater than the second, and vice versa. But the TOB is a function of the WOB, which again depends on the position of the drill bit. Thus, the axial and torsional vibrations become coupled at the bit when the latter is in contact with the formation; an iterative scheme is necessary to solve the simultaneous equations of axial and torsional vibrations. Frequency Dependence of Damping Matrices. While deriving the expression of the damping matrix, it is assumed that the matrix depend; only on the dominant frequency of vibration. In evaluating the expression at a certain step, the dominant mode has to be assumed before solving the system of equations. After solving it, the dominant mode can be checked from the modal expansion of the displacement vector. In case of disagreement, the steps are to be repeated. The analytical model thus affords the option of a meaningful investigation of the complex phenomenon of bit lift-off. It is evident that the detection of the critical rotary speeds depends on the discretization scheme. Priority should be given to the scheme which can capture, at least approximately, all the resonant frequencies falling within the operating range. In this approach, the formation surface has only one spatial frequency, that corresponding to a trilobed formation. This represents the most widely accepted formation surface profile in case of drilling with tricone bits. The necessity of having a polyharmonic profile is not felt in modeling the interaction among the axial vibrations of the drillstring, the bit lift-off, and the lobe amplitude modulation. Thus, the frequency of excitation is only three times the frequency of the rotary speed and higher multiples are neglected. A polyharmonic profile can be incorporated, in which case, of course, the optimum discretization scheme has to be changed accordingly. The model can be enhanced to reduce the extent of physical idealization. 7

The obvious extension is the incorporation of the mud pressure fluctuation and the bending vibrations. The model can be elaborated by introducing the formation impedance. Also, the friction from side wall can be included. Since the solution is based on numerical integration, it can accommodate nonlinear friction models. The lobe amplitude modulation model can be developed by considering the cone geometry and the details of the bit-teethformation interaction. For adequate calibration, the model has to be tested for drillstrings with more complicated geometric properties and drilling under several operating conditions and formations. It can then be used for realtime feedback to the driller. The interested reader can find a detailed description of this modelling approach in [110, 118].

1.3 1.3.1

Torsional Vibrations Preliminary Remarks

Downhole measurements show that the application of a constant rotary speed at the surface does not necessarily translate into a steady rotational motion of the drill-bit. In fact, the downhole torsional speed typically shows largeamplitude fluctuations in time. This discrepancy in the rotational speed of the drillstring is due to the large torsional flexibility of the drilling assembly [114], and the presence of torsional modes. Drillstring torsional vibrations remained undetected for a long time, perhaps because of the large inertia of the rotary table. The table acts almost as a clamped connection at the top that strongly attenuates torsional vibration modes travelling upward from the drill-bit. Nevertheless, significant dynamic torque fluctuation can be experienced at the rig floor [34]. The torsional vibration modes of drilling assemblies can be classified in two categories: transient and stationary. Transient vibrations correspond to localized variations of drilling conditions, as encountered, for example, when the lithology changes. Stationary vibrations may also develop at least for an extended time segment; certainly the most extensively investigated such effect is the stick-clip pattern. Indeed, the bit stick-slip is arguably the tor8

sional vibration pattern that has the most influence on the drilling process especially in highly deviated wells. Accordingly, it has received much attention from the engineering community. Drillstring torsional vibrations, like their axial counterparts, may hinder drilling. Torsional vibrations can lead to excessive loadings resulting in equipment wear, joint failure, or damage of the drill-bit [14, 48]. Previous literature reviews on this subject include [46, 65, 71, 96, 119, 124].

1.3.2

The Stick-Slip Phenomenon

An important class of torsional vibrations is associated with the stick-slip behavior of the drill-bit. Stick-slip is a self excited torsional vibration induced by the nonlinear relationship between the friction induced torque and the angular velocity at the bit [59]. It can produce rotational speeds as high as ten times the nominal rotary speed, as well as total standstill of the bit. The stick-slip phenomenon has been observed to occur during as much as 50% of the drilling time [44]. An early investigation of the friction induced torsional vibrations, and stick-slip of the drill-bit has been reported in [6]. Since then, the stick-slip behavior of drill-bits has been extensively examined both analytically and experimentally. The drill-bit might come to a standstill due to a sudden WOB increase, or combined effects of significant drag, a tight hole, severe “dog-legs”, and “key-seatings”. The static friction which must be overcome for starting again the drill-bit rotation can be significantly higher than the normal Coulomb friction acting on the drilling assembly. Since the rotary table or the topdrive system are consistently turning, the drillstring stores torsional energy and twists up. When the available torque can finally overcome the static friction, the rotational energy is suddenly released and the drill-bit comes loose. It whips at very high angular velocities,sometimes as fast as ten times the nominal rotational speed [44]. As the elastic torsional energy of the assembly decreases, so does its rotational speed. Then, the drill-bit ultimately comes to a standstill again and the whole cycle repeats itself. Therefore, the stickslip phenomenon is a self-excited mechanism, that is, the excitation forces are produced by the motion of the system itself and the nonlinear nature of the downhole friction [29, 72, 76, 89]. However, the phenomenon requires 9

certain drilling conditions before it can develop. Stick-slip of the bit will not ordinarily occur if the drillstring is shorter than a critical length [38, 76]. The critical length of the assembly is a function of the rotary speed of the string, the dry friction, and the system’s viscous damping [75]. During stick-slip, assuming a constant rotary rate, the longer the drilling assembly is the more severe the torsional vibrations are. As the rotary speed approaches the critical speed, the stick-slip frequency approaches the torsional natural frequency of the drillstring. Further, novel models to investigate the self-excited vibrations of the drillstring have been presented in [63, 90, 105, 106]. Stick-slip may result in extensive bit wear, backward rotation, severe shock loadings of the drillstring, fatigue, and eventually failure of drilling equipment [44, 115]. It also decreases the ROP by 25%, typically, perhaps due to the nonlinear relationship between the drilling rate and the rotational speed of the drill-bit [124]. Also, the whipping and high speed rotation of the drill-bit in the slip phase can generate severe axial and lateral vibrations of the BHA that may result in drillstring connection failures. At the surface, the stick-slip phenomenon is sometimes characterized by a groaning noise and sawtooth-like variations, of large amplitude, of the applied torque [44, 68, 124]. Possible remedies include greater drillstring stiffness, higher BHA inertia, increased rotational speed, and a reduced difference between static and dynamic frictions [38, 124]. Measurement-while-drilling tools make it possible to detect the stick-slip phenomenon and identify its severity while drilling, thereby allowing real time adoption of curative operations. Control of the rotational behavior of a drilling assembly can be achieved by varying the rotary speed or the WOB, modifying mud properties (to alter downhole friction), and changing the type of drill-bit or the configuration of the BHA [115]. In a parallel direction, an increasingly popular solution to stick-slip introduced in the 1980s is to increase the drillstring damping by means of an active damping system. This system reduces the torque fluctuations and torsional drillstring vibrations affecting in this manner the stick-slip conditions. The underlying concept is to reduce the amplitude of the downhole rotational vibrations using a closed circuit that provides torque feedback. The feedback is used by the rotational drive which slows down the rotary rate when the torque increases and speeds it up when the torque decreases. This system is often called impedance control system or soft torque system 10

[39, 41, 59, 60, 115, 124]. A related procedure is given in [123]. Note that, the effectiveness of H∞ control in suppressing stick-slip oscillations has been studied in [111]. Further, an optimal state feedback control that can be effective in suppressing stick-slip oscillations once they are initiated has been suggested in [134]. Finally, a control approach based on modeling error compensation has been proposed in [103].

1.3.3

Modeling Perspective

The wave equation has been used to describe the torsional behavior of drilling assemblies [3, 11] since the early 1960’s. Another approach assuming a frictioninduced torsional drillstring vibration mechanism which leads to self-excited downhole vibrations related phenomena and, particularly, the stick-slip of the bit [6] was introduced in the early 1980’s. Fourier transforms have also been used to compute torsional resonant frequencies [53]. Further, the problem has been modeled as a single-degree-of-freedom (SDOF) system [38, 68, 76, 124], multiple-degree-of-freedom system [135]and as a continuous system [14, 94].

1.3.4

Continuous Model

Torsional drilling vibrations are primarily associated with PDC bits [23] because these bits are associated with high downhole friction coefficients that can result in the stick-slip oscillations of the bit. Briefly, the stick-slip phenomenon consists of the repeated winding–unwinding behavior of the assembly about its axis. Because of the damaging consequences of the stick-slip phenomenon effort has been devoted in understanding, and predicting the torsional vibrations of drillstrings. The governing differential for the torsional vibration of the assembly is similar to that governing its axial vibrations. This explains why early investigations often considered the two vibration modes simultaneously (e.g. see [3, 8, 9, 37]). The torsional behavior of the assembly is described by ∂ 2θ ∂ 2θ − JG = gt (x, θ, t) , (1.6) ∂t2 ∂x2 where J is the polar moment of inertia of the considered cross-section of drillstring, θ is the angular displacement of that section, ct is the damping coefficient for torsional vibrations, G is the shear modulus of the drilling ρJ

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assembly material, and gt (x, θ, t) is the torsional applied load. The product JG quantifies the torsional stiffness of the system. A rigorous derivation of Eq. 1.6 can be found in [28].

1.3.5

Stick-Slip Modeling

This section presents an approach to analytical modeling of the stick-slip; it relies primarily on the model presented by Dawson et al. in 1987 [38]. This approach uses a SDOF representation of the drillstring in which an massless torsional spring of stiffness k models the entire length of the drilling assembly. The rotary table drives the system at the surface at a constant speed Ω. Therefore, the equation of motion is ˙ + kφ = kΩt , I φ¨ + cr φ˙ + F (φ)

(1.7)

where φ is the angular displacement of the BHA, cr is the coefficient of viscous damping, k is the torsional stiffness of the drillstring, I is the polar ˙ denotes the moment of inertia with respect to the rotation axis, and F (φ) friction-induced forces. Eq. 1.7 may be normalized by the moment of inertia, yielding ˙ + ω 2 φ = ω 2 Ωt , φ¨ + 2ζω0 φ˙ + f (φ) 0 0

(1.8)

where ω0 =

p

cr k/I and ζ = √ . 2 kI

(1.9)

˙ of the form Assume f (φ) ( f1 − ˙ = f (φ) f2

f1 −f2 ˙ φ V0

0 6 φ˙ < V0 , V0 6 φ˙

(1.10)

that is, a piecewise linear model with a change at φ˙ = V0 . The parameters f1 , f2 , and V0 in Eq. 1.10 depend on the physical characteristics of the drilling assembly. Note that this simple model considers the reduction of the friction when the system switches from a static to kinetic state. Next, to investigate the stick-slip behavior of the assembly, one considers two distinct time segments. The first phase extends from t = 0 to t = t1 , 12

which the time at which the angular velocity of the drillstring equals V0 . During this time, the bit is in the slipping phase; its corresponding equation of motion is f1 − f2 ˙ φ¨1 + 2ζω0 φ˙1 + f1 − φ1 + ω02 (φ1 − Ωt) = 0 , V0

(1.11)

or, equivalently, φ¨1 + 2ζ1 ω0 φ˙1 + ω02 φ1 = ω02 Ωt − f1 ,

(1.12)

where the introduction of the damping ratio ζ1 simplifies the notation. Note that f1 − f2 . (1.13) 2V0 ω0 For the slipping phase, the equation of motion as described in Eq. 1.12 is subject to the initial conditions ( φ1 (t)|t=0 = −f1 /ω02 . (1.14) φ˙1 (t)|t=0 = 0 ζ1 = ζ −

The solution of Eq. 1.12 depends on the value of ζ1 . Specifically, for ζ1 < 1, the solution of the equation of motion is ([38], [28]) φ1 (t) = [c1 sin ωd t + c2 cos ωd t] e−ζ1 ω0 t −

f1 + 2ζ1 ω0 Ω + Ωt , ω02

(1.15)

where q Ω(2ζ12 − 1) 2ζ1 Ω 1 − ζ12 ω0 , c1 = and c2 = . ω0 ω0 Similarly, for ζ1 > 1, the solution of Eq. 1.12 is ωd =

h √2 i √ f1 + 2ζ1 ω0 Ω 0 ζ1 −1ω0 t 0 − ζ12 −1ω0 t φ1 (t) = c1 e + c2 e e−ζ1 ω0 t − + Ωt , ω02 (1.16) where 13

p p Ω(−2ζ12 + 2ζ1 ζ12 − 1 + 1) Ω(2ζ12 + 2ζ1 ζ12 − 1 − 1) 0 p p = and c2 = . 2ω0 ζ12 − 1 2ω0 ζ12 − 1

c01

Taking the first time derivative of φ1 (t) in Eq. 1.15, or Eq. 1.16 yields the rotational speed of the BHA, φ˙1 (t). By definition, the time t1 is the instant at which φ˙1 (t) = V0 . Therefore, taking the first time derivative of Eq. 1.15, or Eq. 1.16 and solving for the smallest roots of either equation yields t1 . Then, substituting t1 back in this equation yields the value of the displacement of the BHA. Next, one is interested in the sticking phase of the pattern for which the equation of motion of the BHA may be written φ¨2 + 2ζω0 φ˙2 + ω02 φ2 = ω02 Ωt − f2 ,

(1.17)

with the initial conditions φ2 (t)|t=0 = φ1 (t)|t=t1 and φ˙2 (t)|t=0 = V0 .

(1.18)

The solution of Eq. 1.17 is [38] φ2 (t) = e−ζω0 t (c3 cos ωb t + c4 sin ωb t) −

f2 + 2ζω0 Ω + Ωt ω02

(1.19)

where ωb =

p

1 − ζ 2 ω0 ,

and · ¶¸ µ 1 f + 2ζω Ω f + 2ζω Ω 2 0 2 0 c3 = φ˙1 (t1 )+ and c4 = V0 − Ω + ζω0 φ˙1 (t1 ) + . ω02 ωb ω02 As for the slipping phase, taking the first time derivative of Eq. 1.19 and solving for the smallest root of φ˙2 (t) = V0 yields the time t2 at which the rotational velocity of the BHA equals V0 again. Beyond t2 , Eq. 1.11 governs the BHA displacement again.

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After the BHA is stuck, its velocity remains null and the magnitude of the displacement φ increases linearly until φ = −f1 /ω02 . Therefore, the torque originating from the twisting of the drillstring is T = kφ where k is the torsional stiffness that Eq. 1.7 introduced.

1.4

Lateral Vibrations

1.4.1

Preliminary Remarks

Lateral vibrations also known as transverse, bending, or flexural vibrations are widely recognized as the leading cause of drillstring and BHA failures [22, 85, 126]. Paradoxically, the impact of drillstring lateral modes of vibration remained unrecognized for a considerable time period because most lateral vibrations do not travel to the surface, even in vertical wells [23]. Furthermore, lateral vibrations are dispersive and of frequencies higher than those of their torsional counterparts. Accordingly, they attenuate rapidly while propagating towards the surface [97]. Therefore, they are difficult to be detected based on surface measurements alone. Developments of downhole measurement techniques, especially MWD tools, have helped capturing the significance of these vibrations and their impact on equipment failures [126]. Various downhole mechanisms can induce lateral oscillatory modes, including primarily bit–formation and drillstring–bore-hole interactions. Weighton-bit fluctuations may also give rise to lateral instabilities due to linear axial–lateral coupling [126]. Also, the initial curvature of the BHA can result in lateral vibrations [97, 126]. Many studies have addressed the harmful effects of the lateral vibrations to the BHA [2, 16, 22, 26, 43, 85, 86, 108, 126]. Further, lateral vibrations cause severe damage to the bore-hole wall [58, 81] and affect the drilling direction [84]. Also, lateral vibrations can result in precessional instabilities [23]. Moreover, lateral vibrations may initiate bore-hole formation patterns resulting in axial and torsional drill-bit vibrations [35]. Despite their inherent damaging nature, flexural vibrations can be used in a positive manner by providing directional control at the drill-bit and by increasing the ROP [23, 61]. 15

Previous literature surveys on this subject include [21, 46, 96].

1.4.2

The Whirling Phenomenon

An important subset of lateral vibrations is the whirling of the bottom-hole assembly. Whirling is a condition where the instantaneous center of rotation moves about the bit face as the bit rotates [129] and can be forward, backward or chaotic. Whirling develops when the dynamic forces applied at the bit cause its center of rotation to move as the bit rotates [13, 126]. Several factors can induce whirling. Mass imbalance, such as that created by MWD tools, or an initially bent BHA, together with high compressive loads generated by WOB can produce some eccentricity of the drilling assembly. As drill-collars rotate, this eccentricity gives rise to a dynamic imbalance. The center of mass is then sent off the centerline of the drillstring, resulting in bent drill-collars. The magnitude of the centrifugal force acting, in a D’Alembert’s sense, at the center of mass of the collar is proportional to the initial eccentricity, the square of the rotation rate, and the mass of the collar [65, 126]. Numerical solutions of the “eigenproblem” for whirling modes for a range of whirl speeds and an attempt to correlate the numerical predictions with observation by using a small experimental rig have been reported in [27]. Further, the tuning of the rotation rate of the drill-collars with their natural frequencies can induce part of the BHA to bow out of its natural shape and perform forward synchronous whirl. Maximum bending deflections occur at rotary rates close to the lateral natural frequencies of the drill-collars, called critical rate. This critical rotary speed is modified by fluid-added mass, stabilizer clearance, and stabilizer friction, of which the respective influences are complex [58]. Further, the amplitude of vibrations due to bit whirl increases with the formation strength for both PDC and RC bits. Finally, supplementary factors affecting the whirl behavior include viscous fluid damping, gravity, coupling of axial with torsional and lateral vibration modes, and wall contact [57]. Bit whirl is quite harmful for drag bits with PDC inserts [13, 129]. Tricone bits, on the other hand, do not suffer as extensively from whirling because they penetrate the formation, thereby reducing their sideway motion [65]. Drillstring components experiencing whirling cause a series of problems. 16

Whirling is a major cause of reduced ROP and early failure of downhole equipment [81]. Further, it contributes significantly to drill-collar wear and connection fatigue. Moreover, whirl induced lateral displacements of the assembly can result in severe and repeated contacts with the bore-hole wall, which can lead to surface abrasion of the drilling equipment, and deterioration of the well condition. Most of the BHA operates in compression making it a region where buckling and whirling are likely to occur. Severe whirling occurrences can be observed on the rig floor by the lateral motion of the travelling block, and the whipping of the draw-works. The BHA whirling nonetheless, remains difficult to detect, while fatigue accumulates and eventually results in failure of the equipment. The forward synchronous whirl or forward whirl of a drill-collar occurs when that section rotates around the bore-hole with the same direction as the drillstring rotation generated by the rotary table [57]. During forward whirl, the same side of the collar is in continuous contact with the bore-hole wall. The mechanism is therefore a cause of flat-spots on collar joints [126]. Forward whirl may develop during normal drilling operations. It is usually induced by an out of balance mass, although it is unlikely to develop if the mass eccentricity is less then the stabilizer clearance [57]. Also, the friction resulting from stabilizers - bore-hole wall contacts introduces instability at certain whirling frequencies, resulting in non-synchronous, self-excited, large amplitude vibrations [125]. Thus, depending on the downhole conditions and drilling parameters, stable forward whirl may develop or evolve into other whirling patterns. For instance, the repeated impacts of the collars on the bore-hole wall can gradually transform forward whirl into backward whirl [58]. The best known kind of non-synchronous collar whirl is the backward whirl. It may occur during normal operations, and develops when the instantaneous center of rotation of a drillstring section lies between the center of the mass of the collar and the bore-hole wall. More specifically, the whirl qualifies as backward if the instantaneous center of rotation travels around the bore-hole in a direction opposite to the driving rotation. Backward whirl can originate from the friction between the stabilizers and the well-bore if it exceeds structural and hydrodynamic damping forces [113]. This leads to 17

backward rolling or slipping motions of the stabilizers that, in turn, can produce a self-excited backward whirling motion of the drill-collars [57]. Further, collars can drive the whirl, if the slip is positive, or resist it, for negative slip. The case of extreme backward whirl is called pure backward whirl. It is the rolling without slipping of the drill-collars on the inside of the well-bore in the direction opposite to that imposed by the rotary table [58]. The friction induced transition from forward to backward whirl leads to perfect backward whirl at the limit of zero mass eccentricity [125]. Backward whirl is a significant threat to drilling assemblies because it superimposes on the forward rotary speed, thereby inducing fluctuating bending moments with periodical changes of sign [57]. These strong moment fluctuations give rise to high amplitude bending stress cycles. Thus, the fatigue life of the drill-collar connections can be significantly shortened when the bending stress cycles accumulate at a rate much greater than the rotary speed [126]. The backward whirl of drill-collars at a frequency close to one of the natural frequencies of the assembly may lead to wall contact and can produce drill-collar precession, that is, a backward rolling motion of the drill-collars along the bore-hole wall. In practice, synchronous whirl cannot develop if the clearance at the stabilizers exceeds the eccentricity of the center of mass of the collars. When forward whirl is impossible, collars may either whirl backward or in a irregular fashion [65]. Extreme cases of non-periodic behaviors are called chaotic whirl, as the motion then depends strongly on initial conditions. The irregular motion is induced by nonlinear fluid forces, stabilizer clearance, and borehole wall interactions. Chaotic motions can also develop for low values of the stabilizer friction [58]. Finally, chaotic whirling of drilling components may comprise minor components of randomness [66].

1.4.3

Modeling Perspective

Lateral vibrations of drillstrings, and modelling techniques for this problem have been the focus of several publications since the mid 1960s. The two common ones involve closed form solutions, and the finite elements discretization. The closed form solutions have been the basis of early analyses. However the great complexity of the problem has limited their applicability. Fortunately, the versatility of the finite elements analysis, and the advent of computers have facilitated the consideration of several parameters involved in the problem. To date, several aspects of drillstring behavior related to lateral 18

vibrations have been studied. Typical studies address natural frequencies determination [19, 24, 50], critical bending stresses calculations [52, 85, 99, 117], stability analysis [127], and lateral displacements prediction of drilling assemblies [46, 133]. Further, some analyses have identified critical failure parameters, and the conditions that trigger a transfer of energy between lateral and rotational modes of vibration [133].

1.4.4

Continuous Model

Small slopes assumption is adopted, and the Euler-Bernoulli beam theory is considered. In this regard, the following hypotheses are made. The plane sections which are normal to the beam axis before deformation remain plane and normal to the beam axis after deformation; this implies that the deformations are due to bending only. Also, the beam is considered elastic, and Hooke’s law relates the stresses and the strains.The Euler-Bernoulli equation is µ ¶ ∂ 2u ∂2 ∂ 2u ρ 2 + 2 EIz 2 = g(x, t), (1.20) ∂t ∂x ∂x where u(x, t), ρ, E, Iz are the lateral displacement, the mass density per volume, the modulus of elasticity and the relevant moment of inertia of the cross section of the beam respectively; g(x, t) denotes the external loading, x refers to the axis of the beam, and t indicates time. Then, the effect of the axial forces is taken into account. The beam is subjected to an axial force P that is considered positive when tensile. The axial force induces an additional moment which modifies the shear-moment relationship and leads to the following differential equation µ ¶ ∂ 2u ∂2 ∂ 2u ∂ 2u ρ 2 + 2 EIz 2 − P 2 = g(x, t). (1.21) ∂t ∂x ∂x ∂x

1.4.5

Whirling Modeling

Many analyses have approached the whirling phenomenon by adopting a twodimensional, single-degree-of-freedom representation of the assembly [58, 65, 126]. This section presents the fundamental principles of this approach; it also explains a procedure to represent the whirling of a drillstring with a 19

multi-degree-of-freedom system.

α

x z

y

A Ω O

Drill-string ω

P

Bore-hole

rc a

Figure 1.1: Coordinate system for whirling investigation (after [113]). Multiple factors can induce whirling. Mass imbalance, such as that created by measurement-while-drilling tools, or an initially bent bottom-hole assembly, together with weight-on-bit-generated high compressive loads, and gravity can produce some eccentricity of the drilling assembly. As drill-collars rotate, this eccentricity gives rise to a dynamic imbalance. The center of mass is then “sent off” the centerline of the drillstring, resulting in bent collars. The magnitude of the centrifugal force acting, in a d’Alembert context, at the center of mass of the collar is proportional to the initial eccentricity, the square of the rotation rate, and the mass of the collar [65, 126]. In these instances, the torsional motion of the drilling assembly generates, in a D’Alambert sense, a force acting at the center of mass of the drill-collar, which is radially directed, and is pointing away from the well-bore center; this force induces bending of the collar section. In a case where the rotational speed of the drilling assembly, Ω, approaches one of its natural fre20

quencies, resonance occurs and the deflection of the collar section increases significantly.Further, the tuning of the rotation rate of the collars with their natural frequencies can make part of the BHA bow out of its natural shape and perform forward synchronous whirl [92]. Maximum lateral deflections occur at rotary rates close to the lateral natural frequencies of the collars, the so-called critical rate. This critical rotary speed is modified by fluidadded mass, stabilizer clearance, and stabilizer friction, of which the respective influences are complex [58]. Furthermore, the amplitude of vibrations due to bit whirl increases with the formation strength for both polycrystalline diamond compact and roller-cone bits [47]. Finally, supplementary factors affecting the whirl behavior include viscous fluid damping (which are velocity-squared dependent drag forces); gravity; coupling of axial, torsional, and lateral vibration modes; and wall contact [57]. SDOF Formulation The whirling motion of drillstrings relates to torsional vibrations in the sense that whirling requires rotation of the assembly to develop. The whirling of BHA components involves motions of these components about the bore-hole central axis. Previous analyses [57, 58, 65, 66, 113] have characterized the whirling of a certain location in the BHA. That is, these approaches pursue a single degree-of-freedom representation of the bottom-hole assembly. Since a primary consequence of whirling is the fatigue failure of drilling equipment, the location that is investigated along the drillstring is ordinarily midway between two stabilizers, where the lateral deflection is the largest. Two- and three-dimensional characterizations have been derived. Assuming a constant rotary speed, the equations of motion of a point equidistant from placed between two stabilizers may be written as [70] m¨ y + cw y˙ + kw y = me0 Ω2 cos(Ωt) ,

(1.22)

and m¨ z + cw z˙ + kw z = me0 Ω2 cos(Ωt) ,

(1.23)

where y and z are the lateral coordinates introduced in Fig. ??, m is the equivalent mass of the collar, cw is the damping coefficient, kw is the collar equivalent lateral stiffness, e0 is the eccentricity of the center of mass, and Ω is the rotational speed of the drilling assembly. Equations 1.22 and 1.23 21

describe the planar motion of the considered point accounting for inertial and damping forces. An elastic restoring term is also considered. Note that the excitation term on the right hand-side of Eq. 1.22 and Eq. 1.23 incorporates the force arising from the rotation of the unbalanced element. A derivation of the equations of motion of a particular point along the drillstring accounting for fluid forces, stabilizer clearance, borehole contact, gravity, and linear and parametric coupling has been provided in [65]. Specifically, introducing the dimensionless polar coordinates (r, θ) of the center of the collar section at the considered location along the drilling assembly, the equations of motion may be written as µ ¶ Ω Ω β¨ r − βrθ + λ r(β − 1) 2θ˙ − λ + βΓr˙ +