How to mathematically model a drill-string: lumped or continuous models?

This work compares the efficacy of lumped and continuous models in simulating drill-string dynamics. We aim to critically explore the inadequacy of lumped models and challenge their perpetuated use in the literature. Lumped models represent the drill-string with a pre-determined number of discrete masses, sometimes as few as one or two. This inherently restricts their ability to capture the distributed nature of drill-strings, compromising their accuracy. In this paper, these limitations are made evident by comparing simulations of lumped models with simulations of continuous models that increase in complexity. First, predictions of the axial behaviour obtained with two models, one lumped and another continuous, are analysed. Second, simulations involving a 3-D fully coupled continuous model are evaluated. The simulations demonstrate the inaccuracy of lumped models. Also, they refute the frequent assumption of representing the bottom hole assembly (BHA) as a single lumped mass. Beyond these results, theoretical arguments based on fundamental wave-reflection phenomena are employed to further support the inadequacy of lumped formulations for this purpose. Third, this paper also discusses the prevalent strategy of validating lumped models through comparisons with laboratory experiments. This approach suffers from being a makeshift validation, as similitude principles are often violated in laboratory setups, resulting in a practice that may have helped perpetuate lumped models in the literature. Consequently, using lumped models for general drill-string dynamics prediction is not appropriate. Recognising their limitations and exploring alternative approaches, such as continuous models, is crucial for accurate drill-string dynamics prediction.