Stability of drillstrings in ultradeep wells: an integrated design model

Abstract

A mathematical model is proposed to describe the critical quasistatic equilibrium of rotating drillstrings with additional centralizers in the lower portion. It is shown that the system of differential equations describing them is stiff. It is solved by decomposition. Drillstrings of different lengths are considered, the critical angular velocities are found, and buckling modes are identified Introduction. Nowadays, the extraction of hydrocarbon fuel from depths of 7 km and more is of pressing importance. Large reserves of such fuels were struck in the regions of the Azov and Black seas (Ukraine). In this connection, new and secondary technologies for development of such fields are developed. Among them are drilling of deep wells, new controlled directional wells, and offshoots in vertical wells [3]. Considering the importance of such technologies for the power industry of Ukraine and the absence of a scientific theory of oil and gas extraction from ultradeep wells, we may conclude that the mathematical simulation of deep-well drillstrings of various configurations is an important scientific and applied problem. Similar problems of the dynamics of thin-walled rods and beams are addressed in [8,9,19].During well drilling, the drillstring is subjected to gravity, torque, and inertial forces exerted by the rotating drill pipe and internal fluid. These factors induce longitudinal, torsional, and flexural vibrations and flexural buckling of the drillstring. These effects may cause sticking of the drillstring, sloughing of the well wall, and instability of the whole system. Since making a modern deep oil or gas well costs over 50 million US dollars and one out of every three wells shows serious compications [4,18], theoretical modeling of these phenomena is of pressing importance. State of the Art in the Problem of Stability of Deep-Well Drillstrings.The main causes of failures during well drilling are instability of the rectilinear configuration of the drillstring, bifurcational buckling, and limiting frictional interaction with well walls. Theoretical modeling of drillstring buckling involves formidable difficulties, of which the principal one is due to the necessity of formulating the Sturm-Liouville problem for a very long drillstring. Since the geometry of drillstrings for deep wells is such that they may be considered like a human hair, their flexural stiffness is low, and the differential equations describing their deformation are stiff. Therefore, many conventional mathematical methods used to integrate the governing equations of drillstring bending converge poorly in this case.In this connection, the stability analysis of drillstrings now involves the consideration of the postbuckling equilibrium with allowance for the contact interaction between the drillstring and the well wall. The problem is substantially simplified by neglecting the rotation, torque, and boundary conditions and assuming that the axial compressive force is constant along the infinite rod.It is assumed that after buckling, the drillstring t...

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