# Study on the Vertical Stability of Drilling Wellbore under Optimized Constraints
## 1 Introduction  
As one of the major fossil energy sources, coal has brought a huge supply to power, steel, chemical, and other industries with the growing global demand for energy [1, 2]. Coal mine wellbore construction remains an important project at present. However, during the process of drilling method construction, after the drilling rig has drilled the well hole, the builders will measure the deviation of the well hole and then place the prefabricated bottom of the well wall and each section of the well wall into the well hole, then connect them with flanges in turn. The resulting well wall structure is a round tubular structure, which is referred to as the wellbore structure by many scholars [3, 4, 5]. The wellbore structure is prone to longitudinal instability during the construction process in drilling methods. Once the wellbore structure loses its stability, it can lead to collapse accidents, causing significant personnel and economic losses [6, 7, 8, 9, 10]. Moreover, the depth of coal resource extraction in western China is gradually increasing, and the construction depth of new shafts is about to reach a historic height [11, 12]. Therefore, the stability of drilling wellbore structures is an urgent engineering problem to be solved. The drilling method is an advanced directional drilling technology that is widely used in energy extraction, mineral exploration, and rescue [13, 14, 15, 16]. In general, the drillingmethod is divided into three stages: floating, floated to the bottom but not filled and fixed, and filled and fixed. The second stage is the weakest stage, and it is easy for the wellbore to experience instability once there is any minor interference. Many scholars have conducted numerous studies on the stability characteristics of this stage. [PERSON] [17; 18] first proposed the problem of structural instability in the wellbore and analyzed structural stability treating it as a slender rod with hinged ends. [PERSON] et al. [19; 20] established the critical depth of the equal cross-section wellbore structure on full water and non-full water by the principle of minimum potential energy and energy method. [PERSON] et al. [21] further established the critical depth of the variable cross-section drilling wellbore structure. [PERSON] et al. [22; 23] established the critical depth of the equal cross-section drilling wellbore structure using catastrophe theory considering the sudden and irreversible characteristics of the wellbore instability process. [PERSON] et al. [24] used numerical calculation methods to analyze the vertical stability of the entire drilling process. [PERSON] et al. [25] monitored the lateral displacement during construction on site. In summary, a relatively large amount of research on drilling wellbore structural stability has been accumulated. However, these studies only focus on the structural stability of traditional construction methods without proposing effective optimization measures to improve structural instability and conducting targeted research on optimization measures. In this study, an optimization measure will be raised, and a targeted theoretical and numerical study on optimization measures will be carried out. Considering that wellbore structure is most prone to instability during the moment of suspended sinking and touching the bottom, the contact conditions at the bottom of the wellbore will directly affect structural stability controlling. So, we raise the idea of pre-throwing cement slurry to the bottom before wellbore falls to the bottom. In this optimized method, cement mortar is injected into the bottom by grouting pipe before the wellbore sinks to the bottom. The injection amount and the speed of counterweight water control the speed of the wellbore sinking. When the wellbore sinks to the bottom under the action of counterweight water, the cement has already been pre-thrown. At this time, the bottom of wellbore structure is placed on the pre-casted cement and firmly restricts the displacement in all directions, forming a fixed contact at the bottom.