Resistivity Response of a Homogeneous Earth with a Contained Vertical Conductor

Using a simple model, we illustrate the influence of a cylindrical conductor on the apparent resistivity for a two-electrode array on a otherwise homogeneous earth. In the formulation, the cylindrical conductor is assumed to be of infinite length. However, by exploiting a simple image plane concept, the solution is applicable to the case where the electrodes are on the surface of a homogeneous half-space model of the earth and the cylindrical conductor extends from the surface vertically downward to infinity. Thus the configuration could correspond, in an ideal sense, to a vertical metal well casing or a similar conductive structure that outcrops to the surface. The calculations indicate the apparent resistivity can be seriously modified by the presence of the cylinder even when the electrodes are located at a distance more than tenfold the cylinder radius. Also, the formulation allows for the presence of an interface impedance at the metal surface of the cylinder. This could be used as a model to examine the induced polarization response.