Distributed model for the drill-string system with multiple regenerative effects in the bit-rock interaction

In this paper, an integrated model is presented to study the axial-torsional dynamics of a distributed drill string. The coupled axial and torsional drill string dynamics typically lead to severe phenomena such as bit-bounce and stick-slip, which are the main causes of failure and efficiency reduction. The top-drive dynamics and the Bottom Hole Assembly (BHA) dynamics are interconnected through the drill pipes. In order to preserve the infinite-dimensional feature of the drill pipes in the mathematical modeling, a distributed model in terms of Delay Differential Algebraic Equations (DDAE) is proposed to represent the dynamics of the drill string. The bit -rock interaction, which consists of the cutting and the frictional components, is responsible for the coupling between the axial and torsional dynamics. A rate-independent bit-rock interaction law is employed for both cutting and frictional components. In order to capture the global bit motion, including multiple regenerative effects, the well-bottom depth function is used to determine the depth of cut. The well-bottom pattern evolution is formulated through algebraic equations rather than Partial Differential Equations (PDEs). A new formulation for the linearized drill string system in terms of Neutral-type Delay Differential Integral Equations (NDDIEs) is proposed to investigate the initiation of the oscillations around the nominal operation. By this formulation, the local stability of the distributed system is studied by determining the right-most eigenvalue of the linearized dynamics. A simulation-based case study reflecting real-life scenarios is presented to investigate the dynamical behavior of the system under different operating conditions.