Iterative maps for the dynamics of percussive drilling

能源在我们的生活中扮演着至关重要的角色，在过去的150年里，人类文明越来越多地使用化石燃料/天然气、煤炭和石油。因此，越来越困难的作业条件，如斜井或水平井的作业条件，成为钻井行业的标准，这就要求井下钻井过程具有更好的有效性和可控性。这一领域的最新研究证实，在钻头-岩石界面施加动态载荷是坚硬地层新型井下钻井技术的基础。实际实现这一点的一种方法是将可调冲击载荷叠加到传统的旋转钻井上。这种方法将允许在广泛的已钻井地层中进行自适应操作，因此提高了切削率，同时降低了工具磨损，是大位移水平钻井的理想选择。在理解如何应用这一哲学的第一步，也是最重要的一步，建立一个稳健的数学模型来描述井眼中发生的动态相互作用。除了冲击钻进模块的动力学可以描述为一个非光滑的非线性常微分方程组外，该模型还必须考虑井眼中的损伤区对钻进模块的动力学有重要影响。自1998年以来，在阿伯丁开展了一项重要的研究方案，其中包括实验和理论研究。这些研究的重点是评估一种名为共振增强钻井的新钻井方法的实用性，在这种钻井方法中，钻头在共振条件下工作，以提高产生可控冲击载荷的效率，从而在井眼中产生可持续的损伤区。共振增强钻井的数学模型也是这些研究的一部分，最新的工作集中在钻井地层的裂缝动力学上，这对准确预测系统行为至关重要。建议将目前的工作更进一步，开发一套稳健的动态断裂模型。这些模型将与钻头的动力学模型相耦合，以分析井筒中的非线性相互作用。开发这样的模型将是该项目的第一个主要任务。冲击钻进迭代图的构建将是第二项主要任务。可以理解的是，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特别有益，因为没有发展良好的数学技术来获得这些系统的解，而且通常甚至在证明解的存在性方面也存在困难。迭代映射的主要优点是，与基于直接数值积分的技术相比，使用迭代映射计算动力响应只需要一小部分时间。同样重要的是，通过构建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数据量的减少对于开发高效的控制系统都是非常有用的。使用构建的迭代图对系统动力学进行分析的目的是制定最优的外部激励模式，特别是将重点放在获得冲击运动的频率和幅度上，以使钻井速度最大化，作为钻井地层特性的函数。

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.