Mathematical model for mechanical buckling of drillstrings within curved bore-holes, A

Abstract

An innovative mathematical model to describe the mechanical buckling of drillstrings within curved bore-holes was developed. The hypergeometric model congregates the curved bore-hole theory and the generalized beam-column theory, both developed and presented in this work. The curved bore-hole theory dictates the conditions a drill string must satisfy to buckle within a curved bore-hole. The generalized beam column theory describes the singular behavior of an inclined beam-column subjected to a distributed weight and axial forces applied on its ends. The differential equation of the inclined beam-column was derived using the energy method and the general solution was written in a closed form using generalized hypergeometric functions. The conditions required by the curved bore-hole theory are then imposed to the general solution of the differential equation. This results in a numerical procedure with which the buckling force can be calculated for any point along a curved bore-hole, called local buckling force. From this force derives the positional buckling force which accounts for the effects of the distributed weight and friction. Curves plotted using these two forces, called buckling graphs, permit a complete analysis of the buckling behavior of the drillstring for constant cross-sectional drillstrings, tapered drillstrings, tapered bore-holes, and multi-curved bore-holes. All this is possible using the P-curve shifting method developed and presented in this work. The buckling graph and the P-curve shifting method constitute a unique tool for designing optimal drillstrings to drill horizontal oil, gas, and geothermal wells. The model was experimentally validated using a medium-scale apparatus designed and constructed in the Colorado School of Mines. Predictions using the hypergeometric model and results of the experiments are presented.