As originally defined for the case of a metallic wire carrying an alternating current, the skin depth is the distance into the wire at which the current density is reduced to 1/e (or 37%) of its value at the surface of the wire.
The term is retained in induction logging as a qualitative indicator of how deeply the magnetic field penetrates into the formation. However, because of the different geometry, the magnetic field penetrates much deeper into the formation in terms of skin depths than for the wire geometry. For the frequency used by some induction logging sondes, one skin depth in a formation of 1 ohm-m resistivity is 140 inches. Skin depth varies in direct proportion to the square root of the resistivity. See skin effect part (1).