# Geo-Electrical Detection of Impermeable Membranes in the Subsurface
## 5 Numerical Simulation of the Lab Experiment  
The data collected from the laboratory experiment were used as validation of the methodology. In a laboratory environment, we cannot create a half space with an electrical insulator at depth. The tank has finite dimensions and each side acts is an electrical insulator. The boundary effects are taken into account in the simulations using COMSOL. The actual geometrical factor is calculated numerically from the simulation. This extra step will not be needed in the field situation, because of the true half-space in the field. In the laboratory case, we must calculate the geometrical factor for our specific situation, where each of the six sides of the tank is an insulator. The model dimensions are shown in Figure 4c. The right part, depicted by the white box, has the dimensions of the actual tank. In order to simulate the response of a membrane at a certain location, we consider five sides of the tank and the position of the tool at each measurement location as known. The only unknown is the 6 th side of the tank where the membrane is located (in Figure 4c). It is this side that is our target. In other words, we want to simulate the contribution of the membrane when its position is unknown, within the known dimensions of the tank. Since the numerical modelling requires a finite space, we increased the modelled length to be about twice the length of the actual one, resulting size of 15.00 m for the simulation tank. In Figure 4, the simulation tank spans the white and the blue box. The X-axis extends from \(-\)8.00 m to +7.00 m. The actual position of the membrane is located at X-location of 0.00 m in the YZ plane. The flow chart of steps for the determination of the position of the membrane is given in Figure 5. The numerical simulation is used to calculate the geometrical factor, the number that multiplied by the measurements in a homogeneous environment (i.e., a water-filled tank) will result in that homogeneous value. The tool measures resistance in Ohm. The geometrical factor is calculated for the simulation tank for all positions of the tool (and thus all electrode locations), starting from +7.00 m until 0.00 m. The different positions of the tool represent different distances between the tool and the membrane. In absence of the membrane the geometrical factor multiplied by the data (synthetic and laboratory) should always result in 21 Ohm\(\cdot\)m, which is the resistivity value for the homogenous medium in our case. The next step is the calculation of the kernels. We use the geometrical factor from the simulation tank and the simulated resistance (in Ohm) from the actual tank and simulate various theoretical positions of a membrane. Thus, the increase of resistivity observed when the tool approaches the boundary is due the presence of the left boundary of the 6 th side on the actual tank. If the measured data match the kernel for a simulated position of the membrane, then the position of the membrane has been found. If the geometrical factor was calculated based on the actual tank, instead of the simulated tank, we would end up obtaining 21 Ohm\(\cdot\)m for all apparent resistivities. Next, the same geometrical factor is applied to the laboratory data. The kernel is calculated for many theoretical positions of the membrane. If the kernel for a given position and the measured data overlap, then the position of the membrane has been found. The difference or correspondence between a kernel and the laboratory data can be defined by misfit function. In this case, we use the root-mean-square error (RMS) as a metrics for the goodness-of-fit. In preliminary analyses we also tested the area between the curves and the Frechet [14] as goodness-of-fit metrics. The RMS performed best and we therefore continued using RMS. Before each movement of the tool closer to the membrane, all new measurements need to be included in the new kernels. The fact that the kernels need to be updated with new measurements is called \"on-the-fly\".