On the convergence of Neumann series for electrostatic fracture response

Journal Article · Wed Nov 21 00:00:00 EST 2018 · Geophysics

DOI:https://doi.org/10.1190/geo2018-0564.1· OSTI ID:1487426

Weiss, Chester Joseph ^[1]; van Bloemen Waanders, Bart G. ^[2]

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1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of New Mexico, Albuquerque, NM (United States)
2. Sandia National Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Here, 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. While generally useful for efficient modeling of mild conductivity perturbations in uncluttered settings, we raise the question of its suitability for situations, such as oilfield, where 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 h 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 demonstrate that the spectral radius of the Neumann series operator grows as ~ 1/h, thus suggesting that in the limit of the continuous problem h → 0, the Neumann series is intrinsically divergent for all conductivity perturbation, regardless of their smallness. The hierarchical finite element methodology itself is critically analyzed and shown to possess the h² 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. Here, the methods used here are demonstrably useful in such circumstances.

• Research Organization: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC04-94AL85000
• OSTI ID: 1487426
• Report Number(s): SAND-2018-8246J; 666434
• Journal Information: Geophysics, Vol. 84, Issue 2; ISSN 0016-8033
• Publisher: Society of Exploration Geophysicists
• Country of Publication: United States
• Language: English

Related Subjects

58 GEOSCIENCES