Global axial–torsional dynamics during rotary drilling

Abstract

We have studied the global dynamics of the bottom hole assembly (BHA) during rotary drilling with a lumped parameter axial–torsional model for the drill-string and a linear cutting force model. Our approach accounts for bit-bounce and stick-slip along with the regenerative effect and is independent of the drill-string and the bit–rock interaction model. Regenerative axial dynamics due to variable depth of cut is incorporated through a functional description of the cut surface profile instead of a delay differential equation with a state-dependent delay. The evolution of the cut surface is governed by a nonlinear partial differential equation (PDE) which is coupled with the ordinary differential equations (ODEs) governing the longitudinal and angular dynamics of the BHA. The boundary condition for the PDE captures multiple regeneration in the event of bit-bounce. Interruption in the torsional dynamics is included by considering separate evolution equations for the various states during the stick period. Finite-dimensional approximation for our coupled PDE-ODE model has been obtained and validated by comparing our results against existing results. Bifurcation analysis of our system reveals a supercritical Hopf bifurcation leading to periodic vibrations without bit-bounce and stick-slip which is followed by solutions involving bit-bounce or stick-slip depending on the operating parameters. Further inroads into the unstable regime leads to a variety of complex behavior including co-existence of periodic and chaotic solutions involving both bit-bounce and stick-slip.

Introduction

Exploration of fossil fuels like oil and natural gas relies heavily on deep drilling systems. Deep drilling systems can be classified as rotary drilling, percussive drilling or a combination of both [1], [2], [3], [4]. However, rotary drilling has emerged as the most commonly used and cost-effective technique for drilling oil wells. Schematic of a typical drilling set-up for rotary drilling is shown in Fig. 1. It consists of a rotary table at the surface with a power transmission which transmits the rotary power required for drilling to the bottom hole assembly (BHA) which carries the drill–bit with the cutter through a series of drill pipes called the drill-string [5]. One of the phenomena plaguing drilling operations is self-excited vibrations which leads to drill-string failure such as fatigue [6], [7]. Therefore it is necessary to understand the dynamics and behavior of the drill-string during vibrations to avoid or reduce failures. Drill-string vibrations can be divided in three broad categories – axial, torsional and lateral which further lead to bit-bounce, stick-slip and whirl phenomena, respectively. There is an intricate coupling between these vibration modes and several studies have been devoted to better understand the full dynamics of the rotary drilling systems [8], [9], [10], [11]. In this paper, we consider the coupled axial and torsional vibrations of a rotary drilling system and explore parameter regimes involving excessive vibrations leading to bit-bounce (loss of contact between the drill–bit and the surface) and/or stick-slip (intermittent seizure of angular motion of the drill–bit).

One of the first approaches towards modeling the rotary drilling process was presented by Bailey and Finnie [12] who obtained the natural frequencies of axial and torsional vibrations of the drill-string from independent one-dimensional wave equation. This was followed by their experimental studies [13] in which they found the measured force, torque, and natural frequencies of the system to have some correspondence with the theoretical study [12] for an appropriate choice of the boundary conditions. Paslay and Bogy [14] specified a sinusoidal motion of the drill–bit as a representative motion during longitudinal vibrations and obtained the required variable bit forces using the mobility method. Dareing and Livesay [15] later found that a distributed damping along the length of the drill-string is critical in obtaining a good match between the theoretical and experimental results for the displacements and forces during axial and torsional vibrations. Lateral vibrations of the drill string and the associated axial buckling load were considered for the first time by Huang and Dareing [16] in 1968 who modeled the drill string as a simply supported beam. Dunayevsky et al. [17] studied lateral vibrations of the drill strings with a fluctuating axial force and found the instabilities corresponding to parametric excitation. Dawson et al. [18] in 1987 were the first to discuss stick-slip phenomenon during torsional drill-string vibrations wherein they obtained the minimum value of rotary speed for sticking to occur. Similarly Spanos et al. [19] were the first to incorporate bit-bounce during excessive longitudinal vibrations of a drill-string with the roller cone bit. They studied the relationship between weight on bit (WOB) and the amplitude of the bit-motion leading to bit lift-off and observed that the rotary speeds corresponding to axial resonant frequencies of the system are critical.

In all the above-mentioned works, the coupling between the different vibration modes of the drill-string has not been considered. There are two major sources of coupling between the various vibration modes in deep drilling systems; the structural nonlinearity associated with large amplitudes and the dependence of the cutting forces on the vibrations. During the drilling operation, the cutting forces have components along the axial as well as tangential directions. The cutting force depends on the depth of cut which itself is determined by the axial vibrations giving rise to the regenerative effect. The tangential components of the cutting forces at the various cutters on the drill–bit combine to give a cutting torque which enters the equation of motion for the torsional vibrations. Hence, the torsional vibrations are inherently coupled with the axial vibrations. There have been several studies incorporating the coupling due to structural nonlinearities but limited ones on coupling due to the inherent nature of the cutting process itself for drilling systems. Our study in this paper falls in the second category. In what follows, we provide a non-exhaustive short account of the relevant works on the coupling due to structural nonlinearities which is followed by studies on the vibrations incorporating the coupling through the cutting forces.

The first study of coupled drill-string vibrations was reported in 1996 by Yigit and Christoforou [8] where the axial–lateral vibrations of the drill-string coupled through the stretching nonlinearity during bending was studied. They found that the coupled system becomes unstable at lower axial loads than the uncoupled system. They extended their coupled vibration studies [9] to include coupled torsional–lateral vibrations of the drill-string using a lumped parameter model with sinusoidal variation in the applied torque and WOB. The variable WOB leads to parametric forcing and at certain forcing frequencies they observed an energy transfer between the torsional and bending modes. Later, Khulief and Al-Naser [10] studied the coupled torsional and bending vibrations of drill-strings with a variable axial stress due to self-weight and constant external torque and WOB using finite elements. They studied the effect of the rotary speed on the forward and backward whirl frequencies for different free lengths of the drill-string and found that depending on the free length, different whirl modes show different sensitivities to variations in the rotary speed. In 2005, Sampaio et al. [11] modeled the coupled axial–torsional vibrations of drill-strings with the consideration of geometric structural nonlinearity and observed the dynamic response of the nonlinear system to differ significantly from the linear one after the first appearance of stick-slip oscillations. There have been more studies with geometric nonlinearity of the drill-string especially considering axial and torsional dynamics but they have ignored the essential coupling arising due to the cutting forces on the drill–bit.

The first study incorporating the inherent coupling between the axial and the torsional dynamics of a drill-string through the cutting forces and torques was reported by Richard et al. [20] in 2007. Their lumped parameter model for the drill-string incorporated interaction forces between the drill–bit and the surface being cut. These interaction forces included both the forces at the cutting edge as well as the normal and frictional forces acting on the wear flat. The regenerative effect due to the axial vibrations results in a delay differential equation which is further complicated by the dependence of the delay on the torsional vibrations making it a state dependent delay equation. However, their model had no axial stiffness and no damping either in the axial or the torsional dynamics which made steady drilling inherently unstable for all operating conditions [21], [22]. Their model was modified by Besselink et al. [23] and Nandakumar and Wiercigroch [21] by including damping and axial stiffness in the system dynamics which resulted in stable steady drilling for some operating conditions. Kovalyshen [24] furthered the linear stability analysis of the coupled axial–torsional model given in [21] and observed that the axial mode is excited only by the regenerative effect while the torsional mode is excited primarily by the friction at the wear flat/rock interaction. In contrast, our analysis in this paper wherein we have neglected the wear flat completely shows the presence of a dominant torsional vibration involving stick-slip motion without excessive axial vibrations depending on the operating conditions. Hence, the regenerative effect alone is enough to explain excessive torsional vibrations involving stick-slip. Another related study is the stability analysis of a discretized system of equations for the axial and torsional motions of a drill-string with the boundary condition at the drill–bit modeled as per the formulation in [20] by Liu et al. [25] in 2014 using the semi-discretization method.

However, none of these studies have reported the full global dynamics of the coupled axial–torsional vibrations incorporating the possibility of bit-bounce. Bit-bounce leads to the multiple regenerative effect for which the state dependent delay formulation is not very well suited since the equation determining the delay to be used in the calculation of the depth of cut needs to be changed for different periods of the bit motion. In this paper, we present an alternate formulation for the regenerative effect during drilling along the lines of the approach developed for turning by Wahi and Chatterjee [26] which accounts for the possibility of bit-bounce and the associated multiple regenerative effects. In this approach, we have an evolution equation for the cut surface between two successive cutters on bit which is coupled with the evolution equation for the axial and torsional vibrations of the drill-string. The effective depth of cut determining the cutting forces and torques is determined from the cut surface profile and the axial vibration of the drill-string. We note that our formulation for the regenerative effect is independent of the drill-string model as well as the bit–rock interaction model which itself comprises of the cutting force model and a model for the forces on the wear flat. To demonstrate the variety of global motions possible for the rotary drilling system, we have chosen the simplest lumped parameter drill-string model (as reported in [21]) and the simplest bit–rock interaction model (as used in [20], [21]) wherein we have ignored the wear flat itself for further simplicity. Effect of more complicated drill-string models and/or friction forces on the wear flat on the global dynamics of the rotary drilling has been left for future work.

The rest of the paper is organized as follows. In Section 2 we present the complete mathematical modeling of rotary drilling with our approach. It includes a brief description of the lumped parameter drill-string model for the coupled axial–torsional vibrations and the bit–rock interaction model along with the modeling of the cutting force and the evolution during the sticking period. Reduced order system of ordinary differential equations (ODEs) corresponding to our approach and their convergence properties is reported in Section 3. Various possible solutions for our system along with some validation results for the stability boundaries and few transient solutions are reported in Section 4. We finally draw some conclusions in Section 5.