  The work in this paper is motivated by the increasing use of electrical and
electromagnetic methods in geoscience problems where steel-cased wells are
present. Applications of interest include monitoring carbon capture and storage
and hydraulic fracturing operations, as well as detecting flaws or breaks in
degrading steel-casings -- such wells pose serious environmental hazards. The
general principles of electrical methods with steel-cased wells are understood,
and several authors have demonstrated that the presence of steel-cased wells
can be beneficial for detecting signal due to targets at depth. However, the
success of a DC resistivity survey lies in the details. Secondary signals might
only be a few percent of the primary signal. In designing a survey, the
geometry of the source and receivers, and whether the source is at the top of
the casing, inside of it, or beneath the casing will impact measured responses.
Also the physical properties and geometry of the background geology, target,
and casing will have a large impact on the measured data. Because of the small
values of the diagnostic signals, it is important to understand the detailed
physics of the problem and also to be able to carry out accurate simulations.
This latter task is computationally challenging because of the extreme geometry
of the wells, which extend kilometers in depth but have millimeter variations
in the radial direction, and the extreme variation in the electrical
conductivity (typically 5-7 orders of magnitude between the casing and the
background geology).
