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This paper presents a rapidly direct algorithm (RDA) to calculate explicit Fréchet derivatives (EFD) for marine controlled-source electromagnetic measurement (MCSEM) in a three-dimensional (1-D) transversely isotropic (TI) formation. By discretizing the Helmholtz equations about coupled potentials on Yee’s staggered grids by the 3-D finite volume method (FVM), we obtain a complex linear system about the unknown potentials excited by a mass of moving electric current sources. To efficiently determine electromagnetic (EM) fields, we introduce an interpolation operator and projection operator per receiver by using the direct solver PARDISO and 3-D Newtown interpolation. Based on this, the perturbation in goal conductivity is expressed as a piece-wise constant function according to block or pixel model. The spatial scattered electric currents will be decomposed into a series of electric current elements distributed on Yee’s grids due to the conductivity perturbation. We then discretize the scattered electric currents by 3-D FVM and obtain the new right-hand terms about the unknown scattered potentials. This allows for fast production of the linear relationship between the changes in the EM fields and the relative conductivity perturbation per block or pixel, ultimately resulting in the EFD of MCSEM responses. Numerical results demonstrate the efficiency and accuracy of this method. The 3-D pixel sensitivities are presented to further investigate the response characteristics of MCSEM in several cases.

In cylindrical multilayered medium, we develop two tool models for electromagnetic (EM) well logging: a metal mandrel winding with a co-axial (tilted) transmitter coil and a tilted (coaxial) receiver coil. The voltages on receivers in those two models are proven to be the same and only zero-order harmonic of EM field need to be considered for voltage calculation. To calculate the voltage, two pseudo-analytical formulae are presented by using the integral of electrical field in spatial and wavenumber domain, respectively. Those two alternative pseudo-analytical formulae can be used to verify calculation accuracy of each other in some scenarios when other numerical methods do not work well. Furthermore, the reflection coefficients of EM fields, consisting of the ratios of the cylindrical functions, are introduced to avoid overflow problems in the numerical integral. Numerical examples corroborate the correctness and stability of the proposed formulae. These formulae help advance the forward modeling and inversion of logging-while-drilling (LWD) azimuthal resistivity measurements and new extra-deep azimuthal resistivity tools (EDAR).

In this paper, we set up an efficient Fréchet derivative algorithm of inversion of multicomponent induction logging (MCIL) data in horizontal layered transversely isotropic (TI) formations based on transmission line method (TLM) in order to simultaneously reconstruct the model vector including both horizontal and vertical conductivities, horizontal interfaces, borehole dipping angle, and tool azimuth in deviated well from the MCIL data. First, MCIL responses in the TI model are efficiently determined by both the spectrum EM fields obtained by TLM and the semianalytical computation of Sommerfeld integrals based on the cubic spline interpolation. Then, the Born approximations are executed to derive the MCIL Fréchet derivatives as the sixfold integrals of the products of two spectrum EM fields in infinite domains. By using the integral characters of the 2-D Dirac function, the sixfold integrals of Fréchet derivatives are further simplified into twofold integrals: One is the Sommerfeld integral in the radial spectrum domain, and the other is the definite integral in the vertical spatial domain. They are efficiently computed by the semianalytical approach similar to the MCIL simulation. Therefore, we obtain the linear equations between changes in the MCIL responses and small perturbations in the model vector. After that, we iteratively modify all of the model parameters to realize the best fit between the input data and the synthetic data of the inverted model by the normalization of the Fréchet derivative and singular value decomposition technique. Finally, we apply numerical results to investigate the characteristics of the Fréchet derivatives and to validate the inversion method and its antinoise ability. [ABSTRACT FROM PUBLISHER]

In this paper, we advance a fast iterative inversion algorithm of multicomponent induction well-logging (MCIL) datum to simultaneously reconstruct a total model vector in a deviated well in the horizontally stratified transversally isotropic (TI) formation (TI medium) with the case-neglecting borehole. The model vector consists of the horizontal and vertical conductivities, the interface depth of each bed, and a borehole dipping angle. Because an electromagnetic field emitted by a horizontal magnetic dipole is no longer axially symmetrical, we first decompose the horizontal magnetic dipole located at borehole axis as the sum of two harmonics only so that the computation of MCIL responses can be entirely transformed into three axially symmetrical problems. Therefore, we solve them through the numerical mode matching method and obtain the semianalytic (SA) solution of an EM field in an arbitrary bed. Then, based on the perturbation principle and spatial distribution of formation conductivity related to the model vector, we set up an efficient SA algorithm of Fréchet derivatives of the MCIL response with respect to the model vector. Then we extract an initial model from synthetic logs for the inversion through a special blocking technique. Finally, we give an iterative process to simultaneously reconstruct all model parameters from the logs through a combination of a normalization and singular value decomposition technique. Numerical tests validate the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient algorithm to simulate the response of a multicomponent induction well-logging (MCIL) tool in a cylindrically layered anisotropic formation (transversely isotropic media). The tool consists of three pairs of orthogonally arranged transmitter-receiver coils for measuring the magnetic field around a borehole for oil and gas exploration. First, we will derive the EM field components to present the coupled transverse magnetic and transverse electric modes in the spectral domain for an arbitrary observation layer from the Maxwell equation and obtain the spectral-domain Green's function. The new local transmission and both the local and generalized reflections for the media are given. After that, we use a cubic spline interpolation to realize the inverse Fourier transformation and obtain the spatial-domain Green's function for the investigation of the response characteristics of the MCIL tool. Finally, we give some numerical results to explore the response characteristics of the tool in several different environments. [ABSTRACT FROM AUTHOR]

The new multiarray induction logging tool consists of eight different spacing three-coil arrays, operates at three frequencies, and measures both in-phase and quadrature signals to provide more information on the distribution of conductivity around the borehole than the conventional induction tool. A horizontally layered medium with a step-profile invasion per bed is used to describe the formation model. In this paper, we will establish a regularized iterative inversion algorithm to simultaneously reconstruct geometric parameter and resistivity per bed from the multiarray induction logging data. During the inversion, the Fréchet derivative matrix with respect to the model parameters is efficiently calculated in terms of the perturbation principle, and the hybrid approach of Maxwell's equations and the two-order Langrange function is used to determine the eigenmode solution in the radial direction in order to further enhance efficiency of forward modeling and calculation of the Fréchet derivative matrix. Normalizations are used to transform the model vector, log data, and the Fréchet matrix into dimensionless variables. Regularization and exponential damping factors are used to enhance inversion stability. We will also analyze and reduce the inversion errors originating from the noise in input data and the errors in bed thicknesses. Synthetic data tests show that inversions are sensitive to errors in the boundary position per bed. These difficulties are largely alleviated when all the model parameters are simultaneously reconstructed. This is true even if starting values of boundary depth are not very close to the exact model and for the input logs that are noise. [ABSTRACT FROM AUTHOR]