Electromagnetic Field Due to a Magnetic Dipole in a Medium Containing Both Planar and Cylindrical Boundaries

An approximate method based on perturbation theory is used to calculate electromagnetic fields due to a vertical magnetic dipole in an Earth formation having both horizontal and cylindrical boundaries. The medium is first assumed to have a constant background permittivity. Any deviation from the background permittivity is treated as a perturbation. Correction terms are then found and expressed in multiple integrals. The first and the second correction terms involve threefold and sixfold integrations, respectively. An efficient integration technique is used to compute these multiple integrals. As an example, this method is used to calculate the field due to a vertical magnetic dipole over a layered Earth containing a cylindrical cavity. In another example, the response of a two-coil induction tool is calculated when it is located in a rugose borehole.