Summary

Under certain restricting assumptions, an electrically energized steel-cased geologic borehole may be modeled as an electrical transmission line. Current waveforms are obtained by solving the governing telegraph equation in the frequency-domain, followed by numerical inverse Fourier transformation. Electric current pulses propagating along the borehole undergo progressive amplitude loss and waveform distortion with distance, arising from leakage of current into the surrounding geology. A major modeling uncertainty involves the proper boundary condition to impose at the end of a borehole transmission line.

Introduction

There appears to be increasing interest in utilizing a steelcased geologic borehole as a giant source electrode for electromagnetic (EM) exploration purposes. The casing, energized either at/near the wellhead or at a deep downhole point, provides a high conductivity pathway for electric current to enter a large subsurface volume. Moreover, high amplitude current can be delivered close to a geophysical feature of interest near the borehole, such as a hydraulic fracturing stage. The present situation motivates further examination of the current-carrying characteristics of a steel-cased borehole.

Several investigators (Kaufmann, 1990; Kaufmann and Wightman, 1993; Schenkel and Morrison, 1994; Wait, 1994, 1995; Bartel, 2014) have studied the problem of electric current flow in a coaxial cylindrical geometry consisting of fluid, casing, and cement, and surrounded by layered geology. The primary focus has been on logging formation resistivity through casing, and a direct current (DC) assumption is often adopted. In this study, we use the electric transmission line analogy originally developed by Kaufman (1990) to calculate full bandwidth time-domain current waveforms along a cased borehole surrounded by a homogeneous geology.

More than 80 years ago, Schelkunoff (1934) provided a derivation of the (frequency-domain) transmission line equations directly from Maxwell’s equations expressed in cylindrical coordinates. A major assumption involved circular symmetry of both medium parameters and sources, and hence of the generated EM wavefields. Thus, the transmission line representation of a current-carrying borehole applies strictly to a vertical well penetrating horizontal layered geology. It does not apply to a well tracking horizontally within a deep geologic formation.

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