Publications

Abstract not provided.

The bulk electrical anisotropy of sedimentary formations is a macroscopic phenomenon whic h can result from the presence of sand/shale laminae and varations in grain size and pore space. Accounting for its effects on induction log response is an ongoing research problem for the w ell-logging communit y since these types of sedimentary stuctures have long been correlated with productive hydrocarbon reservoirs. Presented here is a finite difference method for sim ulatingEM induction in a fully 3D anisotropic medium. This w ork differs from previous modeling efforts in that the electrical conductivity of the formation is represented as a full 3×3 tensor whose elements can vary arbitrarily with position throughout the formation. As an example, we simulate borehole induction tool responses in a crossbedded eolian sandstone to demonstrate the challenge faced by interpreters when electrical anisotropy is neglected.

A fast precondition technique has been developed which accelerates the finite difference solutions of the 3D Maxwell's equations for geophysical modeling. The technique splits the electric field into its curl free and divergence free projections, and allows for the construction of an inverse operator. Test examples show an order of magnitude speed up compared with a simple Jacobi preconditioner. Using this preconditioner a low frequency Neumann series expansion is developed and used to compute responses at multiple frequencies very efficiently. Simulations requiring responses at multiple frequencies, show that the Neumann series is faster than the preconditioned solution, which must compute solutions at each discrete frequency. A Neumann series expansion has also been developed in the high frequency limit along with spectral Lanczos methods in both the high and low frequency cases for simulating multiple frequency responses with maximum efficiency. The research described in this report was to have been carried out over a two-year period. Because of communication difficulties, the project was funded for first year only. Thus the contents of this report are incomplete with respect to the original project objectives.

A resolution study, employing a 3D nonlinear optimization technique, has been undertaken to study the viability of magnetotelluric (MT) measurements to detect and characterize buried facilities that make weapons of mass destruction. A significant advantage of the MT method is that no active source is required because the method employs passive field emissions. Thus measurements can be carried out covertly. Findings indicate it is possible to image WMD facilities, including depth of burial and lateral extent if a sufficient number of measurements are taken on the perimeter of the facility. Moreover if a station measurement can be made directly over the facility then the resolution is improved accordingly. In all cases it was not possible to image the base of the facility with any confidence as well as provide any precise inferences on the facility electrical conductivity. This later finding, however, is really not that critical since knowledge of facility geometry is far more important than knowledge of its conductivity. For the WMD problem it is recommended that MT measurements be made solely with the magnetic field ratios. In this context it would then be possible to deploy with far greater ease small coils about a suspected facility and would allow for the measurements to be conducted in a more covert manner. Before testing such a measurement system in the field, however, it would be necessary to carry out a similar resolution analysis as was done with MT measurements based on electric and magnetic fields. This is necessary to determine sensitivity of the proposed measurement to underground facilities along with needed data coverage and quality. Such a study is indispensable in producing useful reconstructions of underground facilities.

In large scale 3D EM inverse problems it may not be possible to directly invert a full least-squares system matrix involving model sensitivity elements. Thus iterative methods must be employed. For the inverse problem, we favor either a linear or non-linear (NL) CG scheme, depending on the application. In a NL CG scheme, the gradient of the objective function is required at each relaxation step along with a univariate line search needed to determine the optimum model update. Solution examples based on both approaches will be presented.

The method of finite differences has been employed to solve a variety of 3D electromagnetic (EM) forward problems arising in geophysical applications. Specific sources considered include dipolar and magnetotelluric (MT) field excitation in the frequency domain. In the forward problem, the EM fields are simulated using a vector Helmholtz equation for the electric field, which are approximated using finite differences on a staggered grid. To obtain the fields, a complex-symmetric matrix system of equations is assembled and iteratively solved using the quasi-minimum method (QMR) method. Perfectly matched layer (PML) absorbing boundary conditions are included in the solution and are necessary to accurately simulate fields in propagation regime (frequencies>10 MHz). For frequencies approaching the static limit (<10 KHz), the solution also includes a static-divergence correction, which is necessary to accurately simulate MT source fields and can be used to accelerate convergence for the dipolar source problem.

This report has demonstrated techniques that can be used to construct solutions to the 3-D electromagnetic inverse problem using full wave equation modeling. To this point great progress has been made in developing an inverse solution using the method of conjugate gradients which employs a 3-D finite difference solver to construct model sensitivities and predicted data. The forward modeling code has been developed to incorporate absorbing boundary conditions for high frequency solutions (radar), as well as complex electrical properties, including electrical conductivity, dielectric permittivity and magnetic permeability. In addition both forward and inverse codes have been ported to a massively parallel computer architecture which allows for more realistic solutions that can be achieved with serial machines. While the inversion code has been demonstrated on field data collected at the Richmond field site, techniques for appraising the quality of the reconstructions still need to be developed. Here it is suggested that rather than employing direct matrix inversion to construct the model covariance matrix which would be impossible because of the size of the problem, one can linearize about the 3-D model achieved in the inverse and use Monte-Carlo simulations to construct it. Using these appraisal and construction tools, it is now necessary to demonstrate 3-D inversion for a variety of EM data sets that span the frequency range from induction sounding to radar: below 100 kHz to 100 MHz. Appraised 3-D images of the earth`s electrical properties can provide researchers opportunities to infer the flow paths, flow rates and perhaps the chemistry of fluids in geologic mediums. It also offers a means to study the frequency dependence behavior of the properties in situ. This is of significant relevance to the Department of Energy, paramount to characterizing and monitoring of environmental waste sites and oil and gas exploration.

A 3D frequency domain electromagnetic numerical solution has been implemented for sensing buried structures in a lossy earth. Because some structures contain metal, it is necessary to treat them as very good conductors residing in a complicated lossy earth background. To model these scenarios and to avoid excessive gridding in the numerical solution, we assume the structures to be perfectly conducting, which forces the total electric field to zero within the conductor. This is accomplished by enforcing internal boundary conditions on the numerical grid. The numerical solution is based on a vector Helmholtz equation for the scattered electric fields, which is approximated using finite differences on a staggered grid. After finite differencing, a complex-symmetric matrix system of equations is assembled and preconditioned using Jocobi scaling before it is iteratively solved using the quasi-minimum residual (qmr) or bi-conjugate gradient (bicg) methods. For frequencies approaching the static limit (< 10 kHz), the scheme incorporates a static-divergence correction to accelerate solution convergence. This is accomplished by enforcing the divergence of the scattering current within the earth as well as the divergence of the scattered electric field in the air.