Bit-sticking Phenomena in a Multidegree-of-freedom Controlled Drillstring Figure 2: A Simplified Model for a Conventional Vertical Drillstring AB Drilling mud H (N) ϕr • Lp(m) Drill pipes Jp
BHA Bit
Lb(m) ϕb • ϕb • Tb(Nm) Wob(N) Tb Jb
A. Main elements in a conventional vertical drillstring; B. Mechanical model describing the torsional behaviour of a conventional vertical drillstring.30
BHA = bottom-hole assembly; ϕr = angular velocity of the top-rotary system; •
H = Hook on load; Lp = drill pipeline length; Lb = length of BHA; Tb = torque on the bit; •
ϕb = angular velocity of the bit; ϕr = angular displacement of
the top-rotary system; ϕb = angular dsplacement of the bit; Tm = torque given by the electrical motor at the top-rotary system; Tar = viscous damping torque associated with Jr (lubrication); Tb= torque on the bit; Jr = torsional inertia moment of the top-rotary system; Jp = torsional inertia moment of a drill pipe; Jl = torsional inertia moment of the drill collars; Jb = torsional inertia moment of the bit; ct = drill pipe torsional damping coefficient; ctl = dtill collars torsional damping co-efficient; ctb = bit torsional damping co-efficient; kt = drill pipe torsional stiffness; ktb = bit torsional stiffness; ktl = drill collars torsional stiffness; Wob = weight on the bit.
Figure 3: Bit-sticking Phenomena A
1 2 3 4 5 6 7 8
0 010 20 ϕr B
1 2 3 4 5 6 7
0
-1 -2
30
•• ϕb •
Time (seconds) ϕp
40 50 60
Jl Jp
kt kt ktl ktb ct ct ctl ctb ϕb Bit
Drill pipe p-1 Drill pipe p BHA
How Can We Model the Different Complex Behaviours? A Simplified Model
Every mathematical model is an approximation of the real world, and is full of limitations. However, some models are better than others at describing the evolution of certain physical and engineering systems. For analysis and control purposes, simplifications are mandatory in order to keep the equations manageable. Lumped-parameter models lead to more simple analysis and system simulation in comparison with partial derivatives models. A great variety of lumped-parameter models have been proposed to describe the drillstring torsional behaviour and BHA and bit-sticking phenomena. Most of them consider two degrees of freedom (2-DOF), i.e they describe the dynamics of the top-rotary system and the BHA (typically considered as the bit).5,26,27,29,42–45
The Need for a Multidegree–of–freedom Model for the Drillstring
Although 2-DOF models have revealed important characteristics of the drillstring and the downhole conditions, they still do not reflect two
important facts: the drill pipeline length (Lp) increases as the drilling operation makes progress, and the oscillations along the connected drill pipes and the drill collars (just above the bit). Recently, an n-DOF lumped-parameter discontinuous model that reflects these facts has been proposed (see reference 30). The discontinuity is introduced by the bit–rock interaction, which is modelled by means of a dry friction combined with an exponential-decaying law. The model is presented in Figure 2.
From the figure, four kinds of elements are distinguished: the top-rotary system, p drill pipes modelled as linear springs of torsional
stiffness kt and torsional damping ct, the BHA (including the drill collars) and the bit. The drill pipes are connected to the inertias Jr and Jp, corresponding to the inertia of the top-rotary system and to the inertia of each drill pipe. The number of drill pipes can be modified depending on system analysis requirements. The p-th drill pipe is
connected to the drill collars (Jl) by means of ktl and ctl. Finally, the drill collars are connected to the bit (Jb) by means of ktb and ctb. At the bit, a viscous-damping torque and a dry-friction torque are
taken into account. A viscous-damping torque is also considered at the top-drive system.
0 102030 ϕr 40
•• ϕb •
Time (seconds) ϕp
A. Stick-slip situation with u = 8,138 Nm, weight on the bit (Wob) = 74,386N; B. Permanent stuck bit with u = 8,138Nm, Wob = 82,000N.
50 60 70 80
The following assumptions are made: the borehole and the drillstring are both vertical and straight, no lateral bit motion is present, the friction in the pipe connections and between the pipes and the borehole are neglected, the drilling mud is simplified by a viscous-type friction element at the bit, the drilling mud fluids’ orbital motion is considered to be laminar, i.e. without turbulence, the motor dynamics are not considered, the drive torque is supposed to be constant and positive and the drill pipes are considered to have the same inertia. Consequently, the drillstring torsional model takes the following form:
72 EXPLORATION & PRODUCTION – VOLUME 8 ISSUE 2 ϕr • Tar
Jp Jr
Jp kt kt Tm ϕr ct ct
Top-rotary system Drill pipe 1 Drill pipe 2
Three drilling goals – interpreted here as control goals – are highlighted: to obtain a constant rotary velocity at the top-rotary system and the bit, to reduce mechanical vibrations and to maintain optimal operation conditions despite uncertainties and changing operation conditions, in addition to preserving stability and recovering from shock – in other words, to design a robust and resilient drilling process. The dynamic modelling of the drillstring will play a major role in meeting these drilling goals.
Velocities (radians per second)
Velocities (radians per second)
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64 |
Page 65 |
Page 66 |
Page 67 |
Page 68 |
Page 69 |
Page 70 |
Page 71 |
Page 72 |
Page 73 |
Page 74 |
Page 75 |
Page 76 |
Page 77 |
Page 78 |
Page 79 |
Page 80 |
Page 81 |
Page 82 |
Page 83 |
Page 84 |
Page 85 |
Page 86 |
Page 87 |
Page 88 |
Page 89 |
Page 90 |
Page 91 |
Page 92 |
Page 93 |
Page 94 |
Page 95 |
Page 96 |
Page 97 |
Page 98 |
Page 99 |
Page 100 |
Page 101 |
Page 102 |
Page 103 |
Page 104 |
Page 105 |
Page 106 |
Page 107 |
Page 108