Efficient 2.5D electromagnetic modeling using boundary integral equations

Abstract

Fast and accurate simulation of responses of logging-while-drilling (LWD) electromagnetic (EM) tools in complex 2D and 3D formations is very important for reconstruction of the resistivity distribution in proactive geosteering. Currently, real-time interpretation is based on the 1D parametric inversion. Advances in fast simulation beyond the 1D model would open a way for real-time 2D inversion. We developed, implemented numerically, and tested an efficient method for simulation of LWD EM tools in complex 2D formations with an arbitrary 3D position of the tool. The method is based on the boundary integral equations for the tangential components of the field and the Fourier transform, which reduces the problem to a series of 1D integral equations. Computations are carried out simultaneously for the whole set of measurement points with the same matrix, which provides a short computation time per point. We verified the method by comparing our results with those obtained by the well-known explicit 1D method and by commercial software in the 2D case. Numerical results justified that the method is accurate and time efficient. Also, as an example, we simulated the signals of real propagation and azimuthal resistivity tools for a complex 2D formation model with fault. The software based on our method can be useful in planning geosteering jobs in complex formations.

Numerical simulation and dimension reduction analysis of electromagnetic logging while drilling of horizontal wells in complex structures

“…Note that the 2.5D finite difference scheme and matrix assembly process are similar to the 3D method except that ∇ = ( x, y, z) should be replaced by ∇ = x, ik y , z . Finally, the field counterparts in spatial domain can be obtained by applying the IFT to the solved EM fields in spectral domain (Dyatlov et al 2015):…”

“…For some geological structures, the rock properties are continuous in strike but discontinuous in lateral direction within the well logging scope. Therefore, the structures like folds and faults can normally be simplified to 2D models in LWD modeling and simulated effectively and efficiently by the 2.5D algorithm (Dyatlov et al 2015;Rodríguez-Rozas et al 2018;Thiel et al 2018). The basic principle of the 2.5D algorithm is to apply the Fourier transform to convert a 3D problem in spatial domain into a series of 2D problems in spectral domain, solve the fields in spectral domain and convert it back to the spatial domain through the inverse Fourier transform (Xu and Li 2018;Liu et al 2012;Wu et al 2019).…”

A quadrature-free method for simulation and inversion of 1.5D direct current (DC) borehole measurements

“…We start by splitting the electrostatic potential u into a sum of a primary field u p and a secondary field u s . The primary field is a known potential satisfying (5), and the secondary field is an unknown potential satisfying the homogeneous equation:…”

“…In this section, we compare computational times for the classical semi-analytical method (CSAM) vs. those obtained with the proposed EIHTM. For the CSAM, we select a computational time of a single evaluation equal to 0.001 s, which is 10 times smaller than the computational time referenced for the AC case in [5] and (in average) 4.8 times smaller than the time required by using the integral function of Matlab, which we know is suboptimal.…”

Fast inversion of logging-while-drilling resistivity measurements acquired in multiple wells

“…More recently, Ijasan et al (2014) introduced a layer-based inversion workflow to estimate layer-by-layer petrophysical properties. Dyatlov et al (2015) developed a method for simulation of LWD measurements in 2D formations based on the boundary integral equations and the Fourier transform, which reduces the problem to a series of 1D integral equations. He et al (2015) presented a new analytic method to compute the response of tri-axial tools in layered formations.…”