Impedance of a horizontal coil in a borehole: a model for eddy-current bolthole probes

Abstract

Eddy-current induction by an air-cored coil located inside a long cylindrical bolthole in an isotropic conductive medium is examined using a second-order vector potential formalism. Closed-form expressions are presented for (i) the induced electromagnetic fields and currents, and (ii) the change in coil impedance, ΔZ, due to eddy-current induction in the surrounding conductor. In these expressions, the coil characteristics enter via a source function, Ds. Two alternative methods for computing Ds are described: in one, Ds is computed directly, and in the other, Ds is computed using a knowledge of the source magnetic flux density at the surface of the hole. The strengths and weaknesses of the two approaches are discussed. The non-axisymmetric case of a cylindrical coil with its axis perpendicular to the hole axis is examined in detail because of its significance in eddy-current non-destructive evaluation using bolthole probes. An explicit closed-form expression for ΔZ in terms of the coil and bolthole parameters is derived for this important geometry. The validity of the theoretical calculations was tested by comparing the theoretical results with experimental measurements of ΔZ as a function of frequency for a series of air-cored coils inside 10.0 and 6.4 mm diameter boltholes in Al alloy specimens. The theoretical and experimental results are in excellent agreement.