Numerical modeling of electromagnetic coupling effects for phase correction in borehole EIT measurements

Abstract

Spectral Electrical Impedance Tomography (EIT) allows obtaining images of the complex electrical conductivity for a broad frequency range (mHz to kHz). It has recently received increased interest in the field of near-surface geophysics and hydrogeophysics because of the relationships between complex electrical properties and hydrogeological and biogeochemical properties and processes observed in the laboratory with Spectral Induced Polarization (SIP). However, these laboratory results have also indicated that a high measurement accuracy is required for the phase angle of the complex electrical conductivity because many soils and sediments are only weakly polarizable and show small phase angles between 1 and 20 mrad only. It is challenging to achieve this phase accuracy in a broad frequency range for EIT measurements in the field. In the case of borehole EIT measurements, electrode chains with a length of 10 meters or more are typically used. This may lead to undesired inductive coupling between the electric wires used for current injection and potential measurement and capacitive coupling between the electrically conductive cable shielding and the soil. Depending on the measured transfer impedances, both coupling effects can cause large phase errors that have typically limited the frequency bandwidth of field EIT measurements to the mHz to Hz range. So far, potentially useful information from the high frequency range (up to kHz) could not be reliably measured in the field using EIT.<br /> Within this context, the aim of this PhD thesis was to develop correction procedures for inductive and capacitive coupling effects in borehole EIT measurements to enable more accurate phase measurements in the kHz frequency range. Throughout the thesis, an enhanced field EIT measurement system with 40 channels was used. In addition, custom-made electrode chains with eight electrode modules with a spacing of 1 m and a cable length of 25 m were initially used to develop the correction methods. Each electrode module of the electrode chain was equipped with a brass ring electrode, an integrated amplifier for potential measurement, and an integrated switch for current injection. In general, borehole EIT measurements can be divided into two different cases according to the electrode arrangement. In the first case, EIT measurements are performed in a single borehole. Here, a correction procedure was developed to determine the inductive coupling between the wires (i.e. the mutual inductance) in a single electrode chain using a calibration measurement. In this calibration measurement, all electrodes of the chain were short-circuited and the shield was used as the return line, so that the pure mutual inductance between the wire pairs could be measured indirectly. In the second case, EIT measurements are performed in two boreholes. Here, a correction procedure was developed that combines calibration measurements to determine the mutual inductance between wires close to each other (i.e. inside one electrode chain) with model-based estimates of the mutual inductances between wires far from each other (i.e. in two different electrode chains). In addition, a pole-pole matrix formulation was developed to efficiently describe mutual inductances inside and between electrode chains. To separate parasitic inductances associated with the grounding wire and the wire used for short-circuiting the electrodes in the calibration measurement from the mutual inductance associated with the electrode chains, a second calibration measurement with both electrode chains was used. This second calibration was required because these parasitic inductances cannot be compensated in the calculation of the mutual inductance when two electrode chains are used. A correction procedure was also developed to remove the effects of capacitive coupling. Since a priori correction of borehole EIT measurements as in the case of inductive coupling was not possible, this correction procedure relies on the integration of discrete capacitances in the electrical forward model describing the borehole EIT measurements.<br /> The developed correction methods for inductive and capacitive coupling were successfully verified with test measurements under controlled conditions. For EIT measurements with a single electrode chain, a phase accuracy of better than 1 mrad was achieved for frequencies up to 10 kHz. In the case of EIT measurements with two electrode chains, a phase accuracy of 1 mrad was achieved up to 1 kHz. A field evaluation of borehole EIT measurements at the Krauthausen test site also showed a considerable improvement of the phase accuracy by applying the correction methods. Here, it was observed that inductive coupling had a much stronger effect on the phase measurement than capacitive coupling. The complex electrical resistivity determined from 1D inversions of the borehole EIT measurements matched well with the general stratigraphy of the test site. <br /> The correction procedures outlined above were developed specifically for the custom-made EIT system and electrode chains. In a final step, the applicability of the correction procedures to commercially available electrode chains was explored. In particular, the possibility of performing accurate borehole EIT measurements using passive unshielded and shielded electrode chains was evaluated. In addition to the inductive and capacitive coupling, the use of this type of electrode chains requires the consideration of the capacitive load of the cables. It was shown that the phase errors due to the internal structure of the passive shielded electrode chains can be estimated using electrical circuit simulation. It was found that the phase error of the passive shielded electrode chains was about 1.5 mrad bigger as for the custom-made active electrode chains due to different capacitances between the electrode chains and the soil (in a conductive environment). Therefore, borehole EIT measurements with passive shielded electrode chains resulted in a reasonable phase accuracy of 3 mrad at 1 kHz. Finally, it was also confirmed that unshielded electrode chains are not suitable for accurate phase measurements at high frequencies because the induced phase errors cannot be predicted.<br /> Although the results presented in this thesis are promising, there still is room to further improve the correction procedures. In this study, the electromagnetic response of the underground was neglected because of the small size of the electrode layouts in the field measurements and the used frequencies (up to 1 kHz). If a further expansion of the frequency range is required or when longer electrode arrays will be used, this effect will also need to be considered. The modeling of capacitive effects can also be improved. In particular, the spatial resolution of the meshes used for finite element modeling was limited by available computational resources, which could have caused a certain degree of inaccuracy in the modeling. Furthermore, the calibration procedure to determine the additional parasitic inductance of the short-circuiting wires by using a calibration measurement on two nearly identical electrode chains was complex and should be simplified. <br /> Overall, it can be concluded that the developed correction methods for borehole EIT measurements resulted in a high phase accuracy (about 1 mrad at 1 kHz) when the custom-made EIT measurement system was used with active electrode chains. It was also shown that the correction methods to account for inductive and capacitive coupling can be applied to passive shielded electrode chains with a significant improvement in accuracy (about 3 mrad at 1 KHz). Therefore, this work opens up new research avenues where broadband EIT measurements can be used for improved characterization of hydrogeological and biogeochemical properties and processes in the field.