# Geo-Electrical Detection of Impermeable Membranes in the Subsurface
## 5 Numerical Simulation of the Lab Experiment
### Results for Sand and Clay  
The calculation of the kernel requires knowledge of the background resistivity. In the simple homogenous case of the water tank, the kernel could be calculated using the resistivity of the water. In general, however, the background resistivity of the soil is not known. The kernel needs to be built up while more measurement data become available. The \"on-the-fly\" calculation of the kernel in a heterogeneous environment is described in this section. For this experiment, the same laboratory setup as the water tank was used, but this time clay slabs and cubes of various sizes and shapes were embedded in a tank  
Figure 6: Examples of laboratory data for the water tank (red dots) and kernels calculated for the actual position of the membrane (blue line). **Left**: For Mode 1 (near-looking) with increasing tool size from top to bottom. **Right**: For mode 2 (far-looking) with two sizes of a configuration with one remote electrode and one corresponding size with two remote electrodes. The x-axis corresponds to the position of the tool, with the membrane located at x = 0 m.  
filled with sand (Figure 4, top right and Figure 7, left panels). In this case, the detection of the membrane follows a two-step approach for the box \"Kernel calculation\" in Figure 5. Step 1 consists of calculating an updated background resistivity for current position of the tool. Step 2 consists of calculating kernels (i.e., membrane positions) at various distances from the current position of the tool. Next, the flow-chart of Figure 5 is followed. The difference between the kernels and the measured data is minimized to determine the most probable distance between the tool and the membrane. The building up of the background resistivity while the experiment progresses (and the tool approaches the membrane) is shown in Figure 7. The steps which are taken to determine whether it is safe to move the tool closer to the membrane or to determine the membrane's position are illustrated with an example in Figures 8-10. Figure 7: Calculating background resistivity on-the-fly, to produce a new kernel. From top to bottom the tool approaches the membrane which is located at the bottom (y = 0). **Left**: schematic locations of clay slabs and cubes and the position of the tool. **Right**: background resistivity. A movie of the approach of the tool and the built-up of background resistivity information can be found in the [[https://youtu.be/N6v_zZT1](https://youtu.be/N6v_zZT1) pPdN]([https://youtu.be/N6v_zZT1](https://youtu.be/N6v_zZT1) pPdN) (accessed on 1 January 2022). Figure 8: Tool at a fixed position of +1.2 m distance from the membrane, represented by the black vertical line. The blue line represents the data acquired up until that position with a Mode 2 array of size 80 cm (with 1 remote electrode). (**A**–**F**) reflect tests for various simulated positions of the membrane. Each panel represents a simulated position of the membrane (green vertical line) with the corresponding kernel shown as the orange line. The red vertical line indicates the true location of the membrane at x = 0.0. The blue area visualises the misfit between the data (blue line) and kernel (orange line). (**G**) shows the misfit calculated as RMS between the measured data and the kernel with the membrane at the location of the value on the x-axis. In this case, there is no clear global minimal at 0.0 m. This means that the membrane is still far away from the tool and it is safe to push the tool deeper to the next measurement position. Figure 9: Same as Figure 8 with the tool (black vertical line) at 0.6 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 global minimum at \(-0.1\) m.  
The background resistivity and kernels are calculated \"on-the-fly\" using the measurement data that have been acquired till the current tool position.