**1. Introduction**

Electrical resistivity tomography (ERT) applied to geotechnical investigation has developed rapidly in recent years with the beginning of 1D exploration that is mainly used in groundwater and mining exploration. With the advancement of computer technology and forward and inverse computing skills, in the past ten years, 2D exploration has been widely used in the investigation and monitoring of geotechnical engineering, environmental engineering, and groundwater pollution. 3D ERT is gradually starting to be used, however, due to the limitations of the test environment, 3D is the trend of future development, so the application in current engineering is mainly based on 2D ERT [1–7].

Linearity 2D ERT assumes that the geological formation resistivity property is homogeneous in the vertical line direction. When the geological formation does not meet the condition where they homogenize in the vertical line direction, since the electric current is flowing in 3D, the 2D ERT result may be in error, so the distortion that this hypothesis may cause is often ignored in the interpretation of the test results. The 3D effect means that the geological structure outside the 2D resistivity profile will map to the 2D resistivity profile and cause such errors. The boundary effect means that when 2D ERT is measured on a straight line of finite length but the boundary of the survey line encounters terrain changes such as pools, valleys, or underground structures like concrete structures or metal pipelines, it may be mapped to the 2D electrical resistivity profile, causing exploration errors, thus affecting the test results and the accuracy of the interpretation.

The application of ERT in geotechnical investigation has been developed quite completely, and there exist fairly complete procedures from the arrangemen<sup>t</sup> of the test to the inverse computation analysis of the data, but there has not been a detailed assessment of the 3D e ffect and the boundary effect. 2D ERT assumes that the geological formation is a 2D semi-infinite space. However, in the real state of geological formations, the electric current flows in the 3D (X, Y, Z) direction, therefore, the object in the non-2D section has a certain degree of disturbance to the ground resistivity electric field, which causes some irregular resistivity and noise on the 2D profile [8]. Torleif Dahlin explored the application of 2D ERT in the environment and engineering in Sweden. In the in situ test, it was considered that there was a 3D e ffect, and they suggested avoiding this when setting the line [9]. Lin et al. explored the detection of dam leakage by 2D ERT and found that the 3D e ffect was generated by the structure around the line and projecting the feature of the structure onto the testing profile. Therefore, they suggested that the e ffects of the 3D e ffect should be noted when testing [10]. Many scholars often face the boundary e ffect caused by the boundary of the sandbox when conducting an indoor sandbox experiment [11–16]. Mei, Xing-Tai used a homogeneous thickness model made from dough to explore the indoor test method and criteria for 2D ERT. As the limited boundary on the three sides of the sand-box will a ffect the transfer of the electric current, this is inconsistent with the assumption that the theoretical electric current has boundless extended flows in the underground half-space, and will consequently cause the boundary e ffect. After discussion, it was found that the boundary of the sand-box will cause the resistivity to be in an unstable state until the boundary of the bilateral line is more than double spread, where the apparent resistivity will be stable [17].

It can be found from the above literature that the 3D e ffect and the boundary e ffect are the problems that most scholars may face when conducting electrical resistivity tomography. However, few scholars have explored the 3D e ffect and the boundary e ffect in depth. The purpose of this study was to explore the 3D e ffect and boundary e ffect on 2D ERT and to understand the possible e ffects of changes in the geological formation outside the line. In this paper, through the establishment of a numerical model, we simulated 3D models and boundary stratigraphic fluctuation including a model where the extremity line was medium and encountered pipelines aside from the line. We used the parametric variation of the resistivity ratio, electrode spacing, depth of media embedding, and medium size to explore the influence situation of the 3D e ffect and boundary e ffect on 2D ERT. Finally, based on the numerical modeling results, we propose suggestions or precautions for future testing.

## **2. ERT Background**

ERT mainly uses two current poles (C1, C2) and two potential poles (P1, P2) to collect measurements, as shown in Figure 1. The low frequency alternating current accesses the geological formation through current poles. Under di fferent geometrical positions, potential poles and current poles are separately used to measure the potential di fference and current. Since the calculation involves the distribution of the conductive medium in the formation, there are a large number of potential di fference solutions and boundary conditions that are necessary to solve through forward modeling, as shown in Equation (1). The resistivity calculated by forward modeling is called apparent resistivity, which is usually not the true resistivity of the underground electrical formations. The true resistivity can be obtained by calculating the apparent resistivity through a suitable inverse computation [18].

$$\varrho\_a = \frac{V\_{P1} - V\_{p2}}{I} \times \frac{2\pi}{\frac{1}{r1} - \frac{1}{r2} - \frac{1}{r3} + \frac{1}{r4}} = K \frac{\Delta V}{I} \tag{1}$$

where *K* is the geometrical arrangemen<sup>t</sup> parameter of the electrode; Δ *V* is the measured potential di fference; and *I* is the circulating current intensity.

Apparent resistivity means that when taking measurements in real geological formations, the measured resistivity may transform by changing the electrode spacing and position as the resistivity of the formation may be heterogeneous. For example, if we were to lay dozens of electrode rods equally on the Earth's surface in advance, then constantly change the spacing and strafing right to measure by the current poles and potential poles aforementioned, it can obtain a pseudo-section as shown in Figure 1. Finally, the resistivity distribution of the representative geological formation should be obtained through inverse analysis. Inverse analysis generally adopts an optimization method because the resistivity value calculated by inverse computation is close to that of the true resistivity value [19]. The software of inverse analysis makes an initial value guess for the inverse model through the measured pseudo-section, and after presetting the initial inverse model and minimizing the difference between the model's reacting value and the measured data value through multiple operations. The result of the inverse at present is the resistivity distribution of the real geological formation. This study adopted the EarthImager inverse software developed by AGI to perform numerical simulations and calculations [20].

The simulated testing method in this study adopted dipole–dipole electrode arrangement. The electrode sequence of the dipole–dipole array was C2, C1, P1, P2, respectively. The two current poles form a dipole, and the two potential poles form another dipole, where C1C2 = P1P2 = a and C1P1 = na. When the parameter n is gradually increased, the resistivity of the formation changing from shallow to deep can be obtained. The high sensitivity of this arrangemen<sup>t</sup> is concentrated between the paired current poles and the potential poles, so horizontal formation changes are more suitable, while the vertical direction changes poorly.

**Figure 1.** Schematic diagram of the ERT test.
