It is quite difficult to describe drill string vibration during drilling - complications arise from b it/so il interactions, contact forces (shocks and dry friction) and hole geometries, just to mention a few. Bogdanoff and Goldberg (1961) supported that random models w ould better represent this irregular behaviour. Transient analyses, reviewed in section 2.2.2, certainly model the drill string dynamic behaviour more accurately, but they require large computational effort. Mechanical vibrations have a strong effect on drilling performance through the achieved rate of penetration, accelerated bit w ear, string trajectory and so on. Furthermore, dynamic stresses resulting from vibrations are generically blamed for many drill string failures due to the accumulation of fatigue damage (see Dareing (1985) and Aarrestad and Kyllingstad (1988)) but in fact little is definitively known.
This chapter presents some simple (though representative!) m odels to evaluate the contribution of vibration to cyclic loading. Analytical solutions are developed for longitudinal (section 5.1), torsional (section 5.2) and lateral (section 5.3) vibrations. Dareing and Livesay's m odel (1968) is applied to longitudinal and torsional vibrations w ith the assum ption that the bit is subjected to sinusoidal displacem ent at a frequency three times the rotary speed and that dissipative forces are represented by a viscous term. The effect of tool joints in torsional and longitudinal vibrations was shown to be "insignificant" and therefore the segments of pipes can be considered continuous (Bradbury and Wilhoit, 1962). The lateral vibration of vertical drill strings is investigated using results from section 3.1 and from Huang and Dareing (1968), w ith damping forces and wall confinement effect disregarded.
5.1 - Longitudinal Vibration
A solution is initially developed for the drill string arrangement defined in Figure 5.1.1, and then generalised for any drill string configuration (Figure 5.1.2). The governing equation describing the longitudinal displacements derives from the balance of axial forces in an infinitesimal element (Figure 5.1.1). Thus :
- 7 / " ^ + P f + ( P i - P m ) ^ A 2 = 1 , 2 ( 5 . 1 . 1 )
where :
the subscripts f = 1 , 2 refer, respectively, to drill collar and drill pipe
segments ,
Vi = Vj (x,t) is the longitudinal displacem ent,
Ai is the cross-sectional area - A, = 0.7854 (D,^ - ,
Dj and d, are respectively the outside and inside pipe diameters , £, is the Young's modulus ,
Yi is the viscous damping coefficient,
pi is the density of the material,
Pyn is the density of the mud and
g is the acceleration due to gravity .
The damping factors of drill collars and drill pipes are obtained from the logarithm ic decrement of the transient responses of bit and kelly, respectively. The viscous damping model does not exactly represent the actual non-conservative forces - however, it constitutes an approximation to account for the dissipation of energy. Equation (5.1.1) admits static and dynamic solutions, respectively %( x) and Ui(x,t), given by :
Vi (x,t) = Ui(x) + Ui (x,t) (5.1.2)
Substituting equation (5.1.2) in (5.1.1) results in : -j2—
(5.1.3a)
Equation (5.1.3b) is the so called wave equation. The boundary conditions to be applied to the extremities and drill pipe-drill collar connection are given by :
u,(0) = 0 (5.1.4a)
E , ^ ( 0 ) = - ^ ^ (5.1.4b)
/ T \ — A T ^ 2
A j£ j— (Ijj = A2 E2 -t—(Li) (5.1.4c)
dx dx
ïZjfLj) = ÏÏ2(Lj) (5.1.4d)
UiiO, t) = Uo sin cû t (5.1.4e)
A , E , ^ ( L „ t ) + K U2(L2,t) + M ^ ( L „ t ) = 0 (5.1.4f)
dx dt
= A 2 E 2 - ^ ( L i,t) (5.1.4g)
Ujd^rt) = U2(L2,t) (5.1.4H)
where :
WOB is the weight on b it ,
Uo is the amplitude of bit displacem ent,
(0 is the circular frequency ,
ka is the spring stiffness of cables ,
M is the mass of kelly, swivel and travelling block and Li and L2 are the lengths as defined in Figure 5.1.1 .
The boundary conditions (5.1.4a) to (5.1.4d) and (5.1.4e) to (5.1.4h) apply respectively to equations (5.1.3a) and (5.1.3b). Equations (5.1.4a), (5.1.4d), (5.1.4e) and (5.1.4h) are related to displacement constraints and equations (5.1.4b), (5.1.4c), (5.1.4f) and (5.1.4g) are related to balance of forces. For three-cone drill bits the circular frequency co is equal to three times the rotary table speed. The spring stiffness may be determined by measuring the net elongation of cables w hen a known mass is picked up. The amplitude of bit displacement is reported to depend on the particular drilling situation. In this work it is assumed that Uo is either known or obtainable from a "measuring while drilling" device.
Solution of the static equations
Substituting the boundary conditions (5.1.4a) to (5.1.4d) in equation (5.1.3a) leads to :
üi(x) = % (5.1.5a)
ü J x ) Biais, x^ + b 2E, 2% 4- C2 &2 - gLj A ( P l ~ P m )~^2 (Pl - Pm^ A2E2 WOB A2E2 r - ( Pl ~Pm^ ( P l - P m ) — El 'WOB ---+ b.^ ■ " 2 [ El £ 2 J [a e , 'J (5.1.5b) (5.1.5c) (5.1.5d)
Let Pl and p2 be, respectively, the w eight "in mud", per unit length, of drill collar and drill pipe given by the following expressions :
P i = g { p i - P m ) A i = l , 2
Substituting (5.1.6) into equations (5.1.5a) to (5.1.5d) produces
Uj(x) - - W O B x
A, El
^ [(Pi - Pz)Li - WOB] X +
+ L Pl Pl
2AjEi 2A2E2 A2E2 Lj A2 E2 Aj Ei j
Hence the static axial stresses cr, (x) can be obtained by
Ai Ai
A2 A2
Solution of the dynamic equations
Equation (5.1.3b) admits the following solution :
u.(x, t) = Real {B- s i n ( x + b^) } (5.1.6) (5.1.7a) (5.1.7b) (5.1.8a) (5.1.8b) (5.1.9a)
; ; f = - l (5.1.9b)
I t
w h ere B„ bt and A, are complex numbers. The constants B, and h i are
determined by substituting (5.1.9a) in equations (5.1.4e) to (5.1.4h). Thus :
Bi sin bi = - j Uo (5.1.10a)
tan[X2L + b^ = (5.1.10b)
^ ' M c o - k ^
Bj Aj cos(Aj Lj + bj) = ^ 2 X2Cos[X2 Lj + ^2) (5.1.10c)
Aj Ej
Bj sm(Aj Lj + fcj) = B2 sin[X2 Lj + &2) (5.1.lOd)
Equations (5.1.10a) to (5.1.10d) represent a system of four equations and four unknowns whose solution gives Bj, B2, bi and &2. The dynamic axial
stress Oi(x,t) is expressed by :
(Ji(x,t) = E i ^ ( x , t ) = Real {B. A,. cos(A, % + &,) (5.1.11)
G e n e r a l i s a t i o n o f t h e s o l u t i o n
Actual drill strings comprise a number of drill collars, drill pipes and shock-subs, as illustrated in Figure 5.1.2. Once the case of drill collar-drill pipe is understood it becomes simple to extend the solution to general configurations. The solution for static axial stress can be generalised to :
âj(x) = ^ x - (5.1.12a)
A A
5,.(*) = {ffj.i(L,..;)A-i+Pi(*-i-,-3)}/A i = 2,3,...,m (5.1.12b)
where :
m is the number of pipe segments ,
i
4 = P i = s { P i - P m ) A i = l , 2 , . . . m .
Equation (5.1.9a) applies to each segm ent of pipe. The boundary conditions for the top of section i (i = 1, 2,..., m ), i.e., dàx = Ly are :
du, du
Ai Ei —^ (Li, t) - Ai^j (L i, t)
dx dx (5.1.13a)
~ ^f+3 (Lift) if = o o (5.1.13b)
[Ui(Li,t) - Ui^^(Li,t)] = if * ~ (5.1.13c) Combining equations (5.1.13a) and (5.1.13c) gives
®z+l — sin[XiLi + b i ) + cos(A,.E, + ) K.7 + 1 B, (5.1.14a) b,. = ia n - 4 4 + K i ) ~ L A + 1 A + i A + i Ei Ai 2>i K,7+1 (5.1.14b)
and combining equations (5.1.13a) and (5.1.13b) results in
i f ^ 7 + 1 =
4+1 = B,
sm(A,+iE,+&,+i)J (5.1.15a)
L^z+1 A+i^z+1 J
- A, I; (5.1.15b)
The boundary conditions at the bit and top of the drill string produce :
B, = - j u „
sinbj - cosb]
(5.2.2b)
where I is the polar moment of inertia of the kelly and k, is the torsional