Oil and it's products are used constantly in everyday life. This project is concerned with the model

1
Slip Stick Delay Equations of Drill String
Dynamics
By Chris Welch Supervisor Dr Martin Homer
Oil and it's products are used constantly in
everyday life. This project is concerned with the
modelling of the torsional waves that occur
during the oil drilling process, which can lead
to damage of the drilling equipment.
Previous Work Balanov et al 2003 paper contains
an analysis of the torsional waves of a driven
drill string via numerical simulation, using the
NDDE at various values of two parameters in the
system. These parameters were the weight on
bit (downwards force applied to the drill bit),
A, and the speed of rotation of the drill bit, O
. The simulations produced interesting results
including the evolution of a period 2 limit
cycle from a period 1 limit cycle when O was
varied, multi-stability (possibly marking a
transition into chaos) and tori 3. In part of
this project numerical simulations were produced
and the results supported the claims made by
Balanov et al. However the frictional model used
in Balanov et al does not allow for the
possibility of 'slip stick oscillations thus a
new frictional model was introduced.
Results The two frictional models although
similar, gave rise to very different torsional
waves for the same parameters A and O.
Introduction There are three main parts of the
apparatus that need to be considered, these are
illustrated in Figure 1 and outlined below OIL
RIG contains the motor and control
electronics. DRILL STRING a long hollow shaft
which penetrates several kilometres through the
earths surface and holds the drill bit. DRILL
BIT the cutting tool at the end of the drill
string.
Figure 5 Continuous friction model (a) Period 1
Limit cycle A0.65 O0.3, (b) Period 2 limit
cycle A0.65 O0.46
Figure 5 a and b show the rise of a period 2
limit cycle from a period 1 limit cycle. Figure 5
(a) shows a period 1 limit cycle, the plot
repeats after one orbit and follows the same path
in phase space and figure 5(b) shows a period 2
limit cycle, the trajectory of the first half
period travels around once but does not rejoin,
only during the second half period does the
trajectory reconnect and repeat i.e. it takes 2
periods for the trajectory to repeat itself.
Slip Stick Analysis Coulomb friction may be a
more realistic model of the frictional force
rather than the continuous friction model
proposed by Balanov et al. As the drill bit
spins, Coulomb friction opposes the direction of
the velocity, but does not depend upon the
magnitude of the velocity. At zero velocity, the
frictional force can take any value between
certain fixed limits, thus a better model of slip
stick might be obtained by coulomb friction.
Figures 3 and 4 illustrate the differences
between the two frictional models.
Figure 1 Oil Drilling Equipment 1
The drill string is driven by a motor housed
within the oil rig, which applies a torque at the
top of the drill string, causing it to rotate,
making the drill bit rotate and cut through the
rock. As the drill bit cuts non-linear reaction
torques act upon the drill bit due to friction
2. The frictional torques at the drill bit can
reduce the speed of rotation and may even stop
the drill string, causing the drill bit to
stick. As the motor continues to apply a
torque to the drill string, and the motion at the
drill bit is slower due to friction, the drill
string can twist. When the energy within the
drill string is reaches a certain level and
overcomes the frictional force against the motion
of the drill bit, the drill bit slips. This is
known as the slip-stick scenario and can
destroy the drilling equipment 2.
Figure 6 Coulomb friction model (a) Plot for
A0.65 O0.3 for one period, (b) Plot for
A0.65 O0.3 for the final 100 s of the
simulation.
Figure 7 Coulomb friction model (a) Plot for
A0.65 O0.46 for one period, (b) Plot for
A0.46 O0.3 for two periods. (c) Plot for
A0.46 O0.3 for the final 100s of the
simulation.
Figure 6 shows a phase plot that has the
appearance of a period 1 limit cycle, the
trajectory appears to rejoins after one period.
However if the attractor were period 1 it
would follow the same path every consecutive
period. Figure 6(b) shows this is not the case,
as it shows many periods but none are
identical. Demonstrating that the attractor is
not period 1, but may be period 1 with a
ripple of chaos. Figure 7(a) the trajectory
does not rejoin itself after one period but
does seem to rejoin itself after 2 periods
Figure 7(b), However Figure 7(c) illustrates the
system over a longer time period, which possibly
has the characteristics of a torus.
Equation for Modelling Torsional Waves The
equation below models the motion of the torsional
waves
Coulomb friction was applied to the NDDE and
numerical simulations were produced to determine
if a change in the frictional model used would
have any effects on the results produced by
Balanov et al
It is a Neutral Delay Differential Equation
(NDDE), which has a delay in a derivative of the
state variable and a delay in the system state.
Due to these characteristics more initial
conditions are required to solve the equation.
Knowledge is required not only of the current
state, but also of the system state certain time
period ago (t in figure 2), to be able to solve
NDDEs.
Conclusions In this project numerical
simulations of a neutral delay differential
equation were made by implementing a continuous
frictional model, described in Balanov et al
2003, and a coulomb friction which allowed slip
stick to take place. Although the two frictional
models were similar, the systems behaviour and
the resulting self sustained oscillations were
very different for both frictional models. The
magnitude and behaviour of the torsional waves is
very different in each frictional model. This
shows that friction model must be very carefully
chosen when applied to the modelling equation, if
the frictional model used for numerical
simulation is not the same as that of the
frictional relationship between the drill bit and
surface then the data obtained from the numerical
simulation will be incorrect.
Figure 2 How the system obtains the delay value
References 1 Freunderrich C,C. 2005. How Oil
Drilling Works http//science.howstuffworks.com/oi
l-drilling2.htm 2 J Wilkinson. 2004 Slip
Stick Delay Equations of Drill String Dynamics.
University of Bristol Project Thesis 3 Balanov
AG, NB Janson, PVE McClintock, RW Tucker, CHT
Wang. 2003 Bifurcation analysis of a Neutral
delay differential equation modelling the
torsional motion of a driven drill string. Chaos
Solutions and Fractals 15 pgs 381-394