## 2 Working Principle and Model of the PEC System  
Figure 1 shows a PEC system used for borehole casing inspection. The PEC system is suspended on a cable and is controlled by a winch during inspection. Each measurement period corresponds to a depth along the borehole axis. The PEC system is based on the principle of electromagnetic induction. When a direct current is applied to the transmitting coil, a primary magnetic field is formed in a cylindrical layered structure comprising air, the metal casing, cement, and the formation. The current is then turned off instantaneously, and the abrupt change in the magnetic field induces eddy currents in the cylindrical medium. In this paper, the turn-on, turn-off, and on-time of the input current were set to 50 us, 30 us, and 200 ms, respectively. This produces a secondary magnetic field that can be captured by the receiving sensor. The average thickness over the distribution range of the transmitting magnetic field is indicated by the decay rate of the electromagnetic signal.  
The PEC system can be modeled as a probe comprising the transmitting coil and receiving sensor that are located at the center of the borehole casing. The relative magnetic permeability, relative dielectric constant, electrical conductivity, and radius of the \(j\)th layer can be defined as \(\mu_{j}\), \(\varepsilon_{j}\), \(\sigma_{j}\), and \(r_{j}\), respectively, with \(j\) = 1, 2,..., \(J\). Assuming that the transmitting current is \(I\), then the magnetic field at the position of the borehole axis \(z\) with radius \(r\) excited by one turn of the coil with the position \(z_{t}\) can be calculated as follows [23]:\[H_{1}(z_{t},\omega,r,z)=-\frac{Ir_{1}}{\pi}\int_{0}^{\infty}xK_{0}(x_{j}r)I_{1}(x_ {j}r_{1})\cos[\lambda(z-z_{t})]d\lambda \tag{1}\]  
where \(x\) and \(\lambda\) are introduced variables that satisfy \(x_{j}^{2}=\lambda^{2}-k_{j}^{2}\) and \(k_{j}^{2}=\mu_{j}\varepsilon_{j}\omega^{2}-i\mu_{j}\sigma_{j}\omega\). In addition, \(\omega\) denotes the angular frequency, and \(I_{q}(\cdot)\) and \(K_{q}(\cdot)\) are the first and second modified Bessel functions, respectively, of order \(q\). Turning the current off instantaneously causes the magnetic field to decrease suddenly, and an eddy current is then formed on the conductive medium. The electrical parameters, geometric parameters, and boundary conditions of each layer can be used to determine the eddy current field in the \(j\)th layer:  
\[H_{2}(z_{t},\omega,r,z)=\frac{Ir_{1}}{\pi}\int_{0}^{\infty}x_{j}\big{[}-\tau_{ j}K_{0}(x_{j}r)I_{1}(x_{j}r_{1})+C_{j}I_{0}(x_{j}r)-D_{j}K_{0}(x_{j}r)\big{]} \cos[\lambda_{j}(z-z_{t})\big{]}d\lambda_{j} \tag{2}\]  
where \(\tau_{2}=1\), \(\tau_{j\
eq 2}=0\), and \(C_{j}\) and \(D_{j}\) are coefficients related to the electrical and geometric parameters of each layer of the medium.  
According to the \(P\)-order Gaver-Stehfest inverse Laplace transform formula [24], the magnetic field in the time domain can then be expressed as  
\[H(z_{t},t,r,z)=\begin{cases}\frac{\ln 2}{t}\sum\limits_{p=1}^{P}E_{p}H_{1}(z_{t}, p\ln 2/it,r,z)&t<t_{\text{off}}\\ \frac{\ln 2}{t}\sum\limits_{p=1}^{P}E_{p}H_{2}(z_{t},p\ln 2/it,r,z)&t\geq t_{ \text{off}}\end{cases} \tag{3}\]  
where \(E_{p}\) is the Gaver-Stehfest inverse Laplace coefficient, and \(t_{\text{off}}\) denotes the moment when the current is turned off. The magnetic field is excited by the transmitting coil when \(t<t_{\text{off}}\), and it produces the eddy current field when \(t>t_{\text{off}}\). The above model represents the application of PEC systems to borehole casing inspection. However, increasing safety concerns in oil and gas production mean that the traditional single-transmitter and single-receiver probes cannot meet the requirements for challenging downhole environments.  
Figure 1: PEC system for borehole casing inspection.