# 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 continuousmore »

- Authors:

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

- Publication Date:

- 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

666434

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Geophysics

- Additional Journal Information:
- Journal Name: Geophysics

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 58 GEOSCIENCES

### Citation Formats

```
Weiss, Chester Joseph, and van Bloemen Waanders, Bart G.
```*On the convergence of Neumann series for electrostatic fracture response*. United States: N. p., 2018.
Web. doi:10.1190/geo2018-0564.1.

```
Weiss, Chester Joseph, & van Bloemen Waanders, Bart G.
```*On the convergence of Neumann series for electrostatic fracture response*. United States. doi:10.1190/geo2018-0564.1.

```
Weiss, Chester Joseph, and van Bloemen Waanders, Bart G. Wed .
"On the convergence of Neumann series for electrostatic fracture response". United States. doi:10.1190/geo2018-0564.1.
```

```
@article{osti_1487426,
```

title = {On the convergence of Neumann series for electrostatic fracture response},

author = {Weiss, Chester Joseph and van Bloemen Waanders, Bart G.},

abstractNote = {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.},

doi = {10.1190/geo2018-0564.1},

journal = {Geophysics},

number = ,

volume = ,

place = {United States},

year = {2018},

month = {11}

}