The problem of electrical sounding of a medium with ground surface relief is modelled using the integral equations method. This numerical method is based on the triangulation of the computational domain, which is adapted to the shape of the relief and the measuring line. The numerical algorithm is tested by comparing the results with the known solution for horizontally layered media with two layers. Calculations are also performed to verify the fulfilment of the “reciprocity principle” for the 4-electrode installations in our numerical model. Simulations are then performed for a two-layered medium with a surface relief. The quantitative influences of the relief, the resistivity ratios of the contacting media, and the depth of the second layer on the apparent resistivity curves are established.
The paper discusses a method for solving an integral equation for calculating a three-dimensional electric field in a medium with a two-dimensional geometry based on the Fourier transform. The results of the numerical solution of the transformed integral equation and the original integral equation for the medium with the surface relief are compared. The original equation was solved using parallelization technologies on a system with shared memory. A significant performance improvement based on the transformed equations, including in comparison with the parallel version of the program for the original integral equation, is shown.
The paper considers a mathematical model of electrical tomography above the media with local inclusions. Numerical solutions of a system of integral equations for a medium with local inclusion are compared against a numerical implementation of the analytical solution of the problem for a case of a sphere in homogeneous space. The parameters of local inclusion and the depth of heterogeneity are varied. Maps of secondary sources in the ERT (Electrical Resistivity Tomography) probing problem are constructed: for local inclusion in the form of the ellipsoid, an ellipsoid in a homogeneous space (analytical solution of the problem) and for two-layer half-spaces as well. Numerical results are presented, and maps of secondary sources in the cases where the immersed heterogeneity is an insulator and a conductor are computed.
Numerical modeling of the problem of dam monitoring by the Electrical Resistivity Tomography method is carried out. The mathematical model is based on integral equations with a partial Fourier transform with respect to one spatial variable. It is assumed that the measurement line is located across the dam longitude. To approximate the shape of the dam surface, the Radial Basic Functions method is applied. The influence of locations of the water-dam, dam-basement, basement-leakage boundaries with respect to the sounding installation, which is partially placed under the headwater, is studied. Numerical modeling is carried out for the following varied parameters: 1) water level at the headwater; 2) the height of the leak; 3) the depth of the leak; 4) position of the supply electrode; 5) water level and leaks positions are changing simultaneously. Modeling results are presented in the form of apparent resistivity curves, as it is customary in geophysical practice.
Design of electrical monitoring of dams and barriersis an actual task in geophysics. A primary purpose is an exposure of change of structure, erosion, cracks and losses of weir on the early stages. Then it is important to remove and repair a weir and prevent destructions of dike overall. For mathematical modeling of electrical monitoring of dams and barriers, the authors consider the method of ERT. The paper shows a mathematical model of the electrical survey of dams and barriers based on the method of integral equations and the Fourier transform. Numerical calculations for this model are performed. The simulation results for studying the influence of the location of the water-dam boundary with respect to the sounding array are presented. For the purposes of mathematical modeling, two extreme cases were considered: a) a fluid is assumed to be infinitely conductive, b) a fluid is not conductive, i.e. distilled. The effect of a change in the position of the supply electrode at a fixed water level was also studied. The simulation results are presented in the form of apparent resistivity curves, as it is customary in geophysical practice. Distribution of density of secondary charges is also shown for the cases of infinitely conducting and distilledwater.
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