The simulation of finite ERT electrodes using the complete electrode model

Abstract

Direct-current resistivity surveys usually are performed using steel rods of finite extent and grounding resistance. However, in modeling, electrodes are commonly treated as ideal point sources. We present an approach for numerical computation applying the complete electrode model (CEM), which is known from medical imaging. The electrode surface was discretized, and the partial-differential equations were extended by additional relations incorporating a contact impedance and a condition for the current flow through the electrode surface. We verified the modeling of the electrical potential using an analytical solution for a perfectly coupled half-ellipsoid current source. To quantify the influence of a finite electrode, we computed the electrode effect as the ratio between CEM and point-source solution and investigated its dependence on geometry and contact impedance. Surface measurements using rods of typical spatial extent showed electrode effects on the order of the measuring accuracy for an electrode length/spacing ratio lower than 0.2. However, the effects are more significant for closed geometries such as experimental tanks. A comparison with a point approximation for finite electrodes using point-source locations along the electrode axis showed the best agreement, with points at about 60% of the electrode extension. The contact impedance played a minor role for four-point measurements, contributing only a few percent to the electrode effect. In addition to penetrating electrodes, we investigated surface electrodes with galvanic or capacitive coupling, showing electrode effects on the same order as for penetrating electrodes. An inhomogeneous resistivity distribution clearly increased the size of the effects. We also investigate the use of CEM to simulate current injected through steel-cased boreholes. Finally, we applied the approach with buried ring electrodes to calculate effects caused mainly by geometric disturbances from the borehole.

Monitoring hillslope moisture dynamics with surface ERT for enhancing spatial significance of hydrometric point measurements

“…1 k over the whole measuring period. This range is very favourable and does not influence the measurements as numerical studies show (Rücker and Günther, 2011).…”

“…In the numerical computations, electrodes are considered points, which is not the case for the present ratio of length to distance. However, numerical computations with real electrode lengths show that the deviations are negligible, particularly if the points are placed at about half the electrode length (Rücker and Günther, 2011).…”

Optimized arrays for 2D cross‐borehole electrical tomography surveys

“…This ensured that the positions of the electrodes were known to within 1 mm. The finite sizes of the electrodes (about 5 mm) were accounted for by representing each electrode in the inversion model as points located at their geometric centers (Rücker and Günther 2011). In the inversion of the data set, the L-curve method (Farquharson and Oldenburg 2004) was used to estimate the optimum damping factor for each data set.…”

Effect of electrode shape on grounding resistances — Part 1: The focus-one protocol

“…However, available in the literature are analytical solutions for the potentials around a family of spheroidal electrodes: spherical (Sunde, 1949;Wait, 1982;Lile et al, 1997;Hördt et al 2013), prolate (Wait, 1973(Wait, , 1982Igel, 2007;Rücker and Günther, 2011), and oblate spheroidal electrodes (Ingeman-Nielsen and Tomaškovičová, personal communication, 2015). The prolate and oblate spheroidal models are ellipsoids with rotational symmetry around the major and minor axes, respectively, resulting in near-rod-shaped electrodes (prolate) and pill-shaped electrodes (oblate) (see Figure 1).…”

“…Calculation of the theoretical electrode grounding resistance is possible for simple electrode shapes through analytical formulations (Sunde, 1949;Wait, 1982;Ingeman-Nielsen and Tomaškovičová, personal communication, 2015) and for more complex electrode geometries by numerical modeling (Rücker and Günther, 2011). Sunde (1949) describes a method to derive the single electrode grounding resistance based on the pairwise differential resistance measurements of three electrodes, provided that the electrodes are so distantly spaced that mutual resistance effects between the electrodes may be neglected.…”