Semi-analytical model of the axial movements of an oil-well drillstring in vertical wellbores

Abstract

A lumped element model for the axial movement of an oil-well drillstring is presented. In this paper, the model is restricted to vertical holes, where damping is due to skin friction from time dependent Newtonian annular Couette-Poiseuille flow. The drillstring is constructed of pipes with different diameters and the diameter of the hole varies as a function of depth. Under these assumptions, the axial movement anywhere in the drillstring is basically a convolution between the axial movement on the top and a semi-analytical function that is derived in this paper. Expressions are given for transfer functions for downhole movements and pressures (surge and swab).In a vertical drilling situation, the motion is clearly underdamped, even when the hole is tight. The semi-analytical model illuminates various factors that are shown to be important for describing downhole pressure and motion. In particular the effect of added mass, the steady state viscous forces, the Basset viscous forces and the distribution of pipe sizes in the hole. The latter have non-neglectable impacts on where the resonant frequencies are located, how much they are amplified and what happens to the downhole pressure.Together with statistical power spectra of ocean wave patterns and the response amplitude operators for a floating structure, this model illustrates design concerns related to heave motion and how fast one can run the drillstring into the hole. Moreover, because of the computational simplicity of computing the convolution, the model is well suited for a real-time implementation.