Transient response of an induction logging tool in a borehole

The long time transient response is developed for a point coil transmitter (vertical magnetic dipole) located in a wellbore surrounded by a homogeneous formation. A simple algorithm is derived for the apparent conductivity signal based on the difference between the voltage nulls in a pair of receiver coils which are displaced vertically from the transmitter along the axis of the wellbore. A justification of these results is provided by means of a direct numerical integration (double Fourier transform) and also two separate approximate approaches.