A resurgence of interest in the problem of electrical or electromagnetic scattering from thin conductors, either on a line or in a plane, has been motivated in recent years by time-lapse fracture monitoring in the near surface, enhanced geothermal reservoirs, and unconventional hydrocarbon plays. However, finite-element modeling of small electrical features in large computational domains results in a disproportionately large number of elements concentrated in a volumetrically insignificant fraction of the mesh, and it focuses computational resources away from areas of interest elsewhere, such as receiver locations. We have developed a novel hierarchical electrical model is proposed for unstructured tetrahedral finite-element meshes, in which the usual volume-based conductivity on tetrahedra is augmented by facet- and edge-based conductivity on the infinitesimally thin regions between elements. Doing so allows a slender borehole casing of arbitrary shape to be coarsely approximated by set of connected edges on which a conductivity-area product is explicitly defined. Similarly, conductive fractures are approximated by a small number of connected facets that, together, may warp and bend with the mesh topology at no added cost of localized mesh refinement. Benchmarking tests of the direct current (DC) resistivity problem indicate excellent agreement between the facet/edge representations and independent analytic solutions. Consistency tests are also favorable between the facet/edge and volume representations. Building on prior work in DC modeling of a single horizontal well, a multilateral well casing and fracture set is simulated, yielding estimates of borehole casing voltage and surface electric fields measurable with existing sensor technology. Finally, the implications on broadband electromagnetic simulation with the proposed hierarchical conductivity model are discussed — in particular, its utility for describing variations in the magnetic permeability and electrical conductivity.

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