# Research on Wellbore Trajectory Prediction Based on a Pi-GRU Model
## 1 Introduction  
The precise control and accurate prediction of borehole trajectories are critical determinants in directional drilling operations, directly impacting drilling quality, optimizing operational efficiency, reducing costs, and ensuring safety in near-horizontal drilling scenarios. However, regardless of whether wired or electromagnetic wireless measurement-while-drilling (MWD) systems are employed, the sensors are invariably positioned behind screwdrill tools. Moreover, owing to the use of magnetic sensors, a nonmagnetic drill collar (\(\geq\)3 m in length) must be installed between the sensors and screw drill tools to eliminate magnetic interference from the drilling assembly. Consequently, the measurement point remains positioned more than 6 m behind the drill bit [1], creating a substantial blind zone when measuring borehole parameters. The screw motor adjusts the direction of the drill bit through the orientation of its front-end bent sub, with the bottom-hole assembly (BHA) configuration illustrated in Figure 1.  
The traditional methods for predicting wellbore trajectories can be classified into two main categories: mechanical analysis methods and data-driven methods. The existing mechanical analysis methods fall under three main types. (1) Mechanical interaction-based models, such as those proposed by [PERSON] et al. [2], integrate BHA stress distributions [3], deformation characteristics, and bit-rock interaction mechanisms to account for multifactor coupling effects. (2) Build-rate analysis methods, including the three-point geometry method [4], balanced curvature method [5], and limit curvature method [6], predict dip angle changes by calculating the build-up rate. (3) MWD-derived approaches [7], which utilize real-time trajectory parameters derived from MWD systems to forecast subsequent survey points, are typically implemented through techniques such as the average minimum curvature method [8] and radius of curvature method [9]. In addition to the challenges brought by complex geological conditions and real-time trajectory control, dynamic loads on the drill string (such as torsional vibration, axial impact, and lateral impact) also significantly affect drilling efficiency, tool life, and wellbore stability [10]. To mitigate these impacts, researchers have proposed intelligent control systems and mechanical damping solutions. For instance, intelligent controllers are used to conduct real-time monitoring and adaptive response to downhole vibration and shock loads, thereby enhancing drilling stability and reducing equipment wear [11]. With the rise of artificial intelligence, machine learning has made significant progress in image recognition and natural language processing scenarios. Some researchers have attempted to apply internal artificial intelligence algorithms, including machine learning, to predict wellbore trajectories. In 2005, [PERSON] et al. [12] were the first to apply support vector machines to trajectory prediction. They demonstrated through a regression method based on kernel functions that it provided measurable improvements over geometric  
Figure 1: Screw motor drilling tool assembly. methods. In 2008, [PERSON] et al. [13] combined the wavelet transform with a neural network to achieve millimeter-level accuracy in terms of predicting the displacements of drill bits. While this model successfully predicts drill bit displacements, it lacks time series modeling capabilities. In 2023, [PERSON] et al. [14] demonstrated that long short-term memory (LSTM) could outperform the traditional curvature methods in time series learning tasks, but its single-input design neglects the influences of different stratum lithologies on drill bits. In 2024, [PERSON] [15] used the dip angle and azimuth angles of a drilling measurement instrument within the first 12 m as input data to construct a drill bit prediction model with two hidden layers of a BP neural network, and the prediction accuracy was improved over that of practitioners.