1. 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. 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]. By Chris Welch Supervisor: Dr Martin Homer Equation for Modelling Torsional Waves The equation below models the motion of the torsional waves: 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 (τ in figure 2), to be able to solve NDDEs. 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, Ω. The simulations produced interesting results including; the evolution of a period 2 limit cycle from a period 1 limit cycle when Ω 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. 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. 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 Results The two frictional models although similar, gave rise to very different torsional waves for the same parameters A and Ω. 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. FRICTIONAL FORCE VERLOCITY Figure 4: Continuous Friction Figure 1 – Oil Drilling Equipment [1] Figure 5: Continuous friction model (a) Period 1 Limit cycle A=0.65 Ω=0.3, (b) Period 2 limit cycle A=0.65 Ω=0.46 x(t-τ ) x(t ) (a) x(t-τ ) x(t ) (b) Figure 6: Coulomb friction model (a) Plot for A=0.65 Ω=0.3 for one period, (b) Plot for A=0.65 Ω=0.3 for the final 100 s of the simulation. Figure 7: Coulomb friction model (a) Plot for A=0.65 Ω=0.46 for one period, (b) Plot for A=0.46 Ω=0.3 for two periods. (c) Plot for A=0.46 Ω=0.3 for the final 100s of the simulation. FRICTIONAL FORCE VERLOCITY Figure 3: Coulomb Friction Figure 2: How the system obtains the delay value x(t-τ ) x(t) x(t-τ) x(t) (a) (b) x(t-τ) x(t) (a) (b)(c) 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. 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. References [1] Freunderrich C,C. 2005. How Oil Drilling Works http://science.howstuffworks.com/oil-drilling2.htmhttp://science.howstuffworks.com/oil-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 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. Initial Condition System Output τ time perio d later -τ-τ t- τ t