Applying the transmission line theory to study ungrounded horizontal loop self-transients

Ungrounded horizontal loop responses at low frequencies and/or late times can be modeled in terms of an equivalent circuit with lumped elements, but a loop in a general case is a distributed system. At high frequencies and/or early times, the wire in combination with the underlying earth makes a transmission line in which current behaves according to the wave equation. Solving the equation for current turn-off is quite difficult because the primary parameters of the wire–earth system depend, in an intricate way, on earth conductivity (resistivity) and frequency (or time). In modeling the current turn-off process, the loop was simulated as a symmetrical combination of two identical transmission lines with shorted outputs. Modeling was performed in the frequency domain with subsequent transformation into the time domain. Comparison of measured and computed transient self responses showed that good fit requires taking into account (1) interaction of each line with its own image current, (2) mutual inductance of the two lines, and (3) skin effect in the wire. As a result of mutual inductance, the parameters of the lines and, hence, of the whole loop depend on local conductivity, which, at least in principle, may allow one to infer the resistivity of shallow subsurface from current turn-off responses. A real ungrounded horizontal loops lacks symmetry at early times and its magnetic field differs from that predicted by the conventional methods of induction soundings.