# The Electromagnetic Field Analytical Solution Analysis to Downhole Current Injection [PERSON]
###### Abstract  
As the final crucial line of defense against well blowout accidents, the implementation of the relief well scheme facilitates the swiftest possible rescue of the target well, thereby minimizing economic losses and enhancing production efficiency. Among them, the relative distance and orientation determination between the relief well and the accident well becomes the key to the rapid intersection and connection of the two wells. However, the inaccessibility of the accident well due to safety factors makes it challenging to accurately determine the relative position of the two wells. The current injection method has now been proved to be able to achieve relief well detection, and some signal processing methods are necessary for optimizing the detection performance. In the process of signal characterization, analytical methods show many advantages due to their direct relevance to the principles of electromagnetic theory. To provide an effective theoretical basis for the optimization of relief well detection methods, this paper proposes an analytical solution to the corresponding electromagnetic field distribution for current injection detection scheme. Based on the principle of the current injection detection, the signal model is constructed, and the electromagnetic distribution expression is deduced. Then, the adaptability of the analytical solution obtained under different physical parameters is investigated. Moreover, the reliability and validity of the proposed analytical solution are validated by comparing the results with those obtained from finite element numerical simulations, thus providing a theoretical basis for subsequent signal processing and method optimization. c 2025 15 3741 2025 15  
consequences, including environmental pollution and significant economic losses. In this process, the relief well intersects and connects with the target well, allowing high-density pressurized fluid to be injected into the relief well to control the blowout in the accident well [4; 5; 6]. For achieving a rapid and successful connection, it is crucial to determine the relative position between the two wells. In real-world applications, accurately determining the distance between two wells is highly challenging due to the unpredictable and harsh downhole conditions, along with the strict reliability standards required to ensure a successful relief well intersection and minimize potential losses [3]. Traditional inclinometer tools exhibit a notable decline in detection accuracy as drilling depth increases, often failing to meet the necessary ranging precision [7]. To improve accuracy and support real-time control and guidance in drilling operations, several magnetic ranging techniques have been introduced, including passive magnetic ranging (PMR) and active magnetic ranging (AMR) [8; 9; 10; 11]. PMR estimates the relative position of the target well by detecting and analyzing magnetic field disturbances induced by magnetic structures, such as casing or drill pipes, within the target well using surface-based software systems [12]. Since PMR relies on the residual magnetism, it typically requires no additional drilling hardware and is relatively straightforward to operate. However, this method is highly susceptible to environmental factors. When the relief well is located at a considerable distance from the target well or when sections of the casing exhibit minimal residual magnetism, the magnetic field disturbances detected by the magnetometer in the relief well become exceedingly weak. Consequently, the ranging accuracy of PMR tools degrades significantly under such conditions [13]. Unlike PMR, which relies on residual magnetic fields, AMR utilizes an excitation source along with high-sensitivity sensors to generate strong electromagnetic signals. This enhancement allows for greater detection distances and improved accuracy [14; 15].