In this article, to explore the efficiency and precision of the 3-D finite volume method (FVM) for the logging while drilling (LWD) ultradeep resistivity measurements, we compared four different schemes: field-based total-field FVM, coupled potentials total-field FVM, field-based secondary-field FVM, and coupled potentials secondary-field FVM. The fast and accurate discretization of scattered current density in the secondary-field method is another issue we focus on. On the one hand, we improve the discretization accuracy of the scattered current density near the source by extracting the direct waves in the background electric field. On the other hand, based on the dyadic Green’s functions (DGFs) of vector potentials, the number of Sommerfeld integrals in the background electric field is reduced as much as possible through the background field library and interpolation. The numerical results show that the accuracy and stability of the secondary-field method are better than those of the total-field method and the efficiency of the background electric field is greatly improved through the library and interpolation. Based on the premises of the LWD ultradeep resistivity measurements and the direct solver, the accuracy of the field-based and coupled potentials methods is almost the same, however, the field-based method is much more efficient. Overall, we believe that the field-based secondary-field FVM and the direct solver constitute a more efficient modeling scheme with high precision for LWD ultradeep resistivity measurements.

With the development of computation methods and the requirement of data processing, it is often required to execute electromagnetic (EM) simulations in a lot of different complex formation models simultaneously. For this purpose, in this article, we advance a contraction high-order Born approximation (CHBA) of scattered EM fields from arbitrary perturbation in conductivity based on the 3-D finite volume method (FVM) of coupled potentials. We manage to apply the CHBA to efficiently and precisely simulate the logging while drilling (LWD) ultra-deep resistivity measurement in multiple perturbation models based on arbitrarily given anisotropic reference models. First, from the energy conservation of the EM fields, we derive the rigorous contraction operator about the modified scattered EM fields through the variable transformations. After that, the modified scattered EM fields are expanded into an unconditionally convergent series. All terms of the series can be obtained by solving the Helmholtz equation with recursively right-hand terms. Then, we apply the relative residuals of the modified scattered EM fields to determine the truncation order of the series and acquire the reliable CHBA solution. The Helmholtz equation is discretized by the 3-D FVM and solved by the parallel direct sparse solver (PARDISO). We thus obtain the EM fields of multiple sources in the multiple perturbation models simultaneously. Finally, the numerical results validate the algorithm and compare the EM responses in multiple perturbation models.

The article advances a 3-D adaptive regularization nonlinear inversion of the logging while drilling (LWD) ultradeep multicomponent resistivity (LWD-UDMCR) by the Gauss-Newton (GN) method. We manage to reconstruct the pixel-based horizontal and vertical conductivities simultaneously in a goal domain outside an arbitrary dipping borehole. The piecewise constant functions are used to describe the spatial distribution of the block-based and the pixel-based conductivity. The background formation is assumed as the horizontally layered transversely isotropic (TI) media, and the background electromagnetic (EM) fields are determined analytically by the transmission line method (TLM). We then use the 3-D finite volume method (FVM) of secondary field coupled potentials and parallel direct sparse solver (PARDISO) to simulate the tool responses and Fréchet derivatives simultaneously. Through the projection operator and OpenMP parallel technique, we further enhance the computational efficiency of the pixel-based explicit Fréchet derivatives and set up a complete normalization linearized response. After that, the large normal equation from the quadratic objective function is solved by the preconditioned conjugate gradient (PCG) to determine the gradient of the objective function. By properly controlling the maximum component of the gradient per iteration step, we acquire an adaptive regularization factor so that the stabilization of the inversion solution is assured as well as the realization of the best fit of the input data with the modeling logs. Finally, numerical tests validate the algorithm and antinoise ability.

The article proposes an efficient global optimization of adaptive boundary detection from the logging while drilling (LWD) azimuthal electromagnetic (EM) measurements. The goal is to realize real-time geo-steering in 1-D layered transversely isotropic (TI) formation with an arbitrarily deviated borehole. The method includes apparent resistivity (APR) extraction and 0-D inversion, 1-D adaptive regularized iterative inversion, and global optimization. The APR extraction and 0-D inversion are used to quickly determine the initial horizontal and vertical resistivities of the bed where the tool is located. Subsequently, the 1-D regularized inversion is performed to achieve a local minimum solution near an arbitrarily given initial model. For solving the non-unique problem and acquiring the globally optimal solution, several different initial values are selected according to the possible range per model parameter to construct a serial of initial models. The OpenMP parallel technique is applied for simultaneous inversions at all initial models. Multiple inversion solutions may be obtained due to the non-uniqueness. The one with the minimal residual function in all inversion results will become the globally optimal solution. Furthermore, the tool response and its exact explicit Fréchet derivative with respect to each model parameter are analytically calculated, while an adaptive regularization factor ensures a gradual reduction of the objective function. The correction of field data is required to reduce the mandrel effect. The inversion results of synthetic and field data demonstrated that the proposed inversion algorithm efficiently provides the reliable bed boundaries and resistivities around the wellbore.

A new approximation algorithm is developed to compute the electromagnetic (EM) tensor Green’s function in the biaxial anisotropic media based on the Fourier series expansion. First, a rectangular region is chosen as the computation region, in which the EM field can be expressed as a 2-D Fourier series. The Fourier coefficients can be regarded as the EM field in discrete wavenumber domain. We further obtain the EM solution in the spatial domain in the form of Fourier series. Then, we use the finite terms of the Fourier series to approximate the EM field for enhancement of computation efficiency. Because the new method avoids the numerical integration in the infinite wavenumber domain, it is more convenient to implement than other methods based on integral transform. Finally, the spatial distribution of the tensor Green’s function for a transversely isotropic medium is presented to verify the proposed approach. The agreements between the results obtained by the present method and the analytic solution demonstrate the validity and the robustness of our algorithm.