Robust evaluation of stability regions of oil-well drilling systems with uncertain bit-rock nonlinear interaction

Abstract

Torsional vibrations due to stick-slip phenomena may yield large variations in the drilling angular velocity which in turn may cause failures in the drilling process such as damage in drill pipes, due to large torsional deformation, and in drill bits, due to large angular velocities. It has been shown that standard linear proportional-integral (PI) control laws, in which the control torque is proportional to the differences between actual and reference drive table angular velocity and displacement, may lead to an oscillating angular velocity at the drill-bit due to stick-slip phenomenon. Aiming at reducing this effect through properly designed control gains, stability regions were defined for these parameters, and for a given drilling condition, using as a criteria the average deviation from expected/target angular velocity. This is done using a finite element model for the drilling system together with a dry friction model for a simplified representation of the interaction between drill-bit and rock formation. In order to provide robustness for the analysis, given the complexity of actual bit-rock interaction, it is proposed to consider uncertain parameters for the dry friction model. This is done using a stochastic model for the bit-rock interaction torque followed by a Monte Carlo simulation, which then leads to stochastic results for the angular velocity response, when subject to a linear PI control, and, thus, for the angular velocity deviation stability regions. Confidence intervals for the control gains, performance and stability regions are presented. Results suggest the use of a more robust criteria for the design of control parameters leading to a substantial increase in the performance robustness while not affecting much the nominal performance.