Two‐dimensional induction log modeling using a coupled‐mode, multiple‐reflection series method

Abstract

In induction logging, both vertical and radial conductivity variations exist in the logged medium; elementary separation of variables cannot be applied directly to the partial differential equation describing the electromagnetic field in such a medium. A modal expansion method of solution can be designed using modes defined with respect to the vertical coordinate. This method breaks the original two‐dimensional (2-D) problem into a set of independent one‐dimensional (1-D) problems. The receiver signal is decomposed into a direct signal traveling within the borehole and a reflection signal bearing information about the conductivity distribution outside the borehole. The solution is obtained as a coupled‐mode, multiple‐reflection series accounting for repeated mode reflections at the vertical boundaries in the medium. Each term in the series represents the next higher order reflection in a sequence of multiple reflections; results show two terms are sufficient to reach accuracy acceptable for log modeling.

Pseudo-Analytical Modeling for Electromagnetic Well-Logging Tools in Complex Geophysical Formations

“…This is a drawback of the A-NMM, as pointed out in [46]. In [8,27,28,46], an alternative NMM approach was proposed to circumvent this issue in LWD applications. As both transmitting and receiving LWD antennas are placed around a metallic mandrel inside the borehole and typically have the same radius, they are located in the same radial layer regardless of the axial position of the tool in the logging well.…”

“…Also, the radial position of the layers can be independent defined in each waveguide. In contrast others formulations, the radial mode-matching approach presented in [6,8,27,46] requires the number of layer and its radial positions shall be equal for all axial regions.…”

“…In order to simplify the notation, we will temporarily drop the modal subscript np and also the argument of e z,np and h z,np , restoring them later as needed. The radially-dependent coefficients in are given by a linear combination of first-kind Hankel and Bessel function of integer order n. In a compact notation, we can write them as 27) with G n representing H (1) n or J n . The field amplitudesā andb are vectors 2 × 1 to be determined from the boundary conditions, and are given bȳ…”

“…An alternative way to solve three-dimensional (3D) problems is to employ the numerical mode-matching (NMM) techniques, which provides a middle-ground in terms of computational costs between brute-force and pseudo-analytical techniques [6,8,10,15,[22][23][24][25][26][27][28][29][30][31][32]. Typically, the NMM combines the FD or FE method in one coordinate direction with an analytical (modal) expansion in the others transversal directions.…”

Numerical Mode-Matching Method for Tilted-Coil Antennas in Cylindrically Layered Anisotropic Media With Multiple Horizontal Beds

“…These eigenmodes are then matched in the vertical direction. In contrast, [30]- [32] use sinusoidal basis function to represent vertical eigenmodes, which are then matched in the radial direction. An extension of NMM for 3-D formations has been recently considered in [29] based on a 2-D eigenmode expansion and matching in the vertical direction.…”