Title: On the convergence of Neumann series for electrostatic fracture response

Abstract

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 problemmore » 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.« less

Authors:

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

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+ Show Author Affiliations

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)

Publication Date:

Wed Nov 21 00:00:00 EST 2018

Research Org.:

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:

1487426

Report Number(s):

SAND-2018-8246J
Journal ID: ISSN 0016-8033; 666434

Grant/Contract Number:  

AC04-94AL85000

Resource Type:

Accepted Manuscript

Journal Name:

Geophysics

Additional Journal Information:

Journal Volume: 84; Journal Issue: 2; Journal ID: ISSN 0016-8033

Publisher:

Society of Exploration Geophysicists

Country of Publication:

United States

Language:

English

Subject:

58 GEOSCIENCES