Three‐dimensional induction logging problems, Part I: An integral equation solution and model comparisons

Abstract

A 3‐D frequency‐domain solution based on a volume integral equation approach has been implemented to simulate induction log responses. In our treatment of the problem, we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy. The solution process uses a Krylov subspace iteration to solve the scattering equation, which is based on the modified iterative dissipative method. Internal consistency checks and comparisons with mode matching and finite‐difference solutions for vertical borehole models demonstrate the accuracy of the solution. There are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries. To simulate such responses requires the numerical solution of Maxwell's equations in three dimensions along with independent tests to validate the solution approach and its accuracy. In this paper, we compare two independent 3‐D frequency‐domain solutions for the problem, based on finite differences and the integral equation technique. Specific examples in the quasi‐static limit are studied, including a 45° deviated borehole that intersects formation bed boundaries as well as cases where the bedding exhibits transverse anisotropy. All comparisons made in this paper show very good agreement and demonstrate, for the first time, verifiable induction log responses in the presence of deviated boreholes. We also show that responses arising from deviated boreholes can be significant and must be accounted for properly when interpreting induction logs.

The role of 3-D geomagnetic induction in the determination of the ionospheric currents from the ground geomagnetic data

“…In this investigation, a combination of a field separation technique (Pulkkinen et al, 2003b) with a volume-3-D modeling of geomagnetic induction (Avdeev et al, 2002) was introduced. The combination was applied using a unique data set of geomagnetic recordings and a realistic model of the Earth's 3-D conductivity .…”

“…A volume-3-D induction code by Avdeev et al (2002), combined with the new field separation technique introduced by Pulkkinen et al (2003b), provide efficient tools for the treatment given in the paper. The four-step modeling procedure is as follows:…”

An Approximate 3D Computational Method for Real-time Computation of Induction Logging Responses

“…The computation of the response of a logging tool in a 3D-inhomogeneous medium requires a full 3D code based on the Maxwell equations. Although these codes, e.g., contrast-type of integral-equation methods (e.g., Avdeev et al, 2002;Zhang and Liu, 2003;Avdeev and Knizhnik, 2009;Nie et al, 2013), finite-element methods (e.g., Nam et al, 2013), and finitedifference methods (e.g., Newman and Alumbaugh, 2002;Weiss and Newman, 2002;Davydycheva et al, 2003Davydycheva et al, , 2009, are nowadays available or becoming available, the computational burden is too large to carry out computations for different realizations of borehole trajectory and realistic earth models in a time-efficient matter. Hence, an effective approximate model that includes all the necessary physics is required.…”

Electromagnetic Source Equivalence and Extension of the Complex Image Method for Geophysical Applications

“…Using this approach one avoids analytically usually non-tractable and numerically very time-consuming integration of the equations associated with the electromagnetic induction in the ground (for various numerical techniques as applied to different electromagnetic problems, see e.g. [10][11][12][13]). With simplified source geometries this so-called Complex Image Method (CIM) leads to a great advantage in time-critical applications such as space weather forecasts.…”