We investigate a novel application of Fŕechet derivatives for time-lapse mapping of deep, electrically-enhanced fracture systems with a borehole to surface DC resistivity array. The simulations are evaluated for a cased horizontal wellbore embedded in a homogeneous halfspace, where measurements are evaluated near, mid-range, and far from the well head. We show that, in all cases, measurements are sensitive to perturbations centered on the borehole axis and that the sensitivity volume decreases as a function of increased measurement offset from the well head. The sensitivity analysis also illustrates that careful consideration must be taken when developing an electrical survey design for these scenarios. Specifically, we show that positive perturbations in earth conductivity near the wellbore can manifest as both positive and negative measurement perturbations, depending on where the measurement is taken. Furthermore, we show that the transition between the regions along the wellbore of positive and negative contribution results in a "pinch point", representing a region along the wellbore where a given surface measurement is blind to any changes or enhancement of electrical conductivity.
We investigate through numerical simulation the usefulness of DC resistivity data for characterizing subsurface fractures with elevated electrical conductivity by considering a geophysical experiment consisting of a grounded current source deployed in a steel cased borehole. In doing so, the borehole casing behaves electrically as a spatially extended line source, efficiently energizing the fractures with a steady current. Finite element simulations of this experiment for a horizontal well intersecting a small set of vertical fractures indicate that the fractures manifest electrically in (at least) two ways: a local perturbation in the electric potential proximal the fracture set, with limited far-field expression; and, an overall reduction in the electric potential along the entire length of borehole casing due to enhanced current flow through the fractures into the surrounding formation. The change in casing potential results in a measureable effect that can be observed far from fractures themselves, at distances where the local perturbations in the electric potential around the fractures are imperceptible. Under these conditions, our results suggest that far-field, time-lapse measurements of DC potentials surrounding a borehole casing can be reasonably interpreted by simple, linear inversion for a Coulomb charge distribution along the borehole path, including a local charge perturbation due to the fractures. Such an approach offers an inexpensive method for detecting and monitoring the time-evolution of electrically conducting fractures while ultimately providing an estimate of their effective conductivity - the latter providing an important measure independent of seismic methods on fracture shape, size, and hydraulic connectivity.