# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 1 Introduction  
Unlike vertical drilling, which is the conventional drilling method, in directional drilling, drilling is intentionally performed through a non-vertical path to reach an underground target point. This drilling method is widely used to reach a target layer in underground areas where vertical drilling cannot be performed owing to environmental factors such as densely populated areas, mountains, lakes, or environmentally sensitive areas. In the case of offshore oil fields, it is possible to create multiple boreholes on one platform, and they can be closed and drilled in another direction if problems occur during drilling. Because drilling offshore oil fields is more expensive than drilling onshore oil fields, directional drilling methods are more cost-effective than vertical drilling. Figure 1 shows various applications of directional drilling. Figure 2 shows that directional drilling can be divided into five types according to the section: hold section, build-up section, drop off section, side bend section, and horizontal section. The hold section is a straight section; the build-up section, drop off section, and side bend section are curved; and the horizontal section is planar. If the kickoff point (KOP), which is the point where a vertically running directional pipe exhibits directional properties for the first time, becomes closer to the target point, the drilling trajectory will produce a large angle change within a short section, resulting in excessive torque and drag [2]. Excessive torque and drag can create doglegs, which are sudden changes in a direction that appear like a dog's hind legs, and key seating, in which grooves are formed in the wall of the borehole by drill collars or other tools in the borehole. In addition, if the drilling tools and casing are damaged, it can lead to a hugeincrease in drilling costs [3]. Therefore, it is critical to select appropriate KOPs to reduce the distributed load in the drilling trajectory [4]. Furthermore, directional drilling is more significantly affected by the buckling effect than vertical drilling. The buckling effect is a type of deformation of the drilling tube in which the drilling pipe is twisted or bent owing to its weight, frictional force in the borehole, and torque from the rotation of the drill bit. The deflection of the drilling pipe becomes more severe owing to the buckling effect, and this can cause the lock-up phenomenon, which prevents the drilling pipe from moving [5]. Furthermore, the buckling effect can interfere with the control of the drilling direction and effective load transfer within the drilling pipe, causing delays in the drilling time and increasing the drilling cost [6]. Therefore, in this study, we performed optimization to select the shortest path when designing the drilling trajectory considering the risk of buckling. Furthermore, as these problems have multiple objective functions, multi-objective optimization is required [7]. This approach reduces the drilling time as well as the cost. Figure 1: Several applications of directional drilling. Adapted with permission from Ref. [1]. 2014 [PERSON] et al. Figure 2: Trajectory section of directional drilling. To resolve the problems caused by buckling of the pipe, [PERSON] [8] focused on calculation of the load in the directional drilling trajectory and proposed an analytical model. [PERSON] [9] analyzed an actual oil well by comparing the results obtained from an analytical model proposed by [PERSON] [8] for calculating the friction of the well, a commercial program, and the actual drag and torque measurements. In addition, in a study on the buckling of directional drilling for oil wells, [PERSON] and [PERSON] [5] proposed a model for buckling loads. Many researchers have attempted to use multi-objective optimization in relation to drilling pipes [10; 11; 12]. [PERSON] et al. [13] studied the length of the drilling trajectory, which is related to drilling costs. Optimization was performed for the length of the three-dimensional trajectory using a genetic algorithm.