Solving 3d problems of potential theory in piecewise homogeneous media by using indirect boundary and near-boundary element methods

Effective numerical-analytical approaches for solving the direct problem of electrical prospecting for the media with inclusions of arbitrary shape and constant electrical characteristics are suggested. They are based on the combination of a fundamental solution of Laplase’s equation and principal ideas of the method of boundary integral equations and that of collocation.

Using the indirect boundary and near-boundary element methods, numerical-analytical approaches for solving the problems of potential theory in spatial piecewise homogeneous objects under conditions of an ideal contact between their components are developed. Discrete-continuous models for finding the intensities of unknown sources introduced into the boundary and near-boundary elements, and approximated by constants are reduced to the systems of linear algebraic equations resulted from the satisfaction, in a collocation sense, of the boundary conditions and those of an ideal interface contact.

The software implementation of the approaches proposed in a half-space with inclusions of various shapes and electrical conductivity for the electrical profiling method in a 3D problem of dc electrical prospecting is done. The numerical analysis performed for some mathematical models illustrates high accuracy and potential abilities of the methods suggested. The developed algorithms make it possible to calculate the potential and intensity of an electric field in inhomogeneous media which are characterized by nonplanar boundaries and arbitrary, by depth and lateral distribution, stationary current sources.

An influence of conductivity and depths of inclusions, their shapes, a distance between two spherical inclusions on the apparent resistivity calculated by the difference of potentials measured on a half-space surface is investigated. It is shown that information on a potential field obtained on the surface of the object can be used to identify local foreign inclusions.

The proposed approaches could be the basis for solving inverse problems of geophysics and technical diagnostics in developing methods for the identification of foreign inclusions, voids and defects, and determining their conductivity, dimentions, and location.