Iterative maps for the dynamics of percussive drilling

能源在我们的生活中发挥着至关重要的作用，在过去的150年里，文明越来越多地使用化石燃料/天然气、煤炭和石油。因此，越来越困难的操作条件，例如在斜井或水平长距离威尔斯井中的操作条件，成为钻井工业中的标准，并且这需要井下钻井过程的更好的有效性和可控性。该领域的最新研究证实，硬地层井下钻井新技术的基础是在钻头-岩石界面施加动态载荷。实际实现这一点的一种方式是在传统的旋转钻井上叠加可调节的冲击载荷。这种方法将允许在广泛的钻井地层中进行自适应操作，从而提高切削速度，同时减少工具磨损，并理想地适用于大位移水平钻井。一个强大的数学模型的动态相互作用发生在钻孔是第一步，也是最重要的一步，了解如何可以应用这一理念。除了可以描述为非光滑非线性常微分方程系统的冲击钻井模块的动力学之外，该模型还必须考虑对钻井模块的动力学具有重大影响的井眼中的损伤区。自1998年以来，阿伯丁在这一领域开展了一项重要的研究计划，包括实验和理论研究。这些研究集中于评估一种名为共振增强钻井的新型钻井方法的实用性，其中钻头在共振条件下操作以增加产生可控冲击载荷的效率，从而在钻孔中产生可持续的损伤区。共振增强钻井的数学建模也是这些研究的一部分，最新的工作集中在钻井地层的断裂动力学上，这对于准确预测系统行为至关重要。建议通过开发一套强大的动态断裂模型来进一步开展当前的工作。这些模型将与钻头的动力学模型耦合，以分析井眼中的非线性相互作用。开发这种模型将是该项目的第一项主要任务。第二个主要任务是为冲击钻井构建迭代地图。已经理解的是，降维，特别是构造解析迭代映射，对于理解和设计由非线性分段光滑方程描述的系统将是特别有益的，因为没有很好的数学技术来获得这些系统的解，并且通常甚至在证明解的存在性方面也有困难。迭代映射的主要优点是，与基于直接数值积分的技术相比，使用映射计算动态响应所需的时间要少得多。同样重要的是，通过构造迭代映射实现的降维意味着系统分析所需的数据量显著减少。系统行为的快速预测和减少的数据量对于开发有效的控制系统都非常有用。使用构造的迭代映射的系统动力学分析的目的在于制定外部激励的最佳模式，并且特别地，它将集中于获得作为所钻地层属性的函数的使钻速最大化的推进运动的频率和振幅。

Energy plays a vital role in our lives, and during last 150 years civilization has increasing used fossil fuels / gas, coal and oil. As a result more and more difficult operating conditions, such as that in deviated or horizontal long-reach wells, become a norm within the drilling industry, and this requires better effectiveness and controllability of the downhole drilling processes. The latest research in this area confirmed that a basis for novel downhole drilling techniques of hard formations is founded upon imposing dynamic loading at the bit-rock interface. One way of practically realising this is a superposition of adjustable percussive loading on conventional rotary drilling. This method will allow adaptive operation across a wide range of drilled formations, so enhancing cutting rates while reducing tool wear and lending itself ideally to extended-reach horizontal drilling. A robust mathematical model of the dynamic interactions occurring in the borehole is the first and most important step in understanding how this philosophy can be applied. Apart from the dynamics of the percussive drilling module, which can be described as a system of non-smooth nonlinear ordinary differential equations, the model has to account for the damage zone in the borehole having a major influence on the dynamics of the drilling module. A significant research programme in this area comprising experimental and theoretical studies has been carried out at Aberdeen since 1998. These studies have been focussed to assess the practicality of a novel drilling method named as the resonance enhanced drilling, where the drill-bit operates in resonance conditions to increase the efficiency of generating controllable impact loading and consequently to create a sustainable damage zone in the borehole. Mathematical modelling of resonance enhanced drilling has been also part of these studies, and the latest work has been concentrated on the fracture dynamics of drilled formations, which is crucial for an accurate prediction of the system behaviour. It is proposed to take the current work a step further by developing a suite of robust models of the dynamic fracture. These models will be coupled with the dynamic model of the drill-bit in order to analyse the nonlinear interactions in the borehole. The development of such models will be the first major task of the project. Construction of the iterative maps for the percussive drilling will be the second major task. It has been understood that dimension reduction, and in particular construction of analytical iterative maps, would be especially beneficial for understanding and designing of the system described by non-linear piece-wise smooth equations as there are no well developed mathematical techniques for obtaining solutions for these systems and often there are difficulties even in proving the solution existence. The main advantage of iterative maps is that the computation of dynamic responses using the maps takes a fraction of time when compared to the techniques based on direct numerical integration. Also it is important that the dimension reduction achieved by constructing iterative maps means that the amount of data required for the system analysis is significantly decreased. The fast prediction of the system behaviour and reduced amount of data are both very useful for developing efficient control systems. Analysis of the system dynamics using the constructed iterative maps aims at formulation of optimal patterns of the external excitation, and in particular it will be focused on obtaining the frequencies and amplitudes of the percussive motion maximising drilling rates as functions of the drilled formation properties.