Investigation on Spatial Transformation and Proportional Coefficient of Vehicle-Mounted Transient Electromagnetic Detection Environments in Operational Tunnels

Abstract

The vehicle-mounted transient electromagnetic method (VMTEM) has been proposed to detect tunnel internal defects in operational tunnels based on the ideal space state. However, the space environment of tunnel surrounding rock is different compared with conventional application fields, so the interpretation of detection data has certain inadaptability and unreliability. In this paper, three typical space states involved in the detection process of operational tunnels were analyzed. The diffusion law of the transient electromagnetic response signal under the condition of three typical space states was carried out, the proportional coefficient of different space states was determined, and the spatial transformation problem was also determined. Meanwhile, the results obtained by the numerical simulation calculation and numerical derivation calculation were verified by laboratory experiments. The results showed that the correction coefficient of full space to half space was 2.50, and that of three-quarters space to half space was 1.42. The detection process in operational tunnels involves the mutual transformation process of three typical space states. The critical distance that the spatial transformation does not affect is 300 m. In addition, the results of laboratory experiments verified the typical space proportional coefficient and spatial transformation phenomenon well. The results can provide essential ideas for the subsequent space correction of the detection environment in operational tunnels.

1. Introduction

The transient electromagnetic method is widely used because of the sensitivity property to the electromagnetic field signal of a low-resistivity body [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Meanwhile, the transient electromagnetic method in different engineering fields determines the space state of transient electromagnetism, and different space states have different electromagnetic field diffusion phenomena and data inversion. The space state of transient electromagnetism is a half space state in fields of Earth and aviation [1,2,3,4,5,6]. Correspondingly, the space state of transient electromagnetism is full space in the fields of mines and tunnels [7,8,9,10,11,12,13,14,15]. The data inversion can be divided into linear inversion, nonlinear inversion, and apparent resistivity inversion. The apparent resistivity inversion is the most intuitive and relatively fast. The half space transient electromagnetic inversion method has reached a mature and practical stage [4,9,12]. Since the half space transient electromagnetic field theory is relatively mature, most commercial software adopts the half space inversion to explore underground water and minerals with low resistivity, such as EMIT MAXWELL, Terra TEM and TEMINT [12,13,14].

However, the inversion results are quite different from the actual situation when the inversion method is directly applied to the inversion of full space transient electromagnetism [12]. Therefore, many relevant investigations on the full space effect of tunnel transient electromagnetism have been carried out in the full space inversion [7,8,9,10,11,12,13,14,15]. The proportional coefficient of full space transient electromagnetic inversion is proposed [13], so as to realize the correction of the half space inversion method. It is worth noting that the apparent resistivity result is larger without the proportional coefficient, and this leads to erroneous conclusions in the interpretation of field data. According to Qinghua Liang et al. [16], the induced voltage of the full space state is about 2.52 times that of the half space state, and the full space apparent resistivity value is about 1.842 times of the half space apparent resistivity value. As for the space state of the vehicle-mounted transient electromagnetic method (VMTEM), there are completely different space states at the entrance of the tunnel with the presence of surrounding rock conditions. Compared with the half space and full space states, this space state has similarities and differences. Therefore, the particular space states involved in vehicle-mounted transient electromagnetic detection also need to be further investigated.

The space state of the VMTEM was assumed as the full space [14,15]. However, the detection process of the VMTEM is a moving process from the entrance to the exit of operational tunnels, and the space state of tunnel surrounding rock changes dynamically with the position variation of the vehicle-mounted detection system. As for ideal conditions without the low-resistivity water body and uneven distribution, when the vehicle-mounted detection system is located at the entrance of the tunnel, the lower part of the coil system is the uniform surrounding rock medium, the upper-left is the air medium, and the upper-right is the uniform surrounding rock medium. The coil system can only receive electromagnetic induction signals generated by the lower and upper-right surrounding rock, that is, the three-quarters space (3/4 space) effect. When the vehicle-mounted detection system is located in the middle part of an infinite length tunnel, the above and below of the coil system are uniform surrounding rock medium. The coil system can receive electromagnetic induction signals generated by the above and below surrounding rock simultaneously, that is, the full space effect. For half space and full space, a large number of geophysicists have studied the diffusion law of transient electromagnetic fields, such as Huaifeng Sun [6], Wang [7], Jiang [11], Daiming Tan [13] and Haiyan Yang [17]. However, there is no relevant literature on the diffusion law of electromagnetic fields for the 3/4 space, and the correction coefficient of the corresponding space inversion has not been investigated. Therefore, it is necessary to profoundly investigate the diffusion law of electromagnetic fields in the 3/4 space state.

Based on the previous studies on transient electromagnetic space states, this paper analyzed the diffusion characteristics of the transient electromagnetic fields in different space states, especially the 3/4 space state, so as to understand the spatial transformation law of transient electromagnetic fields. Meanwhile, the proportional coefficients of different space states were fully studied, and the correction method and correction coefficient of the VMTEM were proposed, based on numerical simulation calculation and numerical derivation calculation. Then, the correction principle of spatial transformation was also proposed. Finally, the spatial transformation and critical distance were verified by laboratory experiments. The research can provide a scientific basis for the maintenance of tunnel defects by vehicle-mounted transient electromagnetic detection in the future.

2. Diffusion Characteristics and Proportional Coefficients of Space States

2.1. Transient Electromagnetic Theory and Space Environment

Maxwell’s equations are the basic laws for the propagation of macroscopic electromagnetic fields in a vacuum or medium [18,19]. The equations are a set of partial differential equations used to describe the relationship between electric field, magnetic field, electric charge density, and current density [18,20]. The VMTEM in operational tunnels in the medium can be described by Maxwell’s equations. The equations reflect four basic physical laws, as shown in Formulae (1)–(4):

$$\nabla \times E=-\frac{\partial B}{\partial t} (\mathrm{Faraday}’\mathrm{s} \mathrm{Law})$$

(1)

$$\nabla \times H=J+\frac{\partial D}{\partial t} (\mathrm{Ampere}-\mathrm{Maxwell}’\mathrm{s} \mathrm{Law})$$

(2)

$$\nabla ·D=\rho (\mathrm{Gauss}’\mathrm{s} \mathrm{Law} \mathrm{of} \mathrm{Electric} \mathrm{Field})$$

(3)

$$\nabla ·B=0 (\mathrm{Gauss}’\mathrm{s} \mathrm{Law} \mathrm{of} \mathrm{Magnetic} \mathrm{Field})$$

(4)

where $E$ represents the electric field intensity (V/m), $H$ indicates the magnetic field intensity (A/m), $D$ is the electric displacement vector (C/m²), $B$ is the magnetic flux density (Wb/m²), $J$ denotes the current density (A/m²), $\rho$ represents the electric charge density (C/m³), and $t$ is the time of the electromagnetic field (s).

As a complete set of equations to determine the motion of the electromagnetic field and charge and the current system, the following constitutive relations of the electromagnetic field exist between electric field intensity $E$ and electric displacement vector $D$, magnetic field intensity $H$ and magnetic induction intensity $B$, and electric field intensity $E$ and current density $J$, as shown in Formula (5):

$$D=\epsilon E, B=\mu H, J=\sigma E$$

(5)

where $\epsilon$, $\mu$, and $\sigma$ are the dielectric constant (F/m), magnetic permeability (H/m), and electric conductivity (S/m) of the medium, respectively.

Assuming that the research problem of this paper is free space, $\epsilon ={\epsilon }_0=8.854\times {10}^{-12} \mathrm{F}/\mathrm{m}$, and $\mu ={\mu }_0=4\mathsf{\pi }\times {10}^{-7} \mathrm{H}/\mathrm{m}$, where ${\epsilon }_0$ and ${\mu }_0$ are the vacuum dielectric constant (F/m) and vacuum magnetic permeability (H/m), respectively.

Introducing the vector magnetic potential A, defined as $B=\nabla \times A$, the formulae between (1) and (5) can be rewritten, as shown in Formula (6):

$$\sigma \cdot \frac{\partial }{\partial t}A+\nabla \times H=0$$

(6)

The expression of the electric field vector can be obtained by Fourier transform, as shown in Formula (7):

$$E=\mathrm{i}\omega \mu \left[A+\frac{1}{k^2}\nabla \left(\nabla ·A\right)\right]$$

(7)

where $k$ represents the wave number or propagation constant, $\omega$ indicates the electromagnetic frequency (rad/s), and $\mathrm{i}$ denotes the virtual number unit.

When the transient electromagnetic method is applied to detect tunnel defects in operational tunnels, the specific space state and background field exist, which is different from the conventional application field. The detection process is a moving process from the entrance to the exit of operational tunnels. The space state of the VMTEM changes with the variation of the position of the vehicle-mounted detection system, as shown in Figure 1. As for ideal conditions, without the low-resistivity of a water body and uneven distribution, when the vehicle-mounted detection system is located above the infinite stratum, the lower part of the coil system is the uniform surrounding rock medium, and the air medium is above the coil system. The coil system can receive electromagnetic induction signals generated by the surrounding rock below, namely, the half space effect. When the vehicle-mounted detection system is located at the entrance of the tunnel, the lower part of the coil system is the uniform surrounding rock medium, the upper-left is the air medium, and the upper-right is the uniform surrounding rock medium. The coil system can only receive electromagnetic induction signals generated by the lower and upper-right surrounding rock, that is, the 3/4 space effect. When the vehicle-mounted detection system is located in the middle part of the infinite length tunnel, both the above and below of the coil system are the uniform surrounding rock medium. The coil system can receive electromagnetic induction signals generated by the above and below surrounding rock at the same time, that is, the full space effect. Therefore, there are three obvious space states during the application process, namely, the half space state, the 3/4 space state, and the full space state.

2.2. Diffusion Analysis of Transient Electromagnetic Field in Three Space States

Based on the transient electromagnetic method, numerical simulation calculation of half space electromagnetic response under the condition of the three-dimensional uniform medium was carried out. The transient electromagnetic response curves of the numerical models were calculated by the COMSOL Multiphysics (Version 5.4, AC/DC module), the size of the finite element model was set to 400 m × 400 m × 800 m (length × width × height), and the thickness of the infinite element was 50 m. The resistivity of surrounding rock medium was assumed to be 1000 Ω∙m, the transmitting and receiving coils were laid out in the center of the model, the shape of the coil was simplified to a circle, and the main calculation parameters of coils referred to a previous reference in [14].

The magnetic field and induced current density with a turn-off delay of 1 μs in a half space state could be obtained, as shown in Figure 2. Based on the half space model, the parameters of the air medium above the model were set as those of the surrounding rock medium, and the same model size and mesh division was maintained. The numerical simulation calculation of the electromagnetic response in the full space under the condition of the three-dimensional uniform medium was carried out. The magnetic field and induced current density in the full space state are shown in Figure 3. Based on the full space model, the parameters of the upper-left surrounding rock medium were set as those of air medium, and the same model size and mesh division were maintained. The numerical simulation calculation of the electromagnetic response in the 3/4 space under the condition of the three-dimensional uniform medium was carried out. The magnetic field and induced current density in the 3/4 space state are shown in Figure 4.

As shown in Figure 2, the magnetic field presented a symmetrical distribution in the half space, and the magnetic field propagated outward in the radial direction. The magnetic field was mainly distributed near the transmitting coil, and the maximum value of the magnetic field was distributed in the center of the model. In addition, the induced current was symmetrically distributed in the longitudinal direction of the half space. There was no induced current in the air medium with high resistivity above, and the induced current was only produced in the surrounding rock with low resistivity below. The induced current gradually spread below the syncline. The maximum value of the induced current was distributed at both ends of the diffusion cone, and the amplitude of the induced current in the middle part was the smallest.

It can be seen from the results in Figure 3 that the magnetic field presented a symmetrical distribution in the full space state, and the magnetic field propagated outward in the radial direction. The magnetic field was mainly distributed near the transmitting coil, and the maximum value of the magnetic field was distributed in the center. In addition, the induced current was symmetrically distributed in the transverse and longitudinal directions of the full space. The induced current was generated in the surrounding rock with low resistivity above and below, and the induced current spread to the horizontal direction in a symmetrical form. The maximum value of the induced current was distributed in the center of the two circles.

As illustrated in Figure 4, the magnetic field presented an asymmetric distribution in the 3/4 space state. The maximum value of the magnetic field was distributed in the center of the three-dimensional model. The equipotential line on the left side of the magnetic field was far from the upper air medium with high resistivity and tended to move downward. The equipotential line on the right side of the magnetic field was close to the upper surrounding rock medium with low resistivity and tended to move upward. In addition, the induced current gradually propagated to the lower right in the form of a ring. The angle between the ring plane of the induced current and the horizontal plane was approximately 15 degrees. The symmetrical line of the induced current on the left and right sides deviated from the symmetrical line of the surrounding rock in the 3/4 space state, and the amplitude of the induced current on the left side was smaller than that on the right side.

Meanwhile, the Z-direction magnetic field and induced voltage at the center of the model showed linear changes in the double logarithmic coordinates at the three space state conditions, as presented in Figure 5. With an increase in turn-off delay time, the Z-direction magnetic field and induced voltage at the center of the model gradually decayed. Comparing the measurement results under three different space conditions, it could be found that the amplitude of the magnetic field and induced voltage in the full space state was greater than that in the 3/4 space state. The amplitude of the magnetic field and induced voltage in the 3/4 space state was greater than that in the half space state. On the whole, the fluctuation range of the induced voltage amplitude was more extensive than that of the magnetic field amplitude, and the variation curve of the induced voltage was steeper than the variation curve of the magnetic field. Therefore, the variation curve of the induced voltage was more sensitive to the instantaneous turn-off process of the transmitting current, which also showed that the induced voltage was more suitable for the measurement and interpretation of the transient electromagnetic response signal.

2.3. Proportional Coefficients of Different Space States

(a)

Proportional coefficients based on numerical simulation

According to the variation law of electromagnetic response of three space states under the condition of a three-dimensional uniform medium, it can be seen that there was a certain relationship between the full space state, the 3/4 space state, and the half space state under the condition of the uniform medium. Referring to the investigation of the multiple relations of transient electromagnetism in mines, by Qinghua Liang et al. [16], the proportional coefficient of full space to half space is defined as $k_1$, and the proportional coefficient of 3/4 space to half space is defined as $k_2$, as shown in Formula (8). The results of the two proportional coefficients are shown in Figure 6.

$$k_1=\frac{U_{Full}}{U_{Half}} \mathrm{and} k_2=\frac{U_{3/4}}{U_{Half}}$$

(8)

where $k_1$ and $k_2$ are the proportional coefficients of full space and 3/4 space to half space, respectively, which are dimensionless; $U_{Full}$, $U_{3/4}$, and $U_{Half}$ are the induced voltage in the receiving coil under the conditions of full space, 3/4 space and half space, respectively (unit: V).

It can be seen from Figure 6 that the proportional coefficient curve of full space to half space presented a horizontal straight line on the whole with the increase of time delay, and the average value of the horizontal straight line was 2.5. Therefore, the full space electromagnetic response in uniform media was 2.5 times that of the half space electromagnetic response. Similarly, with the increase in time delay, the proportional coefficient curve of 3/4 space to half space also changed as a horizontal line on the whole, and the average value of the horizontal line was 1.42, that is, the 3/4 space electromagnetic response under the uniform medium was 1.42 times that of the half space electromagnetic response. Affected by the quality and size of the model grid, the proportional coefficient of full space and 3/4 space to half space fluctuated to a certain extent in the middle and late periods, but the relative error was less than 2%.

(b)

Proportional coefficients based on theoretical analysis

Based on the analytical solution of Formula (7), when the direction of the magnetic field is perpendicular to the plane where the transmitting coil is located, the component of the full space magnetic field in the normal direction of the transmitting coil under the condition of the uniform medium can be obtained, as shown in Formula (9):

$$H_z=\frac{2M}{4\pi R^3}\left[\phi \left(u\right)-\sqrt{\frac{2}{\pi }}u{\mathrm{e}}^{-\left(u^2/2\right)}\right]$$

(9)

where $H_z$ is the magnetic field component in the Z direction; $M$ is the dipole moment; $\phi \left(u\right)$ is the probability integral, and its expression is as follows: $\phi \left(u\right)=\sqrt{\frac{2}{\pi }}{{\displaystyle \int }}_0^u{e}^{-\left(t^2/2\right)}\mathrm{d}t$.

According to the investigation of the transient electromagnetic field [18,19,20,21], the analytical solution of the half space magnetic field under the condition of the uniform medium can be obtained, namely:

$$H_z=\frac{M}{4\pi R^3}\left[\left(1-\frac{9}{u^2}\right)\phi \left(u\right)+\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{-\left(u^2/2\right)}\left(\frac{9}{u}+2u\right)\right]$$

(10)

According to Formulae (9) and (10), the proportional coefficient of full space to half space can be obtained, and the same coefficient term can be removed, namely:

$$k_1=2\frac{\left[\phi \left(u\right)-\sqrt{\frac{2}{\pi }}u{\mathrm{e}}^{-\left(u^2/2\right)}\right]}{\left[\left(1-\frac{9}{u^2}\right)\phi \left(u\right)+\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{-\left(u^2/2\right)}\left(\frac{9}{u}+2u\right)\right]}$$

(11)

where $k_1$ is the proportional coefficient of full space to half space.

Based on the theoretical investigation, when the dipole source current is just disconnected in the early stage of transient electromagnetic field, $\mathrm{t}\to 0$, $\mathrm{u}\to \infty$, $\phi \left(u\right)\to 1$, ${\mathrm{e}}^{-\left(u^2/2\right)}\to 0$, and ${\mathrm{e}}^{-\left(u^2/2\right)}$ is an infinitesimal of high order relative to $u^2$, which can be substituted into Formula (11) to obtain the proportional coefficient of full space to half space in the early stage, as shown in Formula (12).

$$k_1=2\frac{\left[\phi \left(u\right)-\sqrt{\frac{2}{\pi }}u{\mathrm{e}}^{-\left(u^2/2\right)}\right]}{\left[\left(1-\frac{9}{u^2}\right)\phi \left(u\right)+\frac{9}{u}\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{-\left(u^2/2\right)}+2u\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{-\left(u^2/2\right)}\right]}=2\frac{\left(1-0\right)}{\left(1+0+0\right)}=2$$

(12)

In the late stage of transient electromagnetic field, $u$ is a small parameter (this is equivalent to the late stage, the distance between the observation point and the field source is short or the conductivity of the medium is low), the probability integral $\phi \left(u\right)$ and the exponential function ${\mathrm{e}}^{-\left(u^2/2\right)}$ can be expanded into a series of $u$, as shown in Formula (13). According to the series theory, the first three terms are taken as Taylor’s expansions of probability integral $\phi \left(u\right)$ and exponential function ${\mathrm{e}}^{-\left(u^2/2\right)}$, as shown in Formula (14).

$${\mathrm{e}}^{-\left(u^2/2\right)}=1-\frac{u^2}{2}+\frac{u^4}{8}-R_{n+1}\left(u\right)$$

(13)

$$\phi \left(u\right)=\sqrt{\frac{2}{\pi }}{{\displaystyle \int }}_0^u{e}^{-\left(t^2/2\right)}\mathrm{d}t=\sqrt{\frac{2}{\pi }}\left(u-\frac{u^3}{6}+\frac{u^5}{40}-P_{n+1}\left(u\right)\right)$$

(14)

where $R_{n+1}\left(u\right)$ and $P_{n+1}\left(u\right)$ are the high-order remainder of the exponential function ${\mathrm{e}}^{-\left(u^2/2\right)}$ and the probability integral $\phi \left(u\right)$ respectively, and $n$ is 2.

By abandoning the higher-order error term and substituting Formulas (13) and (14) into Formula (11), the proportional coefficient of full space to half space in the late period can be obtained, as shown in Formula (15):

$$\begin{gathered} k_1=2\frac{\sqrt{\frac{2}{\pi }}\left(u-\frac{u^3}{6}+\frac{u^5}{40}\right)-\sqrt{\frac{2}{\pi }}\left(u-\frac{u^3}{2}+\frac{u^5}{8}\right)}{\left[\left(1-\frac{9}{u^2}\right)\sqrt{\frac{2}{\pi }}\left(u-\frac{u^3}{6}+\frac{u^5}{40}\right)+\sqrt{\frac{2}{\pi }}\left(1-\frac{u^2}{2}+\frac{u^4}{8}\right)\left(\frac{9}{u}+2u\right)\right]} \\ =2\frac{\left(\frac{u^3}{3}-\frac{u^5}{10}\right)}{\left(\frac{4u^3}{15}-\frac{11u^5}{40}\right)}\approx 2\frac{\frac{u^3}{3}}{\frac{4u^3}{15}}=2.5 \end{gathered}$$

(15)

Formula (12) shows that the induced current under the uniform medium has not been diffused when the current is just disconnected. The proportional coefficient of full space to half space is 2.0, theoretically. It can be considered that the electromagnetic field in the full space state is the superposition of the electromagnetic fields in the upper and lower half spaces. According to Formula (15), the proportional coefficient of full space to half space is 2.5, theoretically, during the induced current diffusion. The electromagnetic field in the full space state is not only the superposition of the electromagnetic fields in the upper and lower half spaces, but also includes the additional electromagnetic fields caused by the upper and lower half spaces (called the ‘space effect’). In the numerical simulation, the induced currents in the full space state and the half space state gradually spread outward. The proportional coefficient of full space to half space maintains a slight variation with an average value of 2.5 in the whole process. The proportional coefficients of full space to half space obtained by the two methods are consistent, so it is feasible to investigate the proportional coefficient of full space to half space by means of numerical simulation.

The proportional coefficient of 3/4 space to half space is determined based on the normal direction of the transmitting coil, not the normal direction of the plane where the induced current diffuses. For the half space state, the plane of the induced current diffusion is parallel to the plane of the transmitting coil. For the 3/4 space state, due to the geometric asymmetry, that is, the lack of some space, the plane where the induced current diffusion is formed in the space is deflected. There is a certain angle between the plane where the induced current diffusion is located in the 3/4 space state and the plane where the transmitting coil is located, that is, the plane where the induced current diffusion is located in the 3/4 space state is not parallel to the plane where the transmitting coil is located. Since the vehicle-mounted detection system still measures the induced eddy current in the Z direction, the electromagnetic field component in the Z direction is still taken as the analysis object of different space coefficients. Therefore, the proportional coefficient of 3/4 space to half space is the vertical component in the normal direction of the transmitting coil.

The 3/4 space state is an asymmetric structure, so it is impossible to derive the analytical formula of the secondary field component of the transient electromagnetic field in the 3/4 space state, and it is difficult to obtain the analytical formula of the proportional coefficient of the 3/4 space to half space in theory. Combined with the numerical simulation, the proportional coefficient of full space to half space is verified correctly. Under the same model size and meshing conditions, it is inferred that the numerical simulation results of the proportional coefficient of 3/4 space to half space under the same conditions can also correctly verify the results of the analytical solution. In the numerical simulation, the average value of the proportional coefficient of 3/4 space to half space was 1.42, and it could be ascertained that the electromagnetic field in the 3/4 space state included three parts. The first part was the electromagnetic field generated by the 1/2 space below. The second part was the electromagnetic field generated by the 1/4 space above (the deflection effect caused by space asymmetry). The third part was the additional electromagnetic field generated by the upper and lower two spaces (the space effect of 3/4 space to half space was significantly smaller than that of full space to half space).

In summary, the proportional coefficient of full space and 3/4 space to half space is a constant value, that is, it shows a fixed multiple relationship. In addition, the numerical simulation results of proportional coefficients in different space states were consistent with the theoretical analysis results, which verified the feasibility of numerical simulation. It provided an essential theoretical basis for the subsequent analysis of the spatial transformation problem of VMTEM in operational tunnels.

3. Analysis of Spatial Transformation and Proposal of Critical Distance

To detect the internal defects of the tunnel structure, the vehicle-mounted detection system runs at a specific speed in operational tunnels, that is, the position of the vehicle-mounted detection system dynamically changes, as shown in Figure 1. Taking the tunnel entrance as an example, the medium below the coil system and above the right does not change when the vehicle-mounted detection system enters the tunnel. In contrast, the medium above the left contains both air medium and uniform surrounding rock medium. The coil system can not only receive the electromagnetic induction signals generated by the surrounding rock below and above the right, but also receives the electromagnetic induction signals generated by the surrounding rock above the left. With the operation of the vehicle-mounted detection system, the electromagnetic induction signals generated by the surrounding rock above the left also increase. Therefore, the running state of the vehicle-mounted detection system in the tunnel entrance section is called transient electromagnetic spatial transformation, that is, from the 3/4 space state to the full space state. Similarly, when the vehicle-mounted detection system gradually approaches the exit of the tunnel, there is also a spatial transformation in the transient electromagnetic space, that is, from the full space state to the 3/4 space state. In summary, the running state of the vehicle-mounted detection system is running from the entrance or exit of the tunnel to the exit or entrance of the tunnel when the vehicle-mounted detection system is running in a limited length of the tunnel. From the perspective of transient electromagnetic spatial transformation, the vehicle-mounted detection system first converts from the 3/4 space state to the full space state, and then converts from the full space state to the 3/4 space state. Therefore, the electromagnetic induction signals at different positions need to be comprehensively analyzed to understand the spatial transformation law of the VMTEM.

Based on the complexity and difficulty of the analytical derivation method, the finite element numerical simulation method was used to investigate the transient electromagnetic response law under different distances from the entrance and exit of the tunnel. Due to the investigation of spatial transformation law, the size of the finite element model needed to be set to be large enough to ensure that the transmitting coil was sufficiently far from the boundary, so that the transient electromagnetic field had sufficient space for diffusion. Considering the calculation ability of the computer, the size of the finite element model was set to 4000 m × 4000 m × 8000 m (length × width × height). In the numerical simulation, it was ensured that the electrical parameters of the transmitting coil and the receiving coil were constant, and only the position parameters changed. Other parameters, such as the size of the infinite element and the grid size, could be set according to the parameters in references [14,22]. The upper part of the medium on the left of the model was set as air medium, and the other positions were set as surrounding rock medium. Meanwhile, it was assumed that the influence of the landform characteristics of the tunnel mountain and the operational tunnel section were not considered. Taking the typical distance between the vehicle-mounted detection system and the entrance and exit of the tunnel as an analysis object, the analysis parameters were 0 m, 10 m, 25 m, 50 m, 100 m, 150 m, 200 m, 250 m, and 300 m, respectively. The transient electromagnetic response law under different offset distances was calculated, and the offset distance of 0 m corresponded to the 3/4 space state. In order to highlight the comparability of spatial transformation, the transient electromagnetic response amplitude of different offset distances was divided by the transient electromagnetic response amplitude of the full space state, and the proportional coefficient $K$ under different offset distances was obtained. The calculation results are shown in Figure 7.

As is exhibited from the results in Figure 7, the proportional coefficient curves of the 3/4 space and full space state showed a straight line. The proportional coefficient under different distances had obvious three-stage variation, namely, early, middle and late stages. The upper limit of the proportional coefficient curve was the amplitude of the proportional coefficient in the full space state, and the lower limit of the proportional coefficient curve was the amplitude of the proportional coefficient in the 3/4 space state. Overall, the proportional coefficients of different distance conditions were close to 1.0 in the early stage, and the proportional coefficients in the middle stage were gradually nonlinearly reduced from 1.0 to 0.568. The proportional coefficients in the late stage became gradually close to 0.568. When the offset distance was small, the early duration was relatively short, the proportional coefficient curve of the medium duration was relatively slow, the corresponding duration was relatively large, and the late duration was relatively large. When the offset distance was large, the early duration was relatively long, the proportional coefficient curve of the medium-term was relatively fast, and the corresponding duration was relatively small. It should be noted that the amplitude of the proportional coefficient in the early stage was slightly larger than the amplitude of the proportional coefficient in the full space state under different offset distances. The increase rate was not more than 5.19%, and the influence of the increase rate of this part could be ignored. Therefore, the offset distance determined the distribution of the induced current and the amplitude of the electromagnetic induction signal. The smaller the offset distance was, the closer it was to the 3/4 space state. The larger the offset distance was, the closer it was to the full space state.

When the vehicle-mounted detection system enters and leaves the tunnel, there is an obvious spatial transformation process. Since the inversion of the vehicle-mounted detection system is based on the uniform medium of the full space state, and the spatial transformation process is obviously asymmetric, the spatial transformation process of the vehicle-mounted detection system needs to be corrected. Based on the relevant investigations [13,16,22], the space correction coefficient $\Phi \left(t\right)$ is proposed for the spatial transformation process, as shown in Formula (16):

$$\Phi \left(t\right)=\left\{\begin{array}{c} 1 t\le t_1\\ 1/\left(a_0+a_1\log \left(t\right)+a_2\log {\left(t\right)}^2+···+a_n\log {\left(t\right)}^n\right) t_1<t\le t_2 \\ 1/0.568 t_2<t \end{array}\right.$$

(16)

where $\Phi \left(t\right)$ is the correction coefficient of spatial transformation, $t_1$ and $t_2$ are the boundary points of the three stages, and their value is related to the distance between the vehicle-mounted detection system and the entrance and exit of the tunnel. The ${\mathrm{a}}_0+{\mathrm{a}}_1\log \left(\mathrm{t}\right)+{\mathrm{a}}_2\log {\left(\mathrm{t}\right)}^2+···+{\mathrm{a}}_{\mathrm{n}}\log {\left(\mathrm{t}\right)}^{\mathrm{n}}$ is the polynomial fitting formula of the proportional coefficient curve in the middle term under different offset distances, $a_0$, $a_1$, $a_2$, and $a_n$ are polynomial fitting coefficients, and $n$ is the number of the polynomial fitting. The value $t$ is the time of the vehicle-mounted detection system, and $\log \left(t\right)$ represents the logarithm of time to the base 10.

According to the relevant investigations [15], the average value of the characteristic parameters of the lifting time of the water-bearing abnormal body is −6.83. The average value of the characteristic parameters of the regression time is −4.2. Therefore, the time window range of instruments could be obtained, as shown in Figure 7. When the offset distance was less than 10 m, the electromagnetic induction signal received by the vehicle-mounted detection system needed to be corrected in the time window of the vehicle-mounted instruments. When the offset distance was between 10 m to 300 m, the received electromagnetic induction signal needed to be corrected in part of the time window, that is, it did not need to be corrected in the early stage and needed to be corrected in the middle stage. When the offset distance was greater than 300 m, the received electromagnetic induction signal did not need to be corrected in the whole of the time window of the vehicle-mounted instruments. Therefore, the offset distance that the transient electromagnetic spatial transformation did not affect was defined as the critical distance without considering the topographic characteristics of the tunnel mountain and the influence of the operational tunnel section. Obviously, it could be judged that the critical distance was L = 300 m, through the above analysis. When the offset distance exceeded 300 m, the vehicle-mounted transient electromagnetic spatial transformation process could be ignored.

To sum up, there is a transient electromagnetic spatial transformation in the running state of the vehicle-mounted detection system in operational tunnels, that is, from the 3/4 space state to the full space state, and, then, from the full space state to the 3/4 space state. By analyzing the offset distance, the closer the vehicle-mounted detection system is to the entrance and exit of the tunnel, the closer the electromagnetic induction signal received is to the 3/4 space state. The farther the vehicle-mounted detection system is to the entrance and exit of the tunnel, the closer the electromagnetic induction signal received is to the full space state. Based on the proposed parameters of vehicle-mounted transient electromagnetic instruments, the critical distance that the space does not affect in the transient electromagnetic spatial transformation process is determined to be 300 m.

4. Laboratory Experiment Verification of space state Transformation

4.1. Theory and Design of the Indoor Model Experiment

According to the similarity criterion of the transient electromagnetic method [12,14], the basic similarity criterion is obtained as shown in Formula (17), and the additional similarity criterion is expressed as shown in Formula (18):

$${\sigma }_m=\frac{L^2}{T}{\sigma }_P$$

(17)

where $p$ and $m$ indicate the prototype and model, respectively, ${\sigma }_p$ and ${\sigma }_m$ are the conductivity of the prototype and model, respectively, $L$ represents the geometric similarity ratio between the prototype and the model, and $T$ denotes the time similarity ratio between the prototype and the model.

$$\frac{V_m\left(t\right)/I_m}{V_p\left(t\right)/I_P}=\frac{N_{Tm}{N}_{Rm}}{N_{Tp}{N}_{Rp}}·\frac{l_{Tm}{}^2{l}_{Rm}{}^2}{l_{Tp}{}^2{l}_{Rp}{}^2}·\frac{T^{2}a}{b}$$

(18)

where $V\left(t\right)$ and $I$ are the induced voltage and emission current in different models, respectively. $N_T$ and $N_R$ represent the turn of the transmitting coil and receiving coil, respectively, $l_T$ and $l_R$ denote the length of the transmitting coil and receiving coil, respectively, and $a$ and $b$ are the ratio factor of electric field and magnetic field between the prototype and the model, respectively.

The PROTEM47 transient electromagnetic instrument is a time-domain transient electromagnetic instrument produced by Geonics in Canada, which is suitable for various occasions [12]. The PROTEM47 transient electromagnetic instrument was adopted in the laboratory experiment, so the time of the two models was consistent. The time similarity ratio was equal to 1, and Formula (17) could be simplified as ${\sigma }_m=L^2{\sigma }_P$, which indicated that the geometric similarity was the main control condition of the basic similarity criterion.

Formula (18) described the proportional relationship between the observed data of the prototype and the model. Since the two proportional factors of the electromagnetic field between the prototype and the model were assumed to be a fixed constant, the proportionality coefficient depended on the specific size of the transmitting coil and receiving coil. Therefore, the coil system could not be designed according to the geometric similarity, which brought significant convenience to the design of two coils in the laboratory experiment.

The PROTEM47 transient electromagnetic instrument could convert the induced voltage of the receiving coil into the rate of change of magnetic induction intensity $d_B/d_t$ according to the corresponding input parameters. If the electromagnetic field proportional factor b = 1 was selected, the transformation result $d_B/d_t$ obtained by field detection equaled $d_B^{\prime }/d_t^{\prime }$ in the physical experiment [15]. Using the interface menu of the PROTEM47 transient electromagnetic instrument and inputting the parameters of each device in the experiment according to Formula (18), the experiment data consistent with that in the field could be obtained.

The experiment mainly verified the response difference of transient electromagnetic effect in the full space state, the 3/4 space state, and the half space state. During the operational process of the vehicle-mounted detection system of the high-speed railway tunnel, the space size involved in the spatial transformation of the VMTEM could reach several hundred meters or even several kilometers. In contrast, the size that could be simulated in the laboratory experiment could only reach several meters, which was relatively small. Therefore, a similar model experiment was used to verify the spatial transformation of the VMTEM. The indoor experiment diagram of the spatial transformation problem is shown in Figure 8. In the indoor experiment, the similarity ratio was 1:300, and the internal size of the model was 1.0 m × 1.0 m × 1.0 m, which was used to simulate the surrounding rock with the actual size of 300 m × 300 m × 300 m. To simulate the spatial transformation relationship of VMTEM, high-resistance filler was used to simulate the actual air inside. The size of the high-resistance filler was 0.5 m × 0.5 m × 1.0 m, which was used to simulate the air with the actual size of 150 m × 150 m × 300 m. The model was filled with salt water, and the resistivity of salt water should be 0.01 Ω∙m, in theory. However, due to the saturation of salt water, it was difficult to achieve the theoretical resistivity of salt water in the laboratory experiments, so the resistivity of salt water in the laboratory experiments was approximately 5.0 Ω∙m (indoor temperature was 18 degrees Celsius).

The critical distance was L = 1 m under the scale of 1:300, which was close to the size of the indoor model. Therefore, the transient electromagnetic response intensity of L=150 m could be preliminarily determined, that is, the critical distance of the indoor experiment was L=0.5 m (the model size met the size).

Based on the above two similarity criteria, the variation law of the transient electromagnetic response curve in the prototype could be reflected by the transient electromagnetic response curve in the model, and only a certain coefficient ratio existed between the two conditions. In the laboratory experiment, the geometric similarity ratio was set to 1:300, so the scale of the physical model experiment could be presented to meet the background field requirements of the site electromagnetic environment.

4.2. Experiment Procedure

The PROTEM47 transient electromagnetic instrument is shown in Figure 9. The PROTEM47 transient electromagnetic instrument could observe three components at the same time, and could also measure a single component separately. The turn-off time is one of the important technical indicators of the transient electromagnetic instruments. If the turn-off time is long, shallow signals are lost, the intensity of the secondary field weakened and the detection effect directly affected. The turn-off time of the device depends on the performance of the transmitting, the size of the transmitting current, and the size of the transmitting coil. Theoretically, the turn-off time of the PROTEM47 could be as short as 0.5 μs. In terms of coil design, the coil is made of copper metal wire, and its shape is square, as shown in Figure 10. The coil size was selected according to different requirements. In the indoor experiment, the width of the transmitting coil and the receiving coil were both 5 cm, the number of turns of the transmitting coil and the receiving coil were both 30 turns, the diameter of the transmitting coil and the receiving coil were both 0.51 mm, and the transmitting current was 0.5 A.

The main content and process of this experiment were as follows:

(1)

The half space state: In the similarity experiment, only the saline medium was used to simulate the three-dimensional uniform half space medium. According to the conductivity and volume of the saline medium, a suitable amount of solid salt was added, and the saline height was set to the half-height of the container. After the liquid surface was calm, the transmitting coil and the receiving coil were placed 5 cm above the surface of the saline solution. The plane of the coil system was parallel to the surface of the solution. The transient electromagnetic response data were measured and recorded.

(2)

The full space state: Similar to the response experiment in the half space state, the saline medium was used to simulate the three-dimensional uniform full space medium. According to the conductivity and volume of the saline medium, an appropriate amount of solid salt was added, and the saline height was set to the height of the container. After the liquid surface was calm, the transmitting coil and the receiving coil were placed in the center of the solution space, and the plane of the coil system was parallel to the solution surface. The transient electromagnetic response data were measured and recorded.

(3)

The 3/4 space state: Similar to the response experiment in the half space state, only the saline medium was used to simulate the three-dimensional uniform 3/4 space medium. In the lower-left corner of the model space, the air bag was placed and fixed to take up a quarter of the space. After the conductivity of the saline medium was adjusted to the design value, and the liquid surface was calm, the transmitting coil and receiving coil were placed in the center of the solution space, and the plane of the coil system was parallel to the solution surface. The transient electromagnetic response data were measured and recorded.

(4)

Judgment of critical distance in the 3/4 space state: Based on the experiment of no abnormal body response in 3/4 space state, the transmitting coil and receiving coil were moved 0.1 m, 0.2 m, 0.3 m, 0.4 m, and 0.5 m, in turn. The transient electromagnetic response data were measured and recorded.

Taking the 3/4 space state as an example, the preparation process of the experiment is described in detail. According to the schematic diagram in Figure 8, a polypropylene water tank with a certain water pressure resistance was selected as the experimental container, and EPE pearl cotton foam board was selected as the high-resistance filler. Saline water with a concentration of 5% at 25 degrees Celsius was pre-configured by mixing solid sodium chloride with water. The resistivity of the salt water was approximately equal to 5.0 Ω∙m, so salt water was selected as being similar material for the surrounding rock. The length and the number of turns of the transmitting and receiving coils were taken as 5 cm and 30 turns, respectively. The transmitting current was taken as 0.5 A. The high-resistance filler was placed on one side of the model container, according to the design requirements. The high-resistance filler and the model container are shown in Figure 11. According to the designed solution concentration, 50 kg of solid particle salt was added, and 750 L water was added. In order to prevent the salt water from entering the high-resistance filler, a polyethylene waterproof plastic film was laid outside the high-resistance filler to isolate the high-resistance filler and salt solution, as shown in Figure 12. According to the coil layout in Figure 8, the transmitting coil and receiving coil were placed at the center of the salt solution. The transient electromagnetic equipment was used to measure the electromagnetic intensity in a 3/4 space state. The indoor field layout is shown in Figure 13.

4.3. Model Experiment Results of the Spatial Transformation Problem

Through the indoor saline model experiment, the transient electromagnetic response curves of saline conductivity in the half space state, the 3/4 space state, and the full space state were measured, respectively, as shown in Figure 14.

Figure 14 shows that the amplitude of the electromagnetic induction curve was relatively large before the 10th gate, and the response curve went down fast. It could be judged that the response of the electromagnetic induction curve was mainly due to the salt water conductor. After the 10th gate, the amplitude of the electromagnetic induction curve was close to the level of air noise response, and it could be judged that this section of the electromagnetic induction curve could not be used to verify the vehicle-mounted transient electromagnetic spatial transformation problem. By comparing the response curves of saline models under different space conditions, the electromagnetic response of the full space model was greater than that of the 3/4 space model to the half space model. The electromagnetic response of the 3/4 space model was greater than that of the half space model.

To better illustrate the problem of spatial transformation, the results of the first six-time gates were taken as the analysis object, and the results are shown in the local magnification in Figure 14. It could be demonstrated from the local amplification diagram that the transient electromagnetic response curves of different models showed the linear attenuation law in the double logarithmic coordinates. Comparing the response curves of the saline model under different space conditions, the amplitude of the three curves had a certain proportional relationship under the condition of the first six time gates. By calculating the average value of the ratios of the first six time gates, the ratio of full space to half space was 2.409, and the ratio of 3/4 space to half space was 1.357. In the calculation results of theoretical calculation and numerical simulation, the ratio of full space to half space was 2.53, and the ratio of 3/4 space to half space was 1.42. It can be seen that the space proportional relationship of the indoor saline model experiment was smaller than that of the theoretical calculation and numerical simulation, and the error was less than 5%. Therefore, it could be judged that the indoor saline model experiment verified the results of theoretical calculation and numerical simulation well.

Meanwhile, the apparent resistivity inversion results could be obtained, based on the experiment results of the three ideal space states, as shown in Figure 15.

It can be seen from Figure 16 that the apparent resistivity inversion results basically showed a linear decline in the early and middle periods under a single logarithmic coordinate condition, based on experimental data. On the whole, the apparent resistivity result of the half space inversion was the largest, the apparent resistivity result of the full space inversion was the smallest, and the apparent resistivity result of the 3/4 space inversion was in the middle. It should be noted that the apparent resistivity result was greater than the resistivity of the actual model in the early period, which meant that there were still some differences between the apparent resistivity inversion results and the actual model resistivity. According to the proportional coefficient, the inversion results of the three ideal space states could also be illustrated, as in Figure 15. Taking the first eight-time gates as the analysis object, the full space inversion results based on the proportional coefficient were smaller than those of the experimental data, while the 3/4 space inversion results, based on the proportional coefficient, were larger than those of the experimental data. Through the calculation of different types of inversion data, the average relative errors of the first eight time gates of the inversion results of different methods were 5.32% and 12.21%, respectively, which meant that the inversion error was within the allowable range.

Based on the saline model in the 3/4 space state, the electromagnetic response curves of saline conductors were measured when the transmitting coil and the receiving coil moved at different distances simultaneously. The results are shown in Figure 16.

It can be demonstrated from Figure 16 that there were certain differences in the amplitude of the response curve of the saline model under different offset distances in the early and middle stages. In the middle and late stages, the response curves of the saline model under different offset distances were the same, and the amplitude was close to the level of air noise response. To compare the response differences of saline models under different offset distances, the results of the first eight time gates were taken as the analysis object, and the local amplification diagram of the results is shown in Figure 16. According to the local magnification, the smaller the coil offset distance of the vehicle-mounted detection system was, the closer the amplitude of the induced voltage of the saline conductor was to that of the saline conductor in the 3/4 space state. The larger the coil offset distance was, the closer the amplitude of the induced voltage of the saline conductor was to that of the saline conductor in the full space state.

Through the indoor saline model experiment, there was a process of spatial transformation in the transient electromagnetic space when the vehicle-mounted detection system moved from the tunnel entrance to the middle part of the tunnel, that is, from 3/4 space to full space. The electromagnetic induction amplitude of the surrounding rock background field received by the vehicle-mounted detection system became gradually closer to the electromagnetic induction amplitude of the uniform surrounding rock in the three-dimensional full space state. Therefore, the spatial transformation process of the VMTEM detection system and the critical distance without considering the space influence could be qualitatively verified.

5. Conclusions and Discussion

This paper mainly investigated the diffusion characteristics and proportional coefficients of three ideal space states, analyzed the spatial transformation process of operational tunnels, and put forward the critical distance. A laboratory experiment was also carried out. The main conclusions are as follows:

(1)

The induced current in half space gradually spreads to both ends, and the induced current in full space diffuses to both ends of the horizontal symmetric center. However, the induced current in 3/4 space is not symmetric, and the angle between the induced current in 3/4 space and the horizontal plane is approximately 15°.

(2)

Taking the amplitude of induced voltage in the vertical direction of the transmitting coil as the research object, the space correction method and correction coefficient of the VMTEM are proposed. As for the condition of the uniform medium, the response signal of three space states all exhibit a linear decrease under the double logarithmic coordinate system. The correction coefficient of full space to half space is 2.50, and the correction coefficient of 3/4 space to half space is 1.42.

(3)

The correction of different space states can be realized by substituting the proportional coefficient of the corresponding space at different stages. Aiming at the spatial transformation problem of vehicle-mounted detection systems, the space correction coefficient formula is proposed. Combined with the proposed time window of the vehicle-mounted detection system, the critical distance is determined to be 300 m.

(4)

By analyzing the response values under different space conditions and coil offset conditions, the indoor saline model experiment verified the phenomenon of spatial transformation in transient electromagnetic space. The critical distance, without considering the influence of transient electromagnetic spatial transformation, was also verified.

In the present study, the inhomogeneity of surrounding rock distribution and the presence of low-resistivity water body were not considered for the transient electromagnetic diffusion law of different space states. Meanwhile, the shape of the tunnel surrounding rock was not fully considered in the analysis of the spatial transformation process. Therefore, the proportional coefficients of different spaces were based on ideal conditions, and there was a certain inconsistency with the actual surrounding rock environment. In future study, the surrounding rock environment of this complex condition will be further considered and modified proportional coefficients of different spaces obtained. As for the indoor saline model experiment, the experiment size was only 1.0 m and still quite small, which could not realistically illustrate the actual situation. Besides, the amount of experimental data were not enough, and there may have been some response curve characteristics that were not found. These two aspects of the model experiment need to be improved in further research.