S U M M A R YWe present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process.We use a Gauss-Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented.We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today's standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data.As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.
SUMMARY We present techniques for the efficient numerical computation of the electrical potential with finite element methods in 3‐D and arbitrary topography. The crucial innovation is, firstly, the incorporation of unstructured tetrahedral meshes, which allow for efficient local mesh refinement and most flexible description of arbitrary model geometry. Secondly, by implementation of quadratic shape functions we achieve considerably more accurate results. Exploiting a secondary potential (SP) approach, meshes are downsized significantly in comparison with highly refined meshes for total potential calculation. However, the latter is necessary for the determination of the required primary potential in arbitrary model domains. To start with, we concentrate on the simulation of homogeneous models with different geometries at the surface and subsurface to quantify their influence. This results in a so‐called geometry effect, which is not only a side effect but may be responsible for serious misinterpretations. Moreover, it represents the basis for treating heterogeneous conductivity models with the SP approach, which is especially promising for the inverse problem. We address how the resulting system of equations is solved most efficiently using modern multifrontal direct solvers in conjunction with reordering strategies or rather traditional pre‐conditioned conjugate gradient methods depending on the size of the problem. Furthermore, we present a reciprocity approach to estimate modelling errors and investigate to which degree the model discretization has to be refined to yield sufficiently accurate results.
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.
The LArge Reservoir Simulator (LARS) was developed to investigate various processes during gas hydrate formation and dissociation under simulated in situ conditions of relatively high pressure and low temperature (close to natural conditions). To monitor the spatial hydrate distribution during hydrate formation and the mobility of the free gas phase generated during hydrate dissociation, a cylindrical Electrical Resistivity Tomography (ERT) array was implemented into LARS. The ERT contains 375 electrodes, arranged in 25 circular rings featuring 15 electrodes each. The electrodes were attached to a neoprene jacket surrounding the sediment sample. Circular (2D) dipole-dipole measurements are performed which can be extended with additional 3D cross measurements to provide supplemental data. The data quality is satisfactory, with the mean standard deviation due to permanent background noise and data scattering found to be in the order of 2.12%. The measured data are processed using the inversion software tool Boundless Electrical Resistivity Tomography to solve the inverse problem. Here, we use data recorded in LARS to demonstrate the data quality, sensitivity, and spatial resolution that can be obtained with this ERT array. © 2013 AIP Publishing LLC. [http://dx
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