Advances in European Geothermal Research pp 854-874 | Cite as

# Three-Dimensional Resistivity Modelling by the Integral Equation Method

## Abstract

A numerical model has been constructed to determine the electric field, electric potential, and apparent resistivity above a three-dimensional conductive inhomogeneity buried in a horizontally stratified earth with point sources of current arbitrarily situated at ground level or at depth. The problem is formulated as an integral equation; the electric potential is written as a surface integral in which the domain of integration is only the bounding surface of the inhomogeneity. The integral equation is transformed into a linear system which can be solved numerically.

The numerical modelling procedure is described and two examples of applications are given. The first deals with the possibility of detecting a buried conductive inhomogeneity simulating a magmatic chamber or a geothermal convective cell. Several parameters are considered: depth, dimension, conductivity of the inhomogeneity, conductance of the superficial layer, length and orientation of the transmitting dipole. The most important variables are found to be the depth and conductance of the upper layer. The second application attempts to quantify the influence of a conductive anomaly immediately below the ground level. With these results it has been possible to interpret an electric sounding over a marly syncline which almost completely masks the deeper structures. Several models have been formulated and compared to analog results obtained with an electrolytic tank.

## Keywords

Apparent Resistivity Induce Polarization Stratify Medium Conductive Anomaly Prismatic Body## Preview

Unable to display preview. Download preview PDF.

## References

- Arsac, J., 1961 — Transformation de Fourier et théorie des distributions. Dunod ed., Paris, 343 pp.Google Scholar
- Ooggon, J.H., 1971, Electromagnetic and electrical modelling by the finite element method. Geophysics, v. 36, p. 132–155.CrossRefGoogle Scholar
- Dieter, K. et al., 1969, IP and resistivity type curves for three-dimensional bodies. Geophysics, v. 43, p. 615–632.CrossRefGoogle Scholar
- Hohmann, G.W., 1971 — Electromagnetic scattering by conductors in the earth near a line source current. Geophysics, v. 36, p. 101–131.CrossRefGoogle Scholar
- Hohmann, G.W., 1975, Three-dimensional induced polarization and electromagnetic modeling. Geophysics, v. 40, p. 309–324.CrossRefGoogle Scholar
- Keller, G.V., and Frischknecht, F.C., 1966 — Electrical methods in geophysical prospecting. New York, Pergamon Press, 513 p.Google Scholar
- Keller, G.V., Furgerson, R., Lee, C.Y., Harthill N., and Jacobson, J.J., 1975 — The dipole mapping method. Geophysics, v. 40(3), p. 451–472.CrossRefGoogle Scholar
- Lee, T., 1975 — An integral equation and its solution for some two and three-dimensional problems in resistivity and induced polarization. Geophys. J.R. astr. Soc, v. 42, p. 81–95.CrossRefGoogle Scholar
- Morse, P.M., and Feshbach H., 1953 — Methods of theoretical physics. Mc Graw Hill Book Co. Inc. ed., New York.Google Scholar
- Parry, J.R., and Ward, S.H., 1971 — Electromagnetic scattering from cylinders of arbitrary cross-section in a conductive half-space. Geophysics, v. 36, p. 67–100.CrossRefGoogle Scholar
- Snyder, D.D., 1976, A method for modeling the resistivity and IP response of two-dimensional bodies. Geophysics, v. 41, p. 997–1015.CrossRefGoogle Scholar
- Weidelt, P., 1975, Electromagnetic intrusion in three-dimensional structures. J. Geophys. v. 41, p. 85–109.Google Scholar