Efficient Algorithm of Contraction High-Order Born Approximation on LWD Ultra-Deep Resistivity Measurement in 3-D Anisotropic Formation

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.