2-D and 3-D interpretation of electrical tomography measurements, Part 1: The forward problem

Geophysics ◽  
1999 ◽  
Vol 64 (2) ◽  
pp. 386-395 ◽  
Author(s):  
Vincent Lesur ◽  
Michel Cuer ◽  
André Straub
Keyword(s):  
Electrical Potential ◽  
Computation Time ◽  
Computer Code ◽  
Boundary Integral ◽  
Forward Problem ◽  
Electrical Tomography ◽  
Homogeneous Half Space ◽  
Electrical Potential Field ◽  
Model Geometry ◽  
A Current

We have developed a computer code to model the electrical potential field for borehole‐to‐borehole measurements. This scheme supports a large class of model geometries including 2-D and 3-D structures embedded in a homogeneous half‐space. It enables the computation of the electrical potential at any point due to a direct current injection at any source point within the model. A new boundary integral formulation is used that generates a sparse linear system. The sparsity is exploited in order to optimize the memory size and the computation time needed to solve the forward problem. This formulation is new because two unknown quantities—the electrical potential and a current‐related quantity—are solved for each interface. The numerical accuracy of this code has been extensively tested. Simulations of a simple model geometry are used to gain insight on when 3-D phenomena differ from those of 2-D models.

The application of the mise-a-la-masse electrical technique in Greenstone belt gold exploration

1989 ◽  
Vol 20 (2) ◽  
pp. 113
Author(s):  
L.G.B.T. Polomé
Keyword(s):  
Greenstone Belt ◽  
Electrical Potential ◽  
Gold Deposits ◽  
Iron Formation ◽  
Banded Iron Formation ◽  
Current Electrode ◽  
Barberton Greenstone Belt ◽  
A Current ◽  
Ore Zones ◽  
Gold Bearing

Most of the gold deposits in the Barberton Greenstone belt of South Africa are relatively small and in structurally complex geological areas.The mise-a-la-masse electrical technique, where a current electrode is earthed in a mineralised zone, was used on one of our exploration projects consisting of a sulphides/gold-bearing carbonaceous banded iron formation within a succession of mafic, ultramafic and sedimentary rocks. The technique was successful in delineating individual mineralised units within a broad lithological sequence. During the survey, electrical potential measurements were recorded on surface, in underground drives and in twenty five boreholes. Measurements were also repeated by earthing the mineralised zone in a number of boreholes. Major discontinuities were recognised within the ore zones and used to interpret geological structures. These were then used to define specific units for ore reserve calculations and the application of selected mining techniques.

Modeling and stabilization of current-controlled piezo-electric beams with dynamic electromagnetic field

2020 ◽  
Vol 26 ◽  
pp. 8
Author(s):  
Ahmet Özkan Özer ◽  
Kirsten A. Morris
Keyword(s):  
Electromagnetic Field ◽  
Piezoelectric Materials ◽  
Electrical Potential ◽  
Gauge Condition ◽  
Current Control ◽  
Magnetic Vector Potential ◽  
Bounded Control ◽  
The Cauchy Problem ◽  
A Current ◽  
Piezo Electric

Piezoelectric materials can be controlled with current (or charge) as the electrical input, instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. The magnetic permeability in piezo-electric materials is generally neglected in models. However, it has a significant qualitative effect on properties of the control system such as stabilizability. Besides the consideration of current control, there are several new aspects to the model. Most importantly, a fully dynamic magnetic model is included. Also, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Hamilton’s principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a current (or charge) source at the electrodes. Two sets of decoupled system of partial differential equations are obtained; one for stretching of the beam and another one for bending motion. Since current (or charge) controller only affects the stretching motion, attention is focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb type gauge condition is used. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike voltage actuation, a bounded control operator in the natural energy space is obtained. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models.

The face‐current concept and its application to survey design in electrical exploration

Geophysics ◽  
2003 ◽  
Vol 68 (3) ◽  
pp. 900-910 ◽  
Author(s):  
Carlos A. Mendonça
Keyword(s):  
Survey Design ◽  
Electrical Potential ◽  
Ground Surface ◽  
Electrode Array ◽  
Homogeneous Half Space ◽  
The Face ◽  
Geoelectrical Structure ◽  
Dipole Sources ◽  
Conductivity Contrast

This paper presents a new method to identify the regions over a 3D geoelectrical structure that produce major contributions to the electrical potential established in response to a dc source at the ground surface. The measured potential is represented by a sum of a known primary potential (due to a homogeneous half space) plus an unknown potential caused by conductivity inhomogeneities. Because the primary potential is continuous everywhere, the interfaces with a conductivity contrast act as sources or sinks of currents in order to maintain the continuity of the current density related to the primary flux. These disturbing face currents are responsible for the generation of the secondary potential, and mapping them over a given structure allows us to assess the regions where the secondary potential is generated. In general, the face currents vanish away from the source according to the decay of the primary electric field. For this reason, deeper investigations can be expected when using pole sources because its primary field decays with the inverse of the squared distance, instead of the cubed distance as for dipole sources. For thin sheets, the polarization decay with distance is one order higher than that for large 3D bodies, which makes the detection of a sheet yet more difficult. The quantification of the total face current over the structure for different positions along a profile helps one choose the proper electrode array and determine its optimum length. This is done in two steps: (1) identification of the offset where the dc source provides the highest polarization (face current) on the targeted structure, and (2) determination of the array length by locating the potential electrodes closest to the region with the highest polarization. This second criterion came from an analogy between the face‐current and artificial current sources, where it is intuitively seen that the resulting potential is highest close to the source. The proposed survey design technique is applied to three models commonly used in electrical exploration: a shallow conductive heterogeneity, a buried contact, and a thin conductive sheet.

An alternative way in electrical resistivity prospection: the quasi-null arrays

2019 ◽  
Vol 220 (3) ◽  
pp. 1463-1480
Author(s):  
S Szalai ◽  
K Szokoli ◽  
E Prácser ◽  
M Metwaly ◽  
M Zubair ◽  
...  
Keyword(s):  
Potential Distribution ◽  
Field Experiments ◽  
Electrical Potential ◽  
Field Tests ◽  
Horizontal Resolution ◽  
Homogeneous Half Space ◽  
Joint Interpretation ◽  
Robust Image ◽  
Imaging Properties ◽  
Better Than

SUMMARY While traditional geoelectric array configurations, such as the Wenner–Schlumberger or the dipole–dipole, can provide very good images of 1-D or robust 2-D structures, they are not sufficiently sensitive to those inhomogeneities that have a small effect on the surface electrical potential distribution. The detection and description of such inhomogeneities become possible by applying quasi-null arrays, which provide very small (close to zero) signals above a homogeneous half-space. The imaging properties of the members of an array series containing such arrays, the so-called γ11n arrays (n = 1–7), are demonstrated and compared to those of the most popular traditional arrays. Although the field applicability of the quasi-null arrays has been heavily questioned, it was demonstrated by our quasi-field analogue modelling experiments. The quasi-field tests also validated all of the numerical modelling results as follows: (1) many or all of the γ11n arrays were able to detect prisms and vertical sheets located at depths larger than those detectable by traditional geoelectric arrays, including the optimized Stummer configuration; (2) the horizontal resolution of the γ11n arrays proved to be better than the horizontal resolution of traditional arrays; (3) with n increasing, the γ11n arrays proved to be less sensitive to 1-D, but more sensitive to 2-D bodies. In case of high n values, the γ11n arrays may even be entirely insensitive to any 1-D structure. On the basis of the quasi-field experiments, γ11n arrays are expected to be very efficient to indicate bodies, or variations in time that only have a small impact on the surface electrical potential distribution (e.g. caves, mines, tunnels, tubes, cables, fractures, dykes), or small changes in the subsurface conditions (monitoring of dams or waste deposits). Data acquisition by both a traditional and a γ11n array, individual inversion of their data, and a joint interpretation of the results are recommended to obtain both a robust image and fine details of the subsurface.

Structure of the Séchilienne unstable slope from large-scale three-dimensional electrical tomography using a Resistivity Distributed Automated System (R-DAS)

2019 ◽  
Vol 219 (1) ◽  
pp. 129-147 ◽  
Author(s):  
M Lajaunie ◽  
J Gance ◽  
P Nevers ◽  
J-P Malet ◽  
C Bertrand ◽  
...  
Keyword(s):  
Large Scale ◽  
Electrical Potential ◽  
Three Dimensional ◽  
Automated System ◽  
Data Uncertainty ◽  
Complex Data ◽  
Electrical Tomography ◽  
Data Set ◽  
Resistivity Model ◽  
Unstable Slope

SUMMARY This work presents a 3-D resistivity model of the Séchilienne unstable slope acquired with a network of portable resistivimeters in summer 2017. The instrumentation consisted in distributed measuring systems (IRIS Instruments FullWaver) to measure the spatial variations of electrical potential. 23 V-FullWaver receivers with two 50 m dipoles have been deployed over an area of circa 2 km2; the current was injected between a fixed remote electrode and a mobile electrode grounded successively at 30 locations. The data uncertainty has been evaluated in relation to the accuracy of electrodes positioning. The software package BERT (Boundless Electrical Resistivity Tomography) is used to invert the apparent resistivity and model the complex data set providing the first 3-D resistivity model of the slope. Stability tests and synthetic tests are realized to assess the interpretability of the inverted models. The 3-D resistivity model is interpreted up to a depth of 500 m; it allows identifying resistive and conductive anomalies related to the main geological and hydrogeological structures shaping the slope. The high fracturation of the rock in the most active zone of the landslide appears as a resistive anomaly where the highest resistivity values are located close to the faults. A major drain formed by a fault in the unaltered micaschist is identified through the discharge of a perched aquifer along the conductive zone producing an important conductive anomaly contrasting with the unaltered micaschist.

Regularization of the Boundary Integrals in the Bem Analysis of 3D Potential Problems

2012 ◽  
Vol 29 (3) ◽  
pp. 385-401 ◽  
Author(s):  
Y. C. Shiah ◽  
M. R. Hematiyan ◽  
Y. H. Chen
Keyword(s):  
Boundary Element ◽  
Local Coordinate System ◽  
Computer Code ◽  
Boundary Integral ◽  
Boundary Integrals ◽  
Element Analysis ◽  
Numerical Tests ◽  
Potential Problems ◽  
Boundary Element Analysis ◽  
Proper Values

AbstractIn the conventional boundary element analysis, near-singularities are present in the associated boundary integral equation for problems involving ultra-thin media. For this case, any conventional numerical schemes will fail to yield proper values for the integrals. In this paper, the boundary integrals of the boundary element method for 3D potential problems are fully regularized by the technique of integration by parts under the local coordinate system. The fully regularized integrands are expressed as very explicit formulations that can be easily programmed into a computer code. Numerical tests carried out for a typical case have verified the accuracy of the approach for any orders of small distance between the source and the element under integration.

3-D resistivity inversion using the finite‐element method

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1839-1848 ◽  
Author(s):  
Yutaka Sasaki
Keyword(s):  
Finite Element ◽  
Approximate Method ◽  
Synthetic Data ◽  
Computation Time ◽  
Inversion Method ◽  
Data Sets ◽  
Near Surface ◽  
Partial Derivatives ◽  
Homogeneous Half Space ◽  
Resistivity Inversion

With the increased availability of faster computers, it is now practical to employ numerical modeling techniques to invert resistivity data for 3-D structure. Full and approximate 3-D inversion methods using the finite‐element solution for the forward problem have been developed. Both methods use reciprocity for efficient evaluations of the partial derivatives of apparent resistivity with respect to model resistivities. In the approximate method, the partial derivatives are approximated by those for a homogeneous half‐space, and thus the computation time and memory requirement are further reduced. The methods are applied to synthetic data sets from 3-D models to illustrate their effectiveness. They give a good approximation of the actual 3-D structure after several iterations in practical situations where the effects of model inadequacy and topography exist. Comparisons of numerical examples show that the full inversion method gives a better resolution, particularly for the near‐surface features, than does the approximate method. Since the full derivatives are more sensitive to local features of resistivity variations than are the approximate derivatives, the resolution of the full method may be further improved when the finite‐element solutions are performed more accurately and more efficiently.

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