## 1 Introduction  
The mineral resources in the shallow part of the Earth have been gradually depleted, and at present, resource development is continuously moving to deeper parts of the Earth [1, 2].With continuous research on the development of unconventional oil and gas resources such as shale gas and shale oil in China [3, 4], there has been unprecedented development of drilling projects deep underground, which are the most direct way to obtain information about the deep part of the Earth [5, 6]. It is now generally accepted that scientific drilling in mainland China should target extraordinarily deep wells from 9000 to 15,000 m in depth [7]. Ultra-deep well drilling technology is an indispensable and important method for the development of resources deep in the Earth. The screw motor, an important power tool in drilling technology, dictates the torque and speed that a screw drilling tool produces. The normal operation and efficiency of the screw motor affect the efficiency and quality of the entire drilling project, which is of great significance for the drilling of ultra-deep wells. However, most screw motors currently use rubber stators for power replenishment by overfitting. The temperature resistance of rubber materials is a limitation as the drilling depth and the downhole temperature continue to increase, especially in ultra-deep wells and extra-deep well projects. The rubber stator of conventional screw drilling tools is unable to meet the drilling requirements for use in the extremely high-temperature and high-pressure environments of ultra-deep wells [8; 9; 10; 11; 12]. Therefore, to better develop ultra-deep well drilling projects, it is necessary to study and develop a screw motor applicable to ultra-deep well drilling environments. This will be meaningful for developing scientific drilling projects, exploring earth science and exploiting resources deep underground. At present, scholars at both home and abroad have studied the theory of the screw rod motor relatively well, and many scholars have conducted in-depth studies on the working principle [13], structural form and linear design [14; 15] of the screw rod motor. Their research extends to the effects of specific motor pitch on efficiency [16], motor seal pressure characteristics [17], motor stator-rotor mating clearance ratio [18] and other aspects. In terms of how to improve the high-temperature resistance of screw motors for use in deep and ultra-deep wells with high-temperature environments, the main focus has been on improving the properties of the rubber for stators [19; 20] and developing equal-wall-thickness screw motors [21; 22]. In terms of improving the rubber material of the stator, the first rubber material used in the screw rod motor was natural rubber. Then, with the development of synthetic rubber materials, silicone rubber [23], nitrile rubber and fluorine rubber were used as screw rod stators in the 1950s to improve performance. Engineers in the former Soviet Union developed a metal-rubber bushing made of nitrile rubber that could guarantee operation at 140 \({}^{\circ}\)C, and Baker Company used a new synthetic rubber that could operate normally at less than 149 \({}^{\circ}\)C [24]. Later, scholars succeeded in improving the thermal stability and mechanical properties of rubber materials by adding other materials, such as alumina particles and glass fibers, to butyl rubber, which enabled the rubber stator to have better performance in terms of resistance to high temperature and corrosion compared to conventional nitrile rubber (NBR) [25]. In the application of equal-wall-thickness screw motors, [PERSON], [PERSON] and [PERSON] of Southwest Petroleum University designed rubber bushings for equal-wall-thickness screw motors and conducted corresponding detailed mechanical property studies and numerical calculations. Additionally, an optimization analysis of the stator-rotor meshing parameters of the equal-wall-thickness screw motor was performed [19; 26] that achieved better results.