Similarity Analysis for Downscaling a Full Size Drill String to a Laboratory Scale Test Drilling Rig

The drill string used in drilling oil and gas wells is a long and slender structure that is confined by the wellbore wall and subject to significant axial, lateral and torsional vibrations while drilling. Detection and mitigation of drill string vibrations are especially important, as vibrations can be hard to detect at the surface, yet cause significant damage to the drill bit, downhole tools and the formation being drilled. Study of the drilling process by downscaling to laboratory conditions is an attractive prospect as it is a cost-efficient alternative to dedicated large-scale testing and it can be instrumented to provide vibration measurements from different locations along the string. However, the extreme geometrical length scales, the complex wellbore trajectories and the large mechanical strains on the drill string lead to challenges when attempting to downscale the drilling process to manageable laboratory conditions. Downscaling based on similarity analysis provides consistent scaling laws for predicting large-scale dynamics based on observations from the downscaled test drilling rig. We perform a similarity analysis for an example full scale drill string that illustrates these challenges in terms of similarity criteria for an equivalent laboratory model of the drill string. We focus particularly on the geometric and mechanical properties of the drill string and consequences of downscaling on the time scale and the forces in the string. We illustrate the downscaling criteria through numerical simulations by solving the governing equations of motion at different scales, and provide recommendations for downscaling based on widely available material types.