Numerical solution of the integral equations of the electromagnetic field in geosteering problems

Abstract

Extra-deep azimuthal resistivity measurements improve the depth of investigation up to 30 m from the wellbore. Interpretation of electromagnetic logging data in the neighbourhood of a well becomes an important technical problem. We present an efficient parallel method for computation of induction tool responses with multiple transmitter–receiver configurations in 2D pixel-based anisotropic model crossed by an arbitrary well trajectory. The cornerstone of the approach is volume integral equation method. We consider the conductivity distribution as a sum of background and anomalous conductivities. Background conductivity is 1D-layered. Anomalous conductivity has arbitrary 2D distribution. Electromagnetic fields are the superposition of background and anomalous fields. Background fields are calculated exactly using rigorous analytical solution for 1D-layered background model. With this approach, the 2D pixel-based model is treated as an extension of the 1D-layered model. The anomalous fields are required only in pixels with conductivity different from the conductivity of the 1D-layered model. Anomalous fields are calculated using convergent series of integral operators [1]. The approach takes into account conductivity anisotropy and allows obtaining both the exact solution and the fast approximate one. The convergence of approximate solution is investigated on some synthetic examples.