# The Electromagnetic Field Analytical Solution Analysis to Downhole Current Injection [PERSON]
###### Abstract
However, AMR applications are constrained to specific conditions and are generally classified into two categories based on the accessibility of the target well, access-dependent AMR (AD-AMR) and access-independent AMR (AI-AMR) [16]. Common AD-AMR systems include the magnetic guidance tool (MGT) [17], the single wire guidance (SWG) tool [18], and rotating magnet ranging service (RMRS) [19]. Although these systems provide precise ranging, they require an excitation source or detection device to be placed within the target well, which is often impractical in the real-world drilling environment. The current injection method, as an effective method for the detection and localization of the relief well, is able to achieve a large detection distance [15]. This method mainly uses the downhole electrode to inject a large current into the stratum, which in turn forms a converging current on the accident well casing, and obtains the relative positions between the relief well and the accident well by measuring the alternating magnetic field generated by the converging current in the relief well [20; 21; 22]. While the effectiveness of the above detection methods has been verified, improving detection accuracy and increasing detection range have become the primary research focuses for the precise positioning of the target well during the relief well operation. Several researchers have carried out simulations and analyses on the relief wells detection systems [23; 24]. [PERSON] et al. [13] analyzed the changing law of converging current and measured magnetic field with geological parameters according to the basic principle of current injection [25]. Further, [PERSON] et al. used a numerical simulation approach to analyze in detail the factors influencing the detection performance of current injection ranging [26]. Moreover, [PERSON] et al. employed the finite element method to simulate the current density and magnetic field distribution in the entire measured area. They also analyzed the impact of electrode placement on detection performance and proposed an optimization scheme along with a high-precision ranging algorithm for practical applications [27]. Although numerical simulation methods are widely used because of their adaptability to complex models, analytical methods perform better in signal characterization because of their direct relevance to electromagnetic problems [28; 29; 30]. Specifically, using analytical methods to solve the electromagnetic response allows for a deeper analysis of the structural characteristics of electromagnetic signals and the behavior of electromagnetic fields. This approach provides a reliable theoretical foundation for further signal processing and method optimization. In light of this, an analytical solution for the electromagnetic field distribution in relief well detection is proposed in this paper. Focusing on the detection principle of current injection, the signal model is developed, and the analytical solution to the electromagnetic problem is derived from theoretical formulas. Based on this, the structural and electromagnetic parameters of the detection system are varied, and the adaptability of the proposed analytical solution, along with the influencing factors, are examined. Finally, the effectiveness of the proposed analytical solution is validated through numerical simulation results with a comparison analysis using finite element methods. The remainder of this paper is organized as follows. In Section 2, the signal model for the current injection relief well detection is introduced, and the derivation of the electromagnetic equations and the acquisition of the analytical solution are completed. In Section 3, the variations in the analytical solution under different physical parameters are demonstrated, and the effectiveness of the proposed method is proved through comparative analyses. Finally, Section 4 concludes the paper.