Complete Analysis of Electrode-Terminated Cables in Two-Layer Conductor BoundedAbove by Air: Current Density, Magnetic Field, and Magnetic Field Gradient

摘要

A complete analysis is given for the current density, magnetic field, andmagnetic field gradient of a point current source located in the upper layer of a two-layer conductor that is bounded above by air. This problem has a long history, and treatments varying degrees of completeness and/or rigor are scattered throughout the older geophysical literature. The intent here is to do a complete, self-contained analysis based on potential theory and Ampere's Law; treat all possible cases for the relative conductivity media, including the special cases of a lower medium that is an insulator or a perfect conductor; a nd provide expressions the magnetic field vector, the current density vector, and the magnetic field gradient tensor in both conducting layers. The effect of performing gradient measurements within an enclosure also is discussed. The principle new results are the inclusion of expressions for the current density and magnetic field in the lower conducting layer, expressions for the magnetic field gradient tensor in both conducting layers, and the analysis of the case of a perfectly conducting lower layer. To complete the set of tools needed to describe a cable terminated by point electrodes, compact general expressions are given for the magnetic field vector and magnetic gradient tensor for a straight current element with arbitrary end points. These are based on the Biot-Savart Law and can be used to calculate the field and gradient contribution from an arbitrary cable joining two or more electrodes by means of superposition. The end results are applicable to such diverse problems as long-wire, electro-prospecting the current distribution around sheet piling protected by active electrodes, and the distributed currents of grounded direct current power systems. (AN).