On the convergence of the Neumann series for electrostatic fracture response

Abstract

The feasibility of Neumann-series expansion of Maxwell’s equations in the electrostatic limit is investigated for potentially rapid and approximate subsurface imaging of geologic features proximal to metallic infrastructure in an oilfield environment. Although generally useful for efficient modeling of mild conductivity perturbations in uncluttered settings, we have raised the question of its suitability for situations such as oilfields, in which metallic artifacts are pervasive and, in some cases, in direct electrical contact with the conductivity perturbation on which the Neumann series is computed. Convergence of the Neumann series and its residual error are computed using the hierarchical finite-element framework for a canonical oilfield model consisting of an L-shaped, steel-cased well, energized by a steady-state electrode, and penetrating a small set of mildly conducting fractures near the heel of the well. For a given node spacing [Formula: see text] in the finite-element mesh, we find that the Neumann series is ultimately convergent if the conductivity is small enough — a result consistent with previous presumptions on the necessity of small conductivity perturbations. However, we also determine that the spectral radius of the Neumann series operator grows as approximately [Formula: see text], thus suggesting that in the limit of the continuous problem [Formula: see text], the Neumann series is intrinsically divergent for all conductivity perturbations, regardless of their smallness. The hierarchical finite-element methodology itself is critically analyzed and shown to possess the [Formula: see text] error convergence of traditional linear finite elements, thereby supporting the conclusion of an inescapably divergent Neumann series for this benchmark example. Application of the Neumann series to oilfield problems with metallic clutter should therefore be done with careful consideration to the coupling between infrastructure and geology. The methods used here are demonstrably useful in such circumstances.