An Alternative Formulation for Modeling Self-Excited Vibrations of Drillstring With Polycrystalline Diamond Compact Bits

Abstract

This work describes an alternative formulation of a system of nonlinear state-dependent delay differential equations (SDDDEs), which governs the coupled axial-torsional vibrations of a 2 DOF drillstring model with a realistic representation of polycrystalline diamond compact (PDC) bits. The regenerative effect associated with the complex cutter layout for such bits can introduce up to 100 state-dependent delays in the equations of motion. This large number of state-dependent delays renders the computational efficiency of conventional solution strategies unacceptable. The regeneration of the bottom-hole surface can alternatively be described by the bit trajectory function, whose evolution is governed by a partial differential equation (PDE). Thus the original system of SDDDEs can be replaced by a nonlinear coupled system of a PDE and ordinary differential equations (ODEs). Via the application of the Galerkin Method, this system of PDE-ODEs is transformed into a system of coupled ODEs, which can readily be solved. The algorithm is further extended to perform a linear stability analysis of the bit motion. The resulting stability boundaries are verified with time-domain simulations. The reported algorithm could, in principle, be applied to a more realistic drillstring model, which may lead to an in-depth understanding of the mitigation of self-excited vibrations through PDC bit designs.