Publications

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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.

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Wellbore integrity is of paramount importance to subsurface resource extraction, energy storage and hazardous waste disposal. We introduce a simple non-invasive technology for casing integrity screening, based on the continuity of electrical current flow. Applying low frequency current to a wellhead, with a distant return electrode, produces a casing current dependent on the properties and depth extent of the well casing as well as the background formation. These currents in-turn generate surface electrical fields that can be captured in a radial profile and be used to analyze properties of the well casing. Numerical modeling results reveal a strong relation of the electric field to the casing properties and depth extent of the well. A small breakage in the casing produces a profile coincident to a cased well with a completion depth above the break. A corroded patch, where the casing conductivity is reduced, also alters the field profiles and its depth may be estimated by comparing to the profile expected from the well completion diagrams. The electric field profiles are also strongly dependent on background resistivity distributions and on whether the well was drilled using water or oil-based drilling fluids. We validate the proof of concept in a field experiment, where we applied currents at the wellheads of two wells with different casing lengths. The two profiles were similar in appearance but offset in amplitude by more than a factor of 5, consistent with the theoretical analysis as well as the 3D modeling results. These results demonstrate that our proposed approach has promise for mapping the general casing condition without well intervention. This approach can be a practical and effective tool for rapidly screening a number of wells before expensive logging-based technologies are employed for casing inspection in detail.

To better understand the factors contributing to electromagentic (EM) observables in developed field sites, we examine in detail through finite element analysis the specific effects of casing completion design. The presense of steel casing has long been exploited for improved subsurface interrogation and there is growing interest in remote methods for assessing casing integrity accross a range of geophysical scenarios related to resource development and sequestration/storage activities. Accurate modeling of the casing response to EM stimulation is recognized as relevant, and a difficult computational challenge because of the casing's high conductivity contrast with geomaterials and its relatively small volume fraction over the field scale. We find that casing completion design can have a significant effect on the observed EM fields, especially at zero frequency. This effect appears to originate in the capacitive coupling between inner production casing and the outer surface casing. Furthermore we show that an equivalent “effective conductivity” for the combined surface/production casing system is inadequate for replicating this effect, regardless of whether the casings are grounded to one another or not. Lastly, we show that in situations where this coupling can be ignored and knowledge of casing currents is not required, simplifying the casing as a perfectly conducting line can be an effective strategy for reducing the computational burden in modeling field-scale response.

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Here, the determination of the geometrical properties of fractures plays a critical role in many engineering problems to assess the current hydrological and mechanical states of geological media and to predict their future states. However, numerical modeling of geoelectrical responses in realistic fractured media has been challenging due to the explosive computational cost imposed by the explicit discretizations of fractures at multiple length scales, which often brings about a tradeoff between computational efficiency and geologic realism. Here, we use the hierarchical finite element method to model electrostatic response of realistically complex 3D conductive fracture networks with minimal computational cost.

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Methods for the efficient representation of fracture response in geoelectric models impact an impressively broad range of problems in applied geophysics. We adopt the recently-developed hierarchical material property representation in finite element analysis (Weiss, 2017) to model the electrostatic response of a discrete set of vertical fractures in the near surface and compare these results to those from anisotropic continuum models. We also examine the power law behavior of these results and compare to continuum theory. We find that in measurement profiles from a single point source in directions both parallel and perpendicular to the fracture set, the fracture signature persists over all distances. Furthermore, the homogenization limit (distance at which the individual fracture anomalies are too small to be either measured or of interest) is not strictly a function of the geometric distribution of the fractures, but also their conductivity relative to the background. Hence, we show that the definition of “representative elementary volume”, that distance over which the statistics of the underlying heterogeneities is stationary, is incomplete as it pertains to the applicability of an equivalent continuum model. We also show that detailed interrogation of such intrinsically heterogeneous models may reveal power law behavior that appears anomalous, thus suggesting a possible mechanism to reconcile emerging theories in fractional calculus with classical electromagnetic theory.

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The recurring problem in electrical and electromagnetic modeling of anthropogenically impacted geologic settings is the need for efficient representation of strong, thin, arbitrarily oriented electrical conductors, such as metal pipes or conductive fractures. The difficulty arises from discretization with roughly equidimensional elements of the governing Maxwell equations over these volumetrically insignificant regions; which by virtue of conductors' thinness, can easily number in the 100's of millions for even simple models. To address this problem, a novel hierarchical electrical model is proposed for unstructured tetrahedral finite element meshes, where the usual volume-based conductivity in tetrahedra is augmented by facet- and edge-based conductivity on the infinitesimally thin regions between elements. This allows a slender borehole casing of arbitrary shape to be approximated by a set of connected edges within the mesh, and on which a conductivity-area product is explicitly defined. Benchmark testing of the direct current (DC) resistivity problem shows excellent agreement between the facet/edge representations and independent analytic solutions. As a practical case, the metallic infrastructure of a mature oilfield in the Kern River Formation is modeled. The oilfield comprises roughly 2 km of surface pipeline and 122 vertical, steel-cased wells, each extending to a depth of 300 m. Numerical results demonstrate strong coupling between surface and downhole conductors and reveal a complex circuit of current flow within the (finite conductivity) steel. This would be difficult to quantify using alternative, approximate methods for accommodating the approximately 30 km of steel casing and surface pipe combined.

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Coincident loop transient induction wireline logging is examined as the borehole analog of the well-known land and airborne time-domain electromagnetic (EM) method. The concept of whole-space late-time apparent resistivity is modified from the half-space version commonly used in land and airborne geophysics and applied to the coincident loop voltages produced from various formation, borehole, and invasion models. Given typical tool diameters, off-time measurements with such an instrument must be made on the order of nanoseconds to microseconds - much more rapidly than for surface methods. Departure curves of the apparent resistivity for thin beds, calculated using an algorithm developed to model the transient response of a loop in a multilayered earth, indicate that the depth of investigation scales with the bed thickness.Modeled resistivity logs are comparable in accuracy and resolution with standard frequency-domain focused induction logs. However, if measurement times are longer than a few microseconds, the thicknesses of conductors can be overestimated, whereas resistors are underestimated. Thin-bed resolution characteristics are explained by visualizing snapshots of the EM fields in the formation, where a conductor traps the electric field while two current maxima are produced in the shoulder beds surrounding a resistor. Radial profiling is studied using a concentric cylinder earth model. Results found that true formation resistivity can be determined in the presence of either oil- or water-based mud, although in the latter case, measurements must be taken several orders of magnitude later in time. The ability to determine true formation resistivity is governed by the degree that the EM field heals after being distorted by borehole fluid and invasion, a process visualized and particularly evident in the case of conductive water-based mud.

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Electromagnetic responses reflect the interaction between applied electromagnetic fields and heterogeneous geoelectrical structures. Here by quantifying the relationship between multi-scale electrical properties and the observed electromagnetic response is therefore important for meaningful geologic interpretation. Furthermore, we present here examples of near-surface electromagnetic responses whose spatial fluctuations appear on all length scales, are repeatable and fractally distributed, suggesting that the spatial fluctuations may be considered as “geologic noise”.