David L. Alumbaugh

2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements

We present 2.5D fast and rigorous forward and inversion algorithms for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements. The forward algorithm is based on a finite-difference approach in which a multifrontal LU decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single-source experiment for each frequency of operation. When the size of the linear system of equations is large, the use of this noniterative solver is impractical. Hence, we use the optimal grid technique to limit the number of unknowns in the forward problem. The inversion algorithm employs a regularized Gauss-Newton minimization approach with a multiplicative cost function. By using this multiplicative cost function, we do not need a priori data to determine the so-called regularization parameter in the optimization process, making the algorithm fully automated. The algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image. To increase the robustness of the algorithm, we also constrain the minimization and use a line-search approach to guarantee the reduction of the cost function after each iteration. To demonstrate the pros and cons of the algorithm, we present synthetic and field data inversion results for crosswell and controlled-source EM measurements.

A geostatistically based inverse model for electrical resistivity surveys and its applications to vadose zone hydrology

[1] A sequential, geostatistical inverse approach was developed for electrical resistivity tomography (ERT). Unlike most ERT inverse approaches, this new approach allows inclusion of our prior knowledge of general geological structures of an area and point electrical resistivity measurements to constrain the estimate of the electrical resistivity field. This approach also permits sequential inclusion of different data sets, mimicking the ERT data collection scheme commonly employed in the field survey. Furthermore, using the conditional variance concept, the inverse model quantifies uncertainty of the estimate caused by spatial variability and measurement errors. Using this approach, numerical experiments were conducted to demonstrate the effects of bedding orientation on ERT surveys and to show both the usefulness and uncertainty associated with the inverse approach for delineating the electrical resistivity distribution using down-hole ERT arrays. A statistical analysis was subsequently undertaken to explore the effects of spatial variability of the electrical resistivity-moisture relation on the interpretation of the change in water content in the vadose zone, using the change in electrical resistivity. Core samples were collected from a field site to investigate the spatial variability of the electrical resistivity-moisture relation. Numerical experiments were subsequently conducted to illustrate how the spatially varying relations affect the level of uncertainty in the interpretation of change of moisture content based on the estimated change in electrical resistivity. Other possible complications are also discussed. INDEX TERMS: 0903 Exploration Geophysics: Computational methods, potential fields; 1869 Hydrology: Stochastic processes; 1875 Hydrology: Unsaturated zone; 1866 Hydrology: Soil moisture; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: geostatistical inverse model, electrical resistivity tomography, vadose zone, resistivitymoisture relation, spatial variability, sequential/successive linear estimator

Image appraisal for 2-D and 3-D electromagnetic inversion

Linearized methods are presented for appraising resolution and parameter accuracy in images generated with 2-D and 3-D nonlinear electromagnetic (EM) inversion schemes. When direct matrix inversion is used, the model resolution and a posteriori model covariance matrices can be calculated readily. By analyzing individual columns of the model resolution matrix, the spatial variation of the resolution in the horizontal and vertical directions can be estimated empirically. Plotting the diagonal of the model covariance matrix provides an estimate of how errors in the inversion process, such as data noise and incorrect a priori assumptions, map into parameter error and thus provides valuable information about the uniqueness of the resulting image. Methods are also derived for image appraisal when the iterative conjugate gradient technique is applied to solve the inverse. An iterative statistical method yields accurate estimates of the model covariance matrix as long as enough iterations are used. Although determining the entire model resolution matrix in a similar manner is computationally prohibitive, individual columns of this matrix can be determined. Thus, the spatial variation in image resolution can be determined by calculating the columns of this matrix for key points in the image domain and then interpolating between. Examples of the image analysis techniques are provided on 2-D and 3-D synthetic crosswell EM data sets as well as a field data set collected at Lost Hills oil field in central California.

On the physics of the marine controlled-source electromagnetic method

We examine the underlying physics of the marine controlled-source electromagnetic (CSEM) method through the use of cross-sectional plots of the vector-current density. A systematic comparison of the cross-sectional current-density distribution within uniform and reservoir-bearing seafloor models reveals that the method induces detectable reservoir responses at the seafloor for source-receiver offsets that are frequency dependent. Higher frequencies generally result in larger anomalous differences between the two models at shorter offsets up to a frequency where induced currents no longer effectively interact with the reservoir due to electromagnetic (EM) attenuation. At zero and low frequencies, the less-attenuated background EM fields mask the reservoir response, although large induced currents are normally incident upon the reservoir. The reservoir response is also masked at larger offsets and/or in shallow environments by theairwave that can be thought of as energy diffusing up and down through the sea...

Three-dimensional induction logging problems, Part 2: A finite-difference solution

A 3-D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite-difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl-free and divergence-free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.

Theoretical and practical considerations for crosswell electromagnetic tomography assuming a cylindrical geometry

An iterative Born imaging scheme is employed to analyze the resolution properties of crosswell electromagnetic tomography. The imaging scheme assumes a cylindrical symmetry about a vertical magnetic dipole source and employs approximate forward modeling at each iteration to update the internal electric fields. Estimation of the anomalous conductivity is accomplished through least-squares inversion. Much of the mathematical formulation of this diffusion process appears similar to the analysis of wavefield solutions, but the attenuation implicit in the complex propagation constant invalidates many of the accepted wavefield criteria for resolution.Images of illustrative models show that vertical resolution improves with increasing frequency and with increased spatial sampling density. In addition, greater conductivity contrasts between the target and the background can result in better resolution. The horizontal resolution depends on the maximum aperture that is employed and with increasing frequency, larger apertures are needed to obtain optimal results. However, the maximum aperture that can be employed, and thus the horizontal resolution, is limited by the rate of attenuation and the noise present in the measurements. Weighting the long-offset data equally with the zero-offset data can improve the resolution if the noise is not a function of the dynamic range of the measurement system. At lower frequencies, the resolution can be improved by measuring both the horizontal and vertical components of the magnetic fields. In addition, multiple frequencies can be employed to improve the resolution for limited aperture measurements.The general applicability of the cylindrically symmetric geometry is examined by comparing the 2-D sensitivity functions to those produced by a 2.5-D model, and by imaging a 3-D body with the 2-D iterative Born scheme. For borehole separations greater than five skin depths it is demonstrated that the measurements, and thus the images, are not affected by the geometry of the conductive zone outside of the interwell plane. Thus the 2-D imaging scheme can be employed in these situations. For borehole separations less than five skin depths, artifacts are produced in the images which will lead to faulty interpretations.

3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach

We present a finite-element time-domain FETD approach for the simulation of 3D electromagnetic EM diffusion phenomena. The finite-element algorithm efficiently simulates transient electric fields and the time derivatives of magneticfields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms. To compute transient electromagnetic fields,theelectricfielddiffusionequationistransformedinto asystemofdifferentialequationsviaGalerkin’smethodwith homogeneous Dirichlet boundary conditions. To ensure numerical stability and an efficient time step, the system of the differential equations is discretized in time using an implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver along with a fill-inreducedorderingtechnique.Whenadvancingthesolution in time, the FETD algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution. To simulate a step-off source waveform, the 3D FETD algorithm also incorporates a 3D finite-element direct current FEDCalgorithmthatsolvesPoisson’sequationusingasecondarypotentialmethodforageneralanisotropicearthmodel. Examples of controlled-source FETD simulations are compared with analytic and/or 3D finite-difference time-domain solutions and are used to confirm the accuracy and efficiencyofthe3DFETDalgorithm.

Three methods for mitigating airwaves in shallow water marine controlled-source electromagnetic data

In the past several years, marine controlled-source electromagnetic (MCSEM) techniques have been applied successfully in deep water (depth > 1 km) for oil and gas exploration. The application of this technology in shallow water is challenged, however, because of “airwaves” that mask the signal from the target reservoir at depth. Based upon the understanding that an airwave is a lateral wave, which can be analytically expressed in a dual-half-space resistivity model, we propose three airwave-mitigation approaches to reduce the effects of these airwaves on MCSEM data. In the EM “x-bucking” approach, the effect of the airwaves can be “bucked” out from two measurements by using the analytic expression of the airwave. The frequency derivative (dE/dFreq) approach takes advantages of the unique characteristics of the airwaves in frequency domain, enhancing the reservoir signals while suppressing the airwave. The magnetotelluric (MT) stripping method uses the plane-wave feature of the airwaves and subtraction of ...

Crosshole electromagnetic tomography: A new technology for oil field characterization

With the advent of crosshole seismic technology in the 1980s, a new generation of high resolution geophysical tools has become available for reservoir characterization. The chief improvement is simply that the tools are deployed in boreholes so measurements take place much closer to the region of interest.

Hydrogeophysical Case Studies in the Vadose Zone

The focus of this chapter is the characterization of the vadose zone, or the unsaturated section of the subsurface, using hydrogeophysical techniques. The regions of water saturation as they relate to the physical properties are shown for reference in Figure 14.1. Characterization below the water table (in the saturated section) is described in Chapter 13 of this volume and will not be discussed in detail here. From a physical properties perspective, the zones of variable saturation above the water table are transitional, and depend upon the soil or rock type and the lateral heterogeneity of the materials. In the vertical direction, the boundaries between all of these zones are dependent upon the types of soil, regolith, or rock that are present; the current and historical climatic conditions; and the regional and local geomorphology of the site. These same factors affect the heterogeneity of the vadose zone in the horizontal (lateral) direction and generally compound the problems of defining the different regions of moisture in the vadose zone.