Electric field and magnetic field reflection and transmission responses generated by a plane wave normally incident onto a finite - thickness geologic layer are mathematically derived and numerically evaluated. A thin layer with enhanced electric current conductivity and/or magnetic permeability is a reasonable geophysical representation of a hydraulic fracture inject ed with a high - contrast proppant pack. Both theory and numerics indicate that backward - and forward - scattered electromagnetic wavefields are potentially observable in a field experiment, despite the extreme thinness of a fracture compared to a typical low - frequency electromagnetic wavelength. The First Born Approximation (FBA) representation of layer scattering, significant for inversion studies, is shown to be accurate for a thin layer with mild medium parameter (i.e., conductivity, permeability, and per mittivity) contrasts with the surrounding homogeneous wholespace. However, FBA scattering theory breaks down for thick layers and strong parameter contrasts. ACKNOWLEDGEMENTS Sandia National Laboratories is a multi - mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy's National Nuclear Security Administration under contract DE - AC04 - 94AL85000. This research is conducted under the auspices of CRADA (Cooperative Research and Development Agreement) SC11/01780.00 between Carbo Ceramics Inc. and Sandia National Laboratories. The author acknowledges former Carbo R&D Vic e - President Mr. Chad Cannan and former SNL Geophysics Department manage r Ms. Amy Halloran for their interest i n and support of this work. Technical discussions with Project Manager and Principal Investigator Dr. Chester J. Weiss of the SNL Geophysics Department greatly benefited this work. Dr. Lewis C. Bartel, formerly with S NL and presently a consultant to Carbo Ceramics, provided many useful and intuitive insights, and is acknowledged as the originator of the concept underpinning a recent patent grant (Aldridge and Bartel, 2016) involving electromagnetic wave scattering.
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.
The need to better represent the material properties within the earth's interior has driven the development of higherfidelity physics, e.g., visco-tilted-transversely-isotropic (visco- TTI) elastic media and material interfaces, such as the ocean bottom and salt boundaries. This is especially true for full waveform inversion (FWI), where one would like to reproduce the real-world effects and invert on unprocessed raw data. Here we present a numerical formulation using a Discontinuous Galerkin (DG) finite-element (FE) method, which incorporates the desired high-fidelity physics and material interfaces. To offset the additional costs of this material representation, we include a variety of techniques (e.g., non-conformal meshing, and local polynomial refinement), which reduce the overall costs with little effect on the solution accuracy.
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.
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.
Aldridge, David F.; Weiss, Chester J.; Knox, Hunter A.; Schramm, Kimberly A.; Bartel, Lewis C.
Under certain restricting assumptions, an electrically energized steel-cased geologic borehole may be modeled as an electrical transmission line. Current waveforms are obtained by solving the governing telegraph equation in the frequency-domain, followed by numerical inverse Fourier transformation. Electric current pulses propagating along the borehole undergo progressive amplitude loss and waveform distortion with distance, arising from leakage of current into the surrounding geology. A major modeling uncertainty involves the proper boundary condition to impose at the end of a borehole transmission line.
A reciprocity theorem is an explicit mathematical relationship between two different wavefields that can exist within the same space - time configuration. Reciprocity theorems provi de the theoretical underpinning for mod ern full waveform inversion solutions, and also suggest practical strategies for speed ing up large - scale numerical modeling of geophysical datasets . In the present work, several previously - developed electromagnetic r eciprocity theorems are generalized to accommodate a broader range of medi um, source , and receiver types. Reciprocity relations enabling the interchange of various types of point sources and point receivers within a three - dimensional electromagnetic model are derived. Two numerical modeling algorithms in current use are successfully tested for adherence to reciprocity. Finally, the reciprocity theorem forms the point of departure for a lengthy derivation of electromagnetic Frechet derivatives. These mathe matical objects quantify the sensitivity of geophysical electromagnetic data to variatio ns in medium parameters, and thus constitute indispensable tools for solution of the full waveform inverse problem. ACKNOWLEDGEMENTS Sandia National Labor atories is a multi - program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy's National Nuclear Security Administration under contract DE - AC04 - 94AL85000. Signif icant portions of the work reported herein were conducted under a Cooperative Research and Development Agreement (CRADA) between Sandia National Laboratories (SNL) and CARBO Ceramics Incorporated. The author acknowledges Mr. Chad Cannan and Mr. Terry Pa lisch of CARBO Ceramics, and Ms. Amy Halloran, manager of SNL's Geophysics and Atmospheric Sciences Department, for their interest in and encouragement of this work. Special thanks are due to Dr . Lewis C. Bartel ( recently retired from Sandia National Labo ratories and now a geophysical consultant ) and Dr. Chester J. Weiss (recently rejoined with Sandia National Laboratories) for many stimulating (and reciprocal!) discussions regar ding the topic at hand.
A Rayleigh wave propagates laterally without dispersion in the vicinity of the plane stress-free surface of a homogeneous and isotropic elastic halfspace. The phase speed is independent of frequency and depends only on the Poisson ratio of the medium. However, after temporal and spatial discretization, a Rayleigh wave simulated by a 3D staggered-grid finite-difference (FD) seismic wave propagation algorithm suffers from frequency- and direction-dependent numerical dispersion. The magnitude of this dispersion depends critically on FD algorithm implementation details. Nevertheless, proper gridding can control numerical dispersion to within an acceptable level, leading to accurate Rayleigh wave simulations. Many investigators have derived dispersion relations appropriate for body wave propagation by various FD algorithms. However, the situation for surface waves is less well-studied. We have devised a numerical search procedure to estimate Rayleigh phase speed and group speed curves for 3D O(2,2) and O(2,4) staggered-grid FD algorithms. In contrast with the continuous time-space situation (where phase speed is obtained by extracting the appropriate root of the Rayleigh cubic), we cannot develop a closed-form mathematical formula governing the phase speed. Rather, we numerically seek the particular phase speed that leads to a solution of the discrete wave propagation equations, while holding medium properties, frequency, horizontal propagation direction, and gridding intervals fixed. Group speed is then obtained by numerically differentiating the phase speed with respect to frequency. The problem is formulated for an explicit stress-free surface positioned at two different levels within the staggered spatial grid. Additionally, an interesting variant involving zero-valued medium properties above the surface is addressed. We refer to the latter as an implicit free surface. Our preliminary conclusion is that an explicit free surface, implemented with O(4) spatial FD operators and positioned at the level of the compressional stress components, leads to superior numerical dispersion performance. Phase speeds measured from fixed-frequency synthetic seismograms agree very well with the numerical predictions.
An efficient numerical algorithm for treating earth models composed of fluid and solid portions is obtained via straightforward modifications to a 3D time-domain finite-difference algorithm for simulating isotropic elastic wave propagation.
Summary: Dispersion and attenuation relations are derived for both the continuous and discrete velocity-memory-stress systems governing 3D anelastic wave propagation in a standard linear solid. Phase speed and attenuation factor curves extracted from these relations enable optimal selection of spatial and temporal gridding intervals to achieve finite-difference algorithm efficiency, while simultaneously minimizing numerical inaccuracy.
In many parts of the United States, as well as other regions of the world, competing demands for fresh water or water suitable for desalination are outstripping sustainable supplies. In these areas, new water supplies are necessary to sustain economic development and agricultural uses, as well as support expanding populations, particularly in the Southwestern United States. Increasing the supply of water will more than likely come through desalinization of water reservoirs that are not suitable for present use. Surface-deployed seismic and electromagnetic (EM) methods have the potential for addressing these critical issues within large volumes of an aquifer at a lower cost than drilling and sampling. However, for detailed analysis of the water quality, some sampling utilizing boreholes would be required with geophysical methods being employed to extrapolate these sampled results to non-sampled regions of the aquifer. The research in this report addresses using seismic and EM methods in two complimentary ways to aid in the identification of water reservoirs that are suitable for desalinization. The first method uses the seismic data to constrain the earth structure so that detailed EM modeling can estimate the pore water conductivity, and hence the salinity. The second method utilizes the coupling of seismic and EM waves through the seismo-electric (conversion of seismic energy to electrical energy) and the electro-seismic (conversion of electrical energy to seismic energy) to estimate the salinity of the target aquifer. Analytic 1D solutions to coupled pressure and electric wave propagation demonstrate the types of waves one expects when using a seismic or electric source. A 2D seismo-electric/electro-seismic is developed to demonstrate the coupled seismic and EM system. For finite-difference modeling, the seismic and EM wave propagation algorithms are on different spatial and temporal scales. We present a method to solve multiple, finite-difference physics problems that has application beyond the present use. A limited field experiment was conducted to assess the seismo-electric effect. Due to a variety of problems, the observation of the electric field due to a seismic source is not definitive.
Spatially-distributed arrays of seismometers are often utilized to infer the speed and direction of incident seismic waves. Conventionally, individual seismometers of the array measure one or more orthogonal components of rectilinear particle motion (displacement, velocity, or acceleration). The present work demonstrates that measure of both the particle velocity vector and the particle rotation vector at a single point receiver yields sufficient information to discern the type (compressional or shear), speed, and direction of an incident plane seismic wave. Hence, the approach offers the intriguing possibility of dispensing with spatially-extended received arrays, with their many problematic deployment, maintenance, relocation, and post-acquisition data processing issues. This study outlines straightforward mathematical theory underlying the point seismic array concept, and implements a simple cross-correlation scanning algorithm for determining the azimuth of incident seismic waves from measured acceleration and rotation rate data. The algorithm is successfully applied to synthetic seismic data generated by an advanced finite-difference seismic wave propagation modeling algorithm. Application of the same azimuth scanning approach to data acquired at a site near Yucca Mountain, Nevada yields ambiguous, albeit encouraging, results. Practical issues associated with rotational seismometry are recognized as important, but are not addressed in this investigation.
Stable and accurate numerical modeling of seismic wave propagation in the vicinity of high-contrast interfaces is achieved with straightforward modifications to the conventional, rectangular-staggered-grid, finite-difference (FD) method. Improvements in material parameter averaging and spatial differencing of wavefield variables yield high-quality synthetic seismic data.
Acoustic wave propagation in a three-dimensional atmosphere that is spatially heterogeneous, time-varying, and/or moving is accurately simulated with a numerical algorithm recently developed under the DOD Common High Performance Computing Software Support Initiative (CHSSI). Sound waves within such a dynamic environment are mathematically described by a set of four, coupled, first-order partial differential equations governing small-amplitude fluctuations in pressure and particle velocity. The system is rigorously derived from fundamental principles of continuum mechanics, ideal-fluid constitutive relations, and reasonable assumptions that the ambient atmospheric motion is adiabatic and divergence-free. An explicit, finite-difference time-domain (FDTD) numerical scheme is used to solve the system for both pressure and particle velocity wavefields. Dependent variables are stored on staggered spatial and temporal grids, and centered FDTD operators possess 2nd-order and 4th-order space/time accuracy. We first present results of a test that shows the accuracy of our algorithm by comparison with analytic formulations. We then present a contrast and comparison of the sound character at a series of distances from a point source activated with a causal source. We are able to investigate the effects of turbulence, complex meteorology (including wind effects), a topographically variable ground surface, and a partially reflective ground surface.
This document is intended to serve as a users guide for the time-domain atmospheric acoustic propagation suite (TDAAPS) program developed as part of the Department of Defense High-Performance Modernization Office (HPCMP) Common High-Performance Computing Scalable Software Initiative (CHSSI). TDAAPS performs staggered-grid finite-difference modeling of the acoustic velocity-pressure system with the incorporation of spatially inhomogeneous winds. Wherever practical the control structure of the codes are written in C++ using an object oriented design. Sections of code where a large number of calculations are required are written in C or F77 in order to enable better compiler optimization of these sections. The TDAAPS program conforms to a UNIX style calling interface. Most of the actions of the codes are controlled by adding flags to the invoking command line. This document presents a large number of examples and provides new users with the necessary background to perform acoustic modeling with TDAAPS.
Findings are presented from the first year of a joint project between the U.S. Army Engineer Research and Development Center, the U.S. Army Research Laboratory, and the Sandia National Laboratories. The purpose of the project is to develop a finite-difference, time-domain (FDTD) capability for simulating the acoustic signals received by battlefield acoustic sensors. Many important effects, such as scattering from trees and buildings, interactions with dynamic atmospheric wind and temperature fields, and nonstationary target properties, can be accommodated by the simulation. Such a capability has much potential for mitigating the need for costly field data collection and furthering the development of robust identification and tracking algorithms. The FDTD code is based on a carefully derived set of first-order differential equations that is more general and accurate than most current sound propagation formulations. For application to three-dimensional problems of practical interest in battlefield acoustics, the code must be run on massively parallel computers. Some example computations involving sound propagation in a moving atmosphere and propagation in the presence of trees and barriers are presented.
The mathematical description of acoustic wave propagation within a time- and space-varying, and moving, linear viscous fluid is formulated as a system of coupled linear equations. This system is rigorously developed from fundamental principles of continuum mechanics (conservation of mass, balance of linear and angular momentum, balance of entropy) and various constitutive relations (for stress, entropy production, and entropy conduction) by linearizing all expressions with respect to the small-amplitude acoustic wavefield variables. A significant simplification arises if the fluid medium is neither viscous nor heat conducting (i.e., an ideal fluid). In this case the mathematical system can be reduced to a set of five, coupled, first-order partial differential equations. Coefficients in the systems depend on various mechanical and thermodynamic properties of the ambient medium that supports acoustic wave propagation. These material properties cannot all be arbitrarily specified, but must satisfy another system of nonlinear expressions characterizing the dynamic behavior of the background medium. Dramatic simplifications in both systems occur if the ambient medium is simultaneously adiabatic and stationary.
The spatial and temporal origin of a seismic energy source are estimated with a first grid search technique. This approach has greater likelihood of finding the global rninirnum of the arrival time misiit function compared with conventional linearized iterative methods. Assumption of a homogeneous and isotropic seismic velocity model allows for extremely rapid computation of predicted arrival times, but probably limits application of the method to certain geologic environments and/or recording geometries. Contour plots of the arrival time misfit function in the vicinity of the global minimum are extremely useful for (i) quantizing the uncertainty of an estimated hypocenter solution and (ii) analyzing the resolving power of a given recording configuration. In particular, simultaneous inversion of both P-wave and S-wave arrival times appears to yield a superior solution in the sense of being more precisely localized in space and time. Future research with this algorithm may involve (i) investigating the utility of nonuniform residual weighting schemes, (ii) incorporating linear and/or layered velocity models into the calculation of predicted arrival times, and (iii) applying it toward rational design of microseismic monitoring networks.