**3. Method**

To understand the effects of the 3D effect and boundary effect on 2D resistivity profiles, this study separately established numerical models including models of pipelines buried next to the survey line and geological formation changes at the survey line boundary. Then, we investigated the possible effects on the 2D resistivity profile through the parametric variation such as the resistivity ratio, electrode spacing, depth of media buried, and medium size.

This study assumed that the resistivity ratio was *n* = *R2*/*R1*, where *R1* is the resistivity in the background soil layer, and *R2* is the soil layer outer resistivity profile or boundary medium. When *n* > 1, the boundary medium of high resistivity (*R2*) has little effect on the low resistivity zone (*R1*). When *n* < 1, the boundary medium of low resistivity (*R2*) has a large influence on the high resistivity zone (*R1*). The main reason is that the current is more concentrated and has an ability to flow better where the low resistivity zone is.

Therefore, this study will investigate the case when *n* < 1, that is, when the boundary medium is low resistivity, for the change of the high resistivity soil layer on the 2D resistivity profile.

#### *3.1. 3D E*ff*ect Model with Pipeline Formation*

To understand the boundary effect that may be caused by geological formations with pipelines, this study established a geological formation model with pipelines as shown in Figure 2. This study will establish and plan a geological formation model and testing parameters from the four aspects of resistivity ratio, spacing of the line and pipelines, depth of pipelines building, and pipeline size in the pipelines and strata.

In the model, the resistivity of the pipelines is *R2*, the soil layer resistivity is *R1*, *R1* > *R2*, and the ratio of the two is n (*n* = *R2*/*R1*). To discuss the influence of the resistivity ratio under this assumption, we fixed R1 as 1000 ohm-m, and set R2 as 10, 50, 100, and 250 ohm-m, respectively. The section size of the pipelines was *dyp* × *dzp*, and the depth of the embedding was *dep* = 2 m (the top of the pipeline). The parallel pipelines around the pipeline setting were 5 to 6 lines, each line was separated *dy* = 3, the spacing to electrode rods was *dx* = 3, and each line had a slight adjustment to the length of line L with the sounding requirement. Table 1 presents the explanatory table of the 3D effect model parameter with the pipeline formation.

**Figure 2.** 3D effect model with the pipeline formation.

**Table 1.** The explanatory table of the 3D effect model parameters with the pipelines.


#### *3.2. Boundary E*ff*ect Model*

To understand the boundary effects that may be caused by geological formation changes on both sides of the survey line, this study explored the effects of the boundary effect on 2D ERT through the changes in the boundary medium using numerical simulations by assuming there was heterogeneous media at the boundary. This study explored the influence difference through a model comparison with a boundary effect and no boundary effect, as shown in Figures 3 and 4.

**Figure 4.** 2.5D boundary effect model.

The boundary effect was first calculated through the apparent resistivity profile of the model with a boundary medium by forward modeling. Then, we calculated the apparent resistivity by inverse calculation to obtain the resistivity profile of the model affected by the boundary medium. To understand the influence of the boundary effect, this study deleted the data points of the boundary medium in the apparent resistivity profile, and then through recalculation to obtain the new resistivity profile section, we compared the difference between the two. The deleted distances of the data points were 0 m, 2 m, 4 m, 6 m, 8 m, and 10 m from the boundary medium.

This study explored the aspects of the resistivity ratio, electrode spacing, the influence distance of the boundary effect, medium depth, and medium size through parametric variation. In the model, it set that the resistivity of the background soil layer was R1, the resistivity of the boundary medium was *R2*, the section size of the medium was *dyp* × *dzp* = 2 × 2 m2, the embedding depth was *dep* = 2 m, the spacing of the electrode rods was *dx* = 2, and the length of the survey line was *L* = 50 m. Table 2 is the explanatory table of the boundary effect model parameters.


**Table 2.** The explanatory table of the boundary effect stratum model parameters.

#### **4. Result and Discussion**

#### *4.1. Pipeline Stratum Model under 3D E*ff*ect*

#### 4.1.1. Resistivity Ratio (*n* = *R2*/*R1*)

To understand the influence of the geological resistivity ratio of the pipeline stratum to the 3D effect, this study set four different kinds of geological resistivity ratio values (*n* = 0.01, 0.05, 0.1, and 0.25). We fixed the *R1* as 1000 ohm-m, and set the *R2* as 10, 50, 100, and 250 ohm-m, respectively, for the discussion on the influence of the resistivity ratio.

This study analyzed the influence through small pipeline sizes (1.5 m × 2 m), and the survey line L1~L5 of the small-scale pipeline was sequentially 1.5, 0, 3, 6, and 9 m away from the boundary of the pipeline, therefore, we adopted L3 (where the horizontal distance from the pipeline was 3 m) to analyze. The result of its resistivity ratio is shown in Figure 5. When the resistivity ratio was less than 0.05, the mapping phenomenon of the 3D effect could be clearly observed, and when the resistivity ratio was more than 0.1, the 3D effect was gradually nonsignificant.

**Figure 5.** The resistivity ratio and the 3D effect influence on the pipeline stratum model.

## 4.1.2. Pipeline Size

To understand the influence of the pipeline size on the 3D effect, this study set two kinds of stratum models with different pipeline sizes, where the section size of the pipelines was 1.5 (W) × 2 (D) m and 3 (W) × 4 (D) m, respectively. As seen in Section 4.1.1, when n was more than 0.1, the 3D effect became weaker. Therefore, this study fixed the resistivity ratio to explore the 3D effect under different conditions of pipeline sizes.

Figure 6 shows the results of testing the 2D ERT near two different pipeline sizes. Due to the pipeline sizes, the distances between the L1~L5 lines of the small-scale pipeline and the boundary of the pipeline were 1.5, 0, 3, 6, and 9 m, respectively, and the distances between the L1~L6 lines of the large size pipeline and the boundary of the pipeline were 0 (the left side of the pipeline), 0 (the right side of the pipeline), 3, 6, 9, and 12 m, respectively. In the case of the small pipeline, there was a nonsignificant 3D effect outside the line of the boundary, but in the large size pipeline, line L4 (6 m away from the edge of the pipeline) shows the mapping results of adjacent pipelines. The results show that the influence of the 3D effect increases with the size of the pipeline.

Furthermore, we observed that the mapping depth was deeper than the actual depth of the pipeline. The results showed that the mapping mechanism was not horizontally mapped, and that the mapping depth was related to the distance between the pipeline and the survey line. When the distance was over 6–9 m, the result showed that it was almost unaffected by the 3D effect.

**Figure 6.** The influence of the pipeline size of the pipeline stratum model on the 3D effect.

## 4.1.3. Embedding Depth

To understand the influence of the pipeline embedding depth on the 3D effect, this study set two kinds of models with different depths of embedded pipeline. The embedding depth (the top of the pipeline) was 0 m (embedding at the surface) and 4 m, respectively. The n value in the model was set as 0.1, the section size of the pipeline was 1.5 (W) × 2 (D) m, and there were five survey lines.

Figure 7 shows the 2D ERT results of the small pipeline with an embedding depth of 0 m and 4 m. Compared with the L1 lines (the horizontal distance from the buried pipeline was 1.5 m), when the embedding depth was 2 m, we could obviously observe the mapping phenomena, but when the embedding depth increased to 4 m, the 3D effect was not as obvious. This shows that as the depth increased, the influence of the 3D effect weakened, and the influence range seems to be related to the distance from the line to the pipeline, and not the horizontal distance.

**Figure 7.** The influence of the pipeline embedding depth in the pipeline stratum model on the 3D effect.

4.1.4. Influence Distance

To find out the possible influence range of the 3D effect in the pipeline stratum, this study used the model that set *n* as 0.1. The section size of the pipeline was 1.5 (W) × 2 (D) m, *dep* = 2 m, *dy* = 3 m, *dx* = 3 m, *L* = 42 m, and the L1~L5 lines were 1.5, 0, 3, 6, and 9 m away from the boundary of the pipeline.

As shown in the image on the left in Figure 7, the profiles at L1 and L3 had a significant 3D effect, whereas the 3D effect of the L4 line profile gradually became less obvious, and in the L5 line profile, there was no 3D effect. Therefore, the 3D effect influence distance was 6 m, and gradually faded after 6 m.

## 4.1.5. Electrode Spacing

To understand the relationship between the electrode spacing of the pipeline stratum and the 3D effect, and whether the electrode spacing could be used as the normalized parameter, this study analyzed the electrode spacing parameter, which was 3 m and 1.5 m, respectively. The model set n as 0.1, the section size of the pipeline was 1.5 (W) × 2 (D) m, *dep* = 2 m (the top of the pipeline), *dy* = 3 m, *dx* = 1.5 m or 3 m, and *L* = 42 m.

We used the profiles of L1~L6 to compare the influence situation and influence distance of the 3D effect with different electrode spacings (*dx* = 1.5, 3). As shown in Figure 8, when the electrode spacing was shortened, the spatial resolution improved. However, the influence range of the 3D effect between different electrode spacings was similar; after the distance from the edge of the pipeline increased to

6 m, the 3D effect gradually became inconspicuous. This result shows that it is not appropriate to use electrode spacing as a normalized parameter for the 3D effect because the influence spacing does not decrease as the electrode spacing decreases.

**Figure 8.** Pipeline stratum model showing the influence between the 3D effect range and electrode spacing.

#### *4.2. Boundary E*ff*ect Model*

#### 4.2.1. Resistivity Ratio (*n* = *R2*/*R1*)

To understand the influence of the boundary effect in different resistivity ratios, this study fixed the background value *R1* as 1000 ohm-m, and assumed that the ratio *n* of *R2* and *R1* were 0.25, 0.1, 0.05, and 0.01, respectively, that is, the model of *R2* resistivity was 250, 100, 50, and 10 ohm-m, respectively.

The results in Figure 9 show that the presence of the medium in line boundary caused a resistivity profile with an unusual resistivity value near the boundary, and the unusual boundary resistivity value was higher than the background value (1000 ohm-m). The study judged that the current encountered the low resistivity medium in the boundary when it flowed through the boundary. This creates a large amount of current that is concentrated in a low-resistivity zone, and causes the resistivity value of the boundary profile to increase abnormally.

In the case of *n* < 1 (*R2*/*R1*), with the increase in the resistivity ratio value n, the influence of the boundary effect gradually decreased. As shown in Figure 9, there was an abnormal resistivity value zone in the resistivity profile when the resistivity ratio value n was under 0.1, and was not obvious until the value n reached 0.25. That is, when resistivity ratio is under about 0.1, there are concerns about the boundary effect, which is similar to the situation of the 3D effect.

**Figure 9.** The influence situation of the boundary effect with different resistivity ratios.

## 4.2.2. Medium Size

To understand the influence of the medium size on the boundary effect, this study fixed the electrode spacing and the medium depth, and used the medium size of 2 × 2 m and 4 × 4 m to explore this further.

The results in Figure 10 show that under the same value n, the addition of the size of the boundary medium does not affect the influence range of the boundary effect; that is, regardless of the size of the boundary medium, the boundary effect is the same, and this situation is different from the 3D effect of the pipeline stratum. It can also be seen in Figure 10 that the influence ranges of the resistivity ratio *n* = 0.1 were the same regardless of the pipe size, and the influence distances were both 8 m.

(**a**) 

**Figure 10.** *Cont.*

**Figure 10.** The influence situation of the boundary effect with different medium sizes. (**a**) medium size 2 × 2; (**b**) medium size 4 × 4.

## 4.2.3. Embedding Depth

In order to understand the influence of different embedding depths on the boundary effect, this study explored the medium that was 2 m and 4 m under the surface, respectively.

The results in Figure 11 show that if the boundary medium was the same size, the influence distance of the boundary medium under the surface 4 m was farther than the one under the surface at 2 m. That is, the deeper the boundary medium, the larger the influence of the boundary effect, which is a different situation to that of the 3D effect of the pipeline boundary. It can be seen in Figure 11 that if the resistivity ratio value n was 1 when *dep* = 2 m, the influence distance of the boundary effect was 8 m, and when *dep* = 4 m, the influence distance of the boundary effect was 10 m. This result can be seen by the data point of the apparent resistivity in the model; when the medium below the surface is

deeper, the deeper data point will be affected, which will cause the influence range of the boundary effect to become wider.

(**a**) 

**Figure 11.** *Cont.*

**Figure 11.** The influence of the boundary effect with different medium depths. (**a**) medium depth 2 m; (**b**) medium depth 4 m.

#### 4.2.4. Influence Distance

To understand the influence distance of the boundary effect, this study set the resistivity ratio *n* = 0.1 and electrode spacing *dx* = 2. These results are shown in the left image in Figure 12, where the resistivity profile obviously had an abnormal resistivity zone within 6 m from the boundary medium, which was obviously affected by the boundary effect. However, after 8 m away from the boundary medium, it was almost unaffected by the boundary effect. This result is similar to the influence distance of the 3D effect.

(**a**) 

**Figure 12.** *Cont.*

**Figure 12.** The influence situation of the boundary effect with different electrode spacing. (**a**) electrode spacing 2 m; (**b**) electrode spacing 4 m.
