Analytical solutions of EM fields due to a dipolar source inside an infinite casing

Abstract

This paper studies the problem of electromagnetic fields observed outside of an infinite metallic casing due to dipolar excitations inside the pipe. Closed-form expressions are derived for the hertz vector potential driving the solution of the boundary value problem. The results indicates that the fields outside the casing are due to a distribution of vertical dipoles that decay in strength away from the true source. Analytical expressions are also obtained for the induced source distribution, as a function of the geometry and electric properties of the system. For the transverse magnetic mode (electric dipole source), the sources represent an electric current channeled vertically along the casing, whereas for the transverse electric mode (magnetic dipole source), they represent a distribution of induced magnetic moments. The strength of the current channeling decays exponentially away from the source, whereas the strength of the induced magnetic moment drops within the first few meters. The expressions obtained for the fields due to a magnetic dipole reveals that the effect of the casing yields a multiplicative complex constant, which attenuates the dipolar-type field. This effect has been observed experimentally in crosswell surveys, but it has never been formally proven. The fields due to an electric dipole excitation describe an inhomogeneous cylindrical surface wave propagating and attenuating in the radial and vertical directions.