# Geo-Electrical Detection of Impermeable Membranes in the Subsurface
## 5 Numerical Simulation of the Lab Experiment
### Results for Sand and Clay
The on-the fly workflow has 3 steps:  
Figure 10: Same as Figure 8 with the tool (black vertical line) at 0.2 m from the membrane (red vertical line). (**A–F**) reflect tests for various simulated positions of the membrane. (**G**) shows the misfit function with a clear global minimum at x = \(-\)0.1 m.  
* First step: For each next position of the tool, a new set of Mode 1 and Mode 2 array measurement data is acquired. * Second step: Next, a tomographic inversion is performed from the surface up to current position of the tool. Inversion software Res3 DiNV is used with dimensions of the simulated tank (i.e., the boundary effects are treated in 3D). This results in an update of the background resistivity (Figure 7, from top to bottom). For simplicity we only show one XY plane in the centre of the tank. Mode 1 arrays are not influenced by the membrane, unless the tool is very close. Mode 1 is sensitive to the background resistivity close to the tool and therefore is most useful to obtain the background resistivity model up to the position of the tool. Mode 2 data, on the other hand, have larger looking ahead capabilities and these data are used calculate the updated kernels. Figure 7 shows the update of the background resistivities for three positions of the tool. When the tool has just been inserted in the soil, there is only a small set of measurements available (Figure 7, top). This small data set is used to perform an inversion for a relatively small volume. When the tool is moved to a closer position (Figure 7, middle), more measurements are available and inversion of the background resistivity of a larger volume is determined. Even closer (Figure 7, bottom), the background resistivity image is nearly complete. The low resistivity zones of the clay slabs and cubes can be discerned. Also visible in the background resistivity image are the two plastic rails that guide the tool (because of the weight) that show up as two high resistivity lines. * Third step: At some point in moving the tool closer to the membrane, it might come too close which might result in puncturing the membrane. Therefore, after each measurement and inversion to obtain the background resistivity, a check is performed whether it is safe to push the tool to the next position. This is done by comparing the updated kernels to the measured data up to the latest position of the tool. After updating the kernels, an attempt is made to determine the position of the membrane relative to the tool. This is illustrated in Figures 8-10, for a decreasing distance from 1.2 to 0.6 to 0.2 m respectively. In theory, the position of the membrane corresponds to the kernel with the minimum misfit between the measured data and the kernel. However, if the tool is still far away from the membrane, too far even for Mode 2 to detect it, there will be no minimum in the RMS. This is the case in Figure 8, which shows the process of the fitting of the kernels. The true position of the membrane is indicated by the red vertical line, the current position of the tool by the black vertical line. The position of the simulated membrane (green vertical line) is moving away from the tool in panels A through F. The kernel corresponding to the simulated membrane position is shown in orange. These kernels were calculated using the background resistivity information from the start of the measurements until the current tool position at 1.2 m. For example, Figure 8A shows the line in orange that would have been measured if the membrane were 0.1 m ahead of the tool, at position 1.1 m. For each simulated position of the membrane, the difference between the kernel (orange) and the data (blue) is calculated. This difference is visualised by the blue area in the plots. The misfit is expressed as the RMS error. For a distance of 0.1 m between the tool and the simulated membrane, (Figure 8A) the misfit is large, showing as a large blue area between the orange and the blue curve.