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.
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.
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.
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 geological settings characterized by self-similarity (e.g. fracture systems; thick and richly textured sedimentary sequences, etc.) we posit that these diagnostics are useful for geological characterization of features far below the typical resolution limit of electromagnetic methods in geophysics.
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 perscribed 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.
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. Although generally useful for efficient modeling of mild conductivity perturbations in uncluttered settings, we have raised the question of its suitability for situations such as oilfields, in which 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 determine that the spectral radius of the Neumann series operator grows as approximately 1/h, thus suggesting that in the limit of the continuous problem h→0, the Neumann series is intrinsically divergent for all conductivity perturbations, 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. The methods used here are demonstrably useful in such circumstances.
Fractures are an interest of many engineering problems. They present complex spatial distributions and hydraulic properties that vary over a wide range of length scales. The multi-length-scale nature as well as the volumetric insignificance of fractures at the filed scale demand an explosive computational effort to account of fractures in standard DC resistivity modeling. Here, we use the hierarchical finite element method (Hi-FEM) to model complex fracture networks in 3D conducting media. The HiFEM method is based on the hierarchy in the electrical properties of 3D geologic media that drastically reduces the computational cost, such that thin conductive fractures can easily be represented by a set of connected 2D facet elements or linear conductive features can be approximated by connected 1D edge elements. Here, we present a demonstrative numerical study of the 3D DC resistivity responses of a complex fractured network consisting of a large number of randomly-oriented fractures. We also simulate the time lapse response of an evolving fracture network as a demonstration of real-time 4D monitoring. Our results indicate that the amplitude and the distribution of DC electric potentials are substantially controlled by fracture properties; moreover, the DC resistivity measurements over a growing fracture network reflect the spatial and the temporal state of the network connectivity.
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.
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.
Electromagnetic responses reflect the interaction between applied electromagnetic fields and heterogeneous geoelectrical structures. Quantifying the relationship between multiscale electrical properties and the observed electromagnetic response is therefore important for meaningful geologic interpretation. We present here examples of near-surface electromagnetic responses whose spatial fluctuations appear on all length scales, are repeatable and fractally distributed, supporting the notion of a 'rough geology' exhibitingmultiscale hierarchical structure. Bounded by end member cases from homogenized isotropic and anisotropic media, we present numerical modelling results of the electromagnetic responses of textured and spatially correlated, stochastic geologic media, demonstrating that the electromagnetic response is a power law distribution, rather than a smooth response polluted with random, incoherent noise as commonly assumed. Our modelling results show that these electromagnetic responses due to spatially correlated geologic textures are examples of fractional Brownian motion. Furthermore, our results suggest that the fractal behaviour of the electromagnetic responses is correlated with degree of the spatial correlation, the contrasts in ground conductivity, and the preferred orientation of small-scale heterogeneity. In addition, the EM responses acquired across a fault zone comprising different lithological units and varying wavelengths of geologic heterogeneity also support our inferences from numerical modelling.
An electromagnetic finite volume forward solver is implemented to create a suite of forward mod- els that provide the expected response for an air-filled buried structure constructed of concrete and rebar. Model parameters considered are the conductivities and thicknesses of a two-layer subsur- face and the nature of VLF plane wave source. By building this suite of models, the results can be packaged into a data set that is both easily callable and requires minimal storage. More importantly, the user is relieved of the time required to manually execute a large number of models. Instead the results are already provided along with an interpolation tool for immediately data access. This document is written in compliance the LDRD reporting requirements for a close-out report on Project 180848. This page intentionally left blank.
Hydraulic fracture stimulation of low permeability reservoir rocks is an established and cross-cutting technology for enhancing hydrocarbon production in sedimentary formations and increasing heat exchange in crystalline geothermal systems. Whereas the primary measure of success is the ability to keep the newly generated fractures sufficiently open, long-term reservoir management requires a knowledge of the spatial extent, morphology, and distribution of the fractures-knowledge primarily informed by microseismic and ground deformation monitoring. To minimize the uncertainty associated with interpreting such data, we investigate through numerical simulation the usefulness of direct-current (DC) resistivity data for characterizing subsurface fractures with elevated electrical conductivity by considering a geophysical experiment consisting of a grounded current source deployed in a steel cased borehole. In doing so, the casing efficiently energizes the fractures with steady current. Finite element simulations of this experiment for a horizontal well intersecting a small set of vertical fractures indicate that the fractures manifest electrically in (at least) two ways: (1) a local perturbation in electric potential proximal to the fracture set, with limited farfield expression and (2) an overall reduction in the electric potential along the borehole casing due to enhanced current flow through the fractures into the surrounding formation. The change in casing potential results in a measurable effect that can be observed far from fractures themselves. Under these conditions, our results suggest that farfield, timelapse measurements of DC potentials can be interpreted by simple, linear inversion for a Coulomb charge distribution along the borehole path, including a local charge perturbation due to the fractures. This approach offers an inexpensive method for detecting and monitoring the time-evolution of electrically conducting fractures while ultimately providing an estimate of their effective conductivity - the latter providing an important measure independent of seismic methods on fracture shape, size, and hydraulic connectivity.
We investigate a novel application of Fŕechet derivatives for time-lapse mapping of deep, electrically-enhanced fracture systems with a borehole to surface DC resistivity array. The simulations are evaluated for a cased horizontal wellbore embedded in a homogeneous halfspace, where measurements are evaluated near, mid-range, and far from the well head. We show that, in all cases, measurements are sensitive to perturbations centered on the borehole axis and that the sensitivity volume decreases as a function of increased measurement offset from the well head. The sensitivity analysis also illustrates that careful consideration must be taken when developing an electrical survey design for these scenarios. Specifically, we show that positive perturbations in earth conductivity near the wellbore can manifest as both positive and negative measurement perturbations, depending on where the measurement is taken. Furthermore, we show that the transition between the regions along the wellbore of positive and negative contribution results in a "pinch point", representing a region along the wellbore where a given surface measurement is blind to any changes or enhancement of electrical conductivity.
In this paper we explore simulated responses of electromagnetic (EM) signals relative to in situ field surveys and quantify the effects that different values of conductivity in sea ice have on the EM fields. We compute EM responses of ice types with a three-dimensional (3-D) finite-volume discretization of Maxwell's equations and present 2-D sliced visualizations of their associated EM fields at discrete frequencies. Several interesting observations result: First, since the simulator computes the fields everywhere, each gridcell acts as a receiver within the model volume, and captures the complete, coupled interactions between air, snow, sea ice and sea water as a function of their conductivity; second, visualizations demonstrate how 1-D approximations near deformed ice features are violated. But the most important new finding is that changes in conductivity affect EM field response by modifying the magnitude and spatial patterns (i.e. footprint size and shape) of current density and magnetic fields. These effects are demonstrated through a visual feature we define as 'null lines'. Null line shape is affected by changes in conductivity near material boundaries as well as transmitter location. Our results encourage the use of null lines as a planning tool for better ground-truth field measurements near deformed ice types.
We investigate through numerical simulation the usefulness of DC resistivity data for characterizing subsurface fractures with elevated electrical conductivity by considering a geophysical experiment consisting of a grounded current source deployed in a steel cased borehole. In doing so, the borehole casing behaves electrically as a spatially extended line source, efficiently energizing the fractures with a steady current. Finite element simulations of this experiment for a horizontal well intersecting a small set of vertical fractures indicate that the fractures manifest electrically in (at least) two ways: a local perturbation in the electric potential proximal the fracture set, with limited far-field expression; and, an overall reduction in the electric potential along the entire length of borehole casing due to enhanced current flow through the fractures into the surrounding formation. The change in casing potential results in a measureable effect that can be observed far from fractures themselves, at distances where the local perturbations in the electric potential around the fractures are imperceptible. Under these conditions, our results suggest that far-field, time-lapse measurements of DC potentials surrounding a borehole casing can be reasonably interpreted by simple, linear inversion for a Coulomb charge distribution along the borehole path, including a local charge perturbation due to the fractures. Such an approach offers an inexpensive method for detecting and monitoring the time-evolution of electrically conducting fractures while ultimately providing an estimate of their effective conductivity - the latter providing an important measure independent of seismic methods on fracture shape, size, and hydraulic connectivity.
The project objective was to detail better ways to assess and exploit intelligent oil and gas field information through improved modeling, sensor technology, and process control to increase ultimate recovery of domestic hydrocarbons. To meet this objective we investigated the use of permanent downhole sensors systems (Smart Wells) whose data is fed real-time into computational reservoir models that are integrated with optimized production control systems. The project utilized a three-pronged approach (1) a value of information analysis to address the economic advantages, (2) reservoir simulation modeling and control optimization to prove the capability, and (3) evaluation of new generation sensor packaging to survive the borehole environment for long periods of time. The Value of Information (VOI) decision tree method was developed and used to assess the economic advantage of using the proposed technology; the VOI demonstrated the increased subsurface resolution through additional sensor data. Our findings show that the VOI studies are a practical means of ascertaining the value associated with a technology, in this case application of sensors to production. The procedure acknowledges the uncertainty in predictions but nevertheless assigns monetary value to the predictions. The best aspect of the procedure is that it builds consensus within interdisciplinary teams The reservoir simulation and modeling aspect of the project was developed to show the capability of exploiting sensor information both for reservoir characterization and to optimize control of the production system. Our findings indicate history matching is improved as more information is added to the objective function, clearly indicating that sensor information can help in reducing the uncertainty associated with reservoir characterization. Additional findings and approaches used are described in detail within the report. The next generation sensors aspect of the project evaluated sensors and packaging survivability issues. Our findings indicate that packaging represents the most significant technical challenge associated with application of sensors in the downhole environment for long periods (5+ years) of time. These issues are described in detail within the report. The impact of successful reservoir monitoring programs and coincident improved reservoir management is measured by the production of additional oil and gas volumes from existing reservoirs, revitalization of nearly depleted reservoirs, possible re-establishment of already abandoned reservoirs, and improved economics for all cases. Smart Well monitoring provides the means to understand how a reservoir process is developing and to provide active reservoir management. At the same time it also provides data for developing high-fidelity simulation models. This work has been a joint effort with Sandia National Laboratories and UT-Austin's Bureau of Economic Geology, Department of Petroleum and Geosystems Engineering, and the Institute of Computational and Engineering Mathematics.
Electromagnetic induction is a classic geophysical exploration method designed for subsurface characterization--in particular, sensing the presence of geologic heterogeneities and fluids such as groundwater and hydrocarbons. Several approaches to the computational problems associated with predicting and interpreting electromagnetic phenomena in and around the earth are addressed herein. Publications resulting from the project include [31]. To obtain accurate and physically meaningful numerical simulations of natural phenomena, computational algorithms should operate in discrete settings that reflect the structure of governing mathematical models. In section 2, the extension of algebraic multigrid methods for the time domain eddy current equations to the frequency domain problem is discussed. Software was developed and is available in Trilinos ML package. In section 3 we consider finite element approximations of De Rham's complex. We describe how to develop a family of finite element spaces that forms an exact sequence on hexahedral grids. The ensuing family of non-affine finite elements is called a van Welij complex, after the work [37] of van Welij who first proposed a general method for developing tangentially and normally continuous vector fields on hexahedral elements. The use of this complex is illustrated for the eddy current equations and a conservation law problem. Software was developed and is available in the Ptenos finite element package. The more popular methods of geophysical inversion seek solutions to an unconstrained optimization problem by imposing stabilizing constraints in the form of smoothing operators on some enormous set of model parameters (i.e. ''over-parametrize and regularize''). In contrast we investigate an alternative approach whereby sharp jumps in material properties are preserved in the solution by choosing as model parameters a modest set of variables which describe an interface between adjacent regions in physical space. While still over-parametrized, this choice of model space contains far fewer parameters than before, thus easing the computational burden, in some cases, of the optimization problem. And most importantly, the associated finite element discretization is aligned with the abrupt changes in material properties associated with lithologic boundaries as well as the interface between buried cultural artifacts and the surrounding Earth. In section 4, algorithms and tools are described that associate a smooth interface surface to a given triangulation. In particular, the tools support surface refinement and coarsening. Section 5 describes some preliminary results on the application of interface identification methods to some model problems in geophysical inversion. Due to time constraints, the results described here use the GNU Triangulated Surface Library for the manipulation of surface meshes and the TetGen software library for the generation of tetrahedral meshes.
We address the electromagnetic induction problem for fully 3D geologic media and present a solution to the governing Maxwell equations based on a power series expansion. The coefficients in the series are computed using the adjoint method assuming an underlying homogeneous reference model. These solutions are available analytically for point dipole source terms and lead to rapid calculation of the expansion coefficients. First order solutions are presented for a model study in petroleum geophysics composed of a multi-component induction sonde proximal to a fault within a compartmentalized hydrocarbon reservoir.
Borehole radar systems can provide essential subsurface structural information for environmental evaluation, geotechnical analysis, or energy exploration. Sandia developed a prototype continuous-wave Borehole Radar (BHR) in 1996, and development of a practical tool has been continuing at a Russian institute under a Sandia contract. The BHR field experiments, which were planned for the summer of 2001 in Russia, provided a unique opportunity to evaluate the latest Sandia algorithms with actual field data. A new three-dimensional code was developed to enable the analysis of BHR data on modest-sized desktop workstations. The code is based on the staggered grid, finite difference technique, and eliminates 55% of the massive storage associated with solving the system of finite-difference linear equations. The code was used to forward-model the Russian site geometry and placement of artificial targets to anticipate any problems that might arise when the data was received. Technical software and equipment problems in the Russian field tests, conducted in August 2001, invalidated all but one of the data sets. However, more field tests with improved equipment and software are planned for 2002, and analysis of that data will be presented in a future report.
IEEE Transactions on Geoscience and Remote Sensing
Everett, Mark E.; Badea, Eugene A.; Shen, Liang C.; Merchant, Gulamabbas A.; Weiss, Chester J.
Electromagnetic induction (EMI) by a magnetic dipole located above a dipping interface is of relevance to the petroleum well-logging industry. The problem is fully three-dimensional (3-D) when formulated as above, but reduces to an analytically tractable one-dimensional (1-D) problem when cast as a small tilted coil above a horizontal interface. The two problems are related by a simple coordinate rotation. An examination of the induced eddy currents and the electric charge accumulation at the interface help to explain the inductive and polarization effects commonly observed in induction logs from dipping geological formations. The equivalence between the 1-D and 3-D formulations of the problem enables the validation of a previously published finite element solver for 3-D controlled-source EMI.