Combining Formation Seismic Velocities while Drilling and a PDE-ODE observer to improve the Drill-String Dynamics Estimation

In this paper we consider the axial motion of a drill-string and the interaction of the drill-bit with the formation. We design an observer that estimates, in real time, the axial speed and force along the drill-string and at the drill-bit using only topside measurements (force and velocity). More generally, our approach enables the design of robust observers for systems of Ordinary Differential Equation coupled with a first order Partial Differential Equation in their actuation path (a class of systems to which belongs the considered drillstring axial dynamics). However, such an observer requires the knowledge of different physical parameters among which the rock intrinsic energy which is a priori unknown. Thus, we combine our observer with an algorithm that provides a near real-rime estimation of the seismic velocities of rocks interacting with the drill-bit, using seismic-while-drilling. The efficiency of the proposed approach is shown through simulation results.