Publications

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Fractured media models comprise discontinuities of multiple lengths (e.g. fracture lengths and apertures, wellbore area) that fall into the relatively insignificant length scales spanning millimeter-scale fractures to centimeter-scale wellbores in comparison to the extensions of the field of interest, and challenge the conventional discretization methods imposing highly-fine meshing and formidably large numerical cost. By utilizing the recent developments in the finite element analysis of electromagnetics that allow to represent material properties on a hierarchical geometry, this project develops computational capabilities to model fluid flow, heat conduction, transport and induced polarization in large-scale geologic environments that possess geometrically-complex fractures and man-made infrastructures without explosive computational cost. The computational efficiency and robustness of this multi-physics modeling tool are demonstrated by considering various highly-realistic complex geologic environments that are common in many energy and national security related engineering problems.

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Electromagnetic (EM) methods are among the original techniques for subsurface characterization in exploration geophysics because of their particular sensitivity to the earth electrical conductivity, a physical property of rocks distinct yet complementary to density, magnetization, and strength. However, this unique ability also makes them sensitive to metallic artifacts - infrastructure such as pipes, cables, and other forms of cultural clutter - the EM footprint of which often far exceeds their diminutive stature when compared to that of bulk rock itself. In the hunt for buried treasure or unexploded ordnance, this is an advantage; in the long-term monitoring of mature oil fields after decades of production, it is quite troublesome indeed. Here we consider the latter through the lens of an evolving energy industry landscape in which the traditional methods of EM characterization for the exploration geophysicist are applied toward emergent problems in well-casing integrity, carbon capture and storage, and overall situational awareness in the oil field. We introduce case studies from these exemplars, showing how signals from metallic artifacts can dominate those from the target itself and impose significant burdens on the requisite simulation complexity. We also show how recent advances in numerical methods mitigate the computational explosivity of infrastructure modeling, providing feasible and real-time analysis tools for the desktop geophysicist. Lastly, we demonstrate through comparison of field data and simulation results that incorporation of infrastructure into the analysis of such geophysical data is, in a growing number of cases, a requisite but now manageable step.

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Well integrity is one of the major concerns in long-term geologic storage sites due to its potential risk for well leakage and groundwater contamination. Evaluating changes in electrical responses due to energized steel-cased wells has the potential to quantify and predict possible wellbore failures, as any kind of breakage or corrosion along highly-conductive well casings will have an impact on the distribution of subsurface electrical potential. However, realistic wellbore-geoelectrical models that can fully capture fine scale details of well completion design and the state of well damage at the field scale require extensive computational e.ort, or can even be intractable to simulate. To overcome this computational burden while still keeping the model realistic, we use the hierarchical finite element method which represents electrical conductivity at each dimensional component (1-D edges, 2-D planes and 3-D cells) of a tetrahedra mesh. This allows well completion designs with real-life geometric scales and well systems with realistic, detailed, progressive corrosion and damage in our models. Here, we present a comparison of possible discretization approaches of a multi-casing completion design in the finite-element model. The e.ects of the surface casing length and the coupling between concentric well casings, as well as the e.ects of the degree and the location of well damage on the electrical responses are also examined. Finally, we analyze real surface electric field data to detect wellbore integrity failure associated with damage.

The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.

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Electrical responses in the vicinity of energized steel-cased well sources offer significant potential for monitoring induced fractures. However, the high complexity of well-fracture-host models spanning multiple length scales compels analysts to simplify their numerical models due to enormous computational costs. This consequently limits our understanding regarding monitoring capabilities and the limitations of electrical measurements on realistic hydraulically fracturing systems. In this paper, we use the hierarchical finite element approach to construct geoelectric models in which geometrically complex fractures and steel-cased wells are discretely represented in 3D conducting media without sacrificing the model realism and computation efficiency. We have discovered systematic numerical analyses of the electrical responses to evaluate the influences of borehole material conductivity and the source type as well as the effects of well geometry, conductivity contrast, source location, fracture growth, and fracture propagation. Furthermore, the numerical results indicate that the borehole material property has a strong control on the electrical potentials along the production and monitoring wells. The monopole source located at a steel-cased well results in a current density distribution that decays away from the source location throughout the well length, whereas the dipole source produces a current density that dominates mainly along the dipole length. Moreover, the conductivity contrast between the fractures and host does not change the overall pattern of the electrical potentials but varies its amplitude. The fracture models near different well systems indicate that the well geometry controls the entire distribution of potentials, while the characteristics of the voltage difference profiles along the wells before and after fracturing are insensitive to the well geometry and the well in which the source is located. Further, the hydraulic-fracturing models indicate that the voltage differences along the production well before and after fracturing have strong sensitivity to fracture growth and fracture set propagation.

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A growing body of applied mathematics literature in recent years has focused on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable to long–range correlations of material properties within an inherently complex, and in some cases self-similar, conducting medium. Rather than considering an exquisitely discretized (and computationally intractable) representation of the medium, the complex and spatially correlated heterogeneity is represented through reformulation of the governing equation for the relevant transport physics such that its coefficients are, instead, smooth but paired with fractional–order space derivatives. Here we apply these concepts to the scalar Helmholtz equation and its use in electromagnetic interrogation of Earth’s interior through the magnetotelluric method. We outline a practical algorithm for solving the Helmholtz equation using spectral methods coupled with finite element discretizations. Execution of this algorithm for the magnetotelluric problem reveals several interesting features observable in field data: long–range correlation of the predicted electromagnetic fields; a power–law relationship between the squared impedance amplitude and squared wavenumber whose slope is a function of the fractional exponent within the governing Helmholtz equation; and, a non–constant apparent resistivity spectrum whose variability arises solely from the fractional exponent. In geologic settings characterized by self–similarity (e.g. fracture systems; thick and richly–textured sedimentary sequences, etc.) we posit that these diagnostics are useful for geologic characterization of features far below the typical resolution limit of electromagnetic methods in geophysics.

The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead ofthe standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in Geophysical Electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid finite element-spectral approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.

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Motivated by the need for improved forward modeling and inversion capabilities of geophysical response in geologic settings whose fine--scale features demand accountability, this project describes two novel approaches which advance the current state of the art. First is a hierarchical material properties representation for finite element analysis whereby material properties can be prescribed on volumetric elements, in addition to their facets and edges. Hence, thin or fine--scaled features can be economically represented by small numbers of connected edges or facets, rather than 10's of millions of very small volumetric elements. Examples of this approach are drawn from oilfield and near--surface geophysics where, for example, electrostatic response of metallic infastructure or fracture swarms is easily calculable on a laptop computer with an estimated reduction in resource allocation by 4 orders of magnitude over traditional methods. Second is a first-ever solution method for the space--fractional Helmholtz equation in geophysical electromagnetics, accompanied by newly--found magnetotelluric evidence supporting a fractional calculus representation of multi-scale geomaterials. Whereas these two achievements are significant in themselves, a clear understanding the intermediate length scale where these two endmember viewpoints must converge remains unresolved and is a natural direction for future research. Additionally, an explicit mapping from a known multi-scale geomaterial model to its equivalent fractional calculus representation proved beyond the scope of the present research and, similarly, remains fertile ground for future exploration.

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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.