[1] The integral equation method has been proven to be an efficient tool to model threedimensional electromagnetic problems. Owing to the full linear system to be solved, the method has been considered effective only in the case of models consisting of a strongly limited number of cells. However, recent advances in matrix storage and multiplication issues facilitate the modeling of horizontally large structures. Iterative methods are the most feasible techniques for obtaining accurate solutions for such problems. In this paper we demonstrate that the convergence of iterative methods can be improved significantly, if the original integral equation is replaced by an equation based on the modified Green's operator with the norm less or equal to one. That is why we call this technique the Contraction Integral Equation (CIE) method. We demonstrate that application of the modified Green's operator can be treated as a preconditioning of the original problem. We have performed a comparative study of the convergence of different iterative solvers applied to the original and contraction integral equations. The results show that the most effective solvers are the BIGGSTAB, QMRCGSTAB, and CGMRES algorithms, equipped with preconditioning based on the CIE method.

The quasi‐linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through an electrical reflectivity tensor λ⁁. In the original formulation of the quasi‐linear approximation, λ⁁ was determined by solving a minimization problem based on an integral equation for the scattering currents. This approach is much less time‐consuming than the full integral equation method; however, it still requires solution of the corresponding system of linear equations. In this paper, we present a new approach to the approximate solution of the integral equation using λ⁁ through construction of quasi‐analytical expressions for the anomalous electromagnetic field for 3-D and 2-D models. Quasi‐analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures. In the last sections of this paper, we extend the quasi‐analytical method using iterations and develop higher order approximations resulting in quasi‐analytical series which provide improved accuracy. Computation of these series is based on repetitive application of the given integral contraction operator, which insures rapid convergence to the correct result. Numerical studies demonstrate that quasi‐analytical series can be treated as a new powerful method of fast but rigorous forward modeling solution.

In this paper we address one of the most challenging problems of electromagnetic (EM) geophysical methods: three-dimensional (3D) inversion of EM data over inhomogeneous geological formations. The difficulties in the solution of this problem are twofold. On the one hand. 3D EM forward modelling is an extremely complicated and time-consuming mathematical problem itself. On the other hand, the inversion is an unstable and ambiguous problem. To overcome these difficulties we suggest using, for forward modelling, the new quasi-analytical (QA) approximation developed recently by Zhdanov et al (Zhdanov M S, Dmitriev V I, Fang Sand Hursan G 1999 Geophvsics at press). It is based on ideas similar to those developed by Habashy et al (Habashy T M, Groom R Wand Spies B R 1993 J. Geophvs. Res. 98 1759-75) for a localized nonlinear approximation, and by Zhdanov and Fang (Zhdanov M S and Fang S 1996a Geophysics 61646-65) for a quasi-linear approximation. We assume that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through a scalar electrical reflectivity coefficient. which is a function of the background geoelectrical cross-section and the background EM field only. This approach leads to construction of the QA expressions for an anomalous EM field and for the Frechet derivative operator of a forward problem. which simplifies dramatically the forward modelling and inversion. To obtain a stable solution of a 3D inverse problem we apply the regularization method based on using a focusing stabilizing functional introduced by Portniaguine and Zhdanov (Portniaguine 0 and Zhdanov M S 1999 Geophvsics 64 874-87). This stabilizer helps generate a sharp and focused image of anomalous conductivity distribution. The inversion is based on the re-weighted regularized conjugate gradient method.

Three‐dimensional electromagnetic inversion continues to be a challenging problem in electrical exploration. We have recently developed a new approach to the solution of this problem based on quasi‐linear approximation of a forward modeling operator. It generates a linear equation with respect to the modified conductivity tensor, which is proportional to the reflectivity tensor and the complex anomalous conductivity. We solved this linear equation by using the regularized conjugate gradient method. After determining a modified conductivity tensor, we used the electrical reflectivity tensor to evaluate the anomalous conductivity. Thus, the developed inversion scheme reduces the original nonlinear inverse problem to a set of linear inverse problems. The developed algorithm has been realized in computer code and tested on synthetic 3-D EM data. The case histories include interpretation of a 3-D magnetotelluric survey conducted in Hokkaido, Japan, and the 3-D inversion of the tensor controlled‐source audio magnetotelluric data over the Sulphur Springs thermal area, Valles Caldera, New Mexico, U.S.A.

Gas shales are economically viable hydrocarbon prospects that have proven to be successful in North America. Unlike conventional hydrocarbon prospects, gas shales serve as the source, seal, and the reservoir rock. Generating commercial production from these unique lithofacies requires stimulation through extensive hydraulic fracturing. The absence of an accurate petrophysical model for these unconventional plays makes the prediction of economic productivity and fracturing success risky.This paper presents an integrated approach to petrophysical evaluation of shale gas reservoirs, specifically, the Barnett Shale from the Fort Worth basin is used as an example. The approach makes use of different formation evaluation data, including density, neutron, acoustic, nuclear magnetic resonance, and geochemical logging data. This combination of logging measurements is used to provide lithology, stratigraphy and mineralogy. It also differentiates source rock intervals, classifies depositional facies by their petrophysical and geomechanical properties, and quantifies total organic carbon. The analysis is also employed to locate optimal completion intervals, zones preferable for horizontal sections, and intervals of possible fracture propagation attenuation. Resistivity image analysis complements the approach with the identification of natural and drilling induced fractures. We compare results from three different wells to show the effectiveness of the method for shale gas characterization.The methodology presented provides a means to understand the geomechanical and petrophysical properties of the Barnett Shale. This knowledge can be used to design a selective completion strategy that has the potential to reduce fracturing expenses and optimize well productivity. Though developed specifically for the Barnett Shale, the underlying ideas are applicable to other thermogenic shale gas plays in North America.