Resistivity Response of a Homogeneous Earth with a Finite-Length Contained Vertical Conductor

The effect of embedded conductors such as well casings on apparent resistivity measurements on the earth's surface is of interest in geophysical exploration. A relatively simple and computationally efficient method of analysis is presented to determine the apparent resistivity error that is caused by the presence of long slender cylindrical conductors embedded vertically into the earth. The conductors may be of arbitrary length, with arbitrary longitudinal resistance and interface resistance. The potential at the well casing surface is found by modeling the casing as a set of finite-length line current sources and obtaining their values by a matrix inversion. The currents thus found are equivalent to the current that actually flows into the different sections of the well casing, and the potential at a field point due to the finite line current sources and the point current source can then be readily found and can be compared to the undisturbed field point voltage. Apparent resistivity profiles calculated for pole-pole, Schlumberger, and dipole-dipole arrays with adjacent well casings are presented.