## 1 Introduction
However, it is undeniable that changing the properties of stator rubber provides very limited improvement of the resistance of motor stators to high temperatures. While equal-wall-thickness screw motors can effectively prevent the build-up of the hysteresis heat of conventional screw motor rubber bushings, they are insignificant for the extremely high-temperature and high-pressure environment of an ultra-deep well. Given this background, in this study, an all-metal screw motor is proposed to be used as the power source in ultra-deep well drilling projects as a way to prevent the effects of high-temperature and high-pressure environments deep underground on the normal operation of screw rod drilling tools. An all-metal screw motor is a motor whose stator and rotor are made of metal and do not contain rubber materials [27]. In recent years, domestic and foreign scholars have conducted research on all-metal motors. In 2014, Baker Hughes in the United States developed a directional drilling system with all-metal screw motors that can adapt to high temperatures of 300 \({}^{\circ}\)C, the first precision-machined all-metal motor in the world [28], then proposed a metal-to-metal (M2M) motor with metal coating between the rotor and stator [29; 30]. In addition, at the 2018 IADC/SPE Drilling Conference, a completed drilling system and measurement-while-drilling (MWD) system capable of continuous operation at 300 \({}^{\circ}\)C for more than 50 h was presented, which could support the development of deep-well subsurface geothermal resources [31]. In China, engineers at Southwest Petroleum University, Xi'an University of Petroleum, and China University of Geosciences have conducted some research on the structural design, theoretical principles, fitting methods, and processing techniques of all-metal motor drilling tools [32; 33]. Kong of China University of Geosciences used simulation software to study the effect of various factors on the leakage of all-metal screw motors, which can be a reference for reducing leakage and improving motor output efficiency [27].Though a series of studies have been conducted by scholars on the flow field, leakage, operating efficiency and contact type of all-metal screw motors, few scholars have studied their deformation characteristics in high-temperature and high-pressure environments, much less their deformation patterns under the extreme conditions of ultra-deep wells, or their structural design. To better apply the all-metal screw motor to the high-temperature and high-pressure drilling projects of ultra deep wells, this study adopted 40 CrNiMo alloy steel as the base material of the motor. First, a design of the fit method and end face linearity of the all-metal screw motor was conducted, and then the variation in clearance after the expansion of the all-metal screw motor in a high-temperature environment and the deformation characteristics in a high-temperature and high-pressure environment were studied through numerical calculations. The results showed that the metal screw motor designed in the paper had stable performance in a high-temperature and high-pressure environment and underwent little deformation and little wear, which proved that the all-metal screw motor is suitable for the extreme environments of high temperature and high pressure in ultra-deep wells and can be adapted to drilling work in ultra-deep wells. The results of this study support the development of all-metal screw motors for applications in extreme drilling environments in ultra-deep wells and provide theoretical guidance and a design basis for screw motors to help promote scientific research as well as exploitation of resources deep underground.