Stability and response of closed loop directional drilling system using linear delay differential equations

A directionally controlled drilling mechanism has a lateral borehole propagation equation of state composed of spatially delayed feedback terms determined by the distribution of contact points between the borehole and drilling mechanism. A lateral borehole propagation equation is developed for a generic class of steering system. After simplification the resulting linear delay differential equation is partially validated against a Finite Element Model (FEM) of borehole propagation. The linear delay differential equations are then incorporated into various closed loop steering configurations to further demonstrate the efficacy of the approach.