Diffraction of axisymmetric waves in a borehole by bed boundary discontinuities

Abstract

ABTRACTThis paper presents the calculation of the diffraction of axisymmetric borehole waves by bed boundary discontinuities. The bed boundary is assumed to be horizontal and the inhomogeneities to be axially symmetric. In such a geometry, an axially symmetric source will produce only axially symmetric waves. Since the borehole is an open structure, the mode spectrum consists of a discrete part as well as a continuum. The scattering of a continuum of waves by bed boundaries is difficult to treat. The approach used in the past in treating this class of problem has been approximate in nature, or highly numerical, such as the finite-element method. We present here a systematic way to approximate the continuum of modes by discrete modes. After discretization, the scattering problem can be treated simply. Since the approach is systematic, it allows derivation of the solution to any desired degree of accuracy in theory; but in practice, it is limited by the computational resources available. We also show that our approach is variational and satisfies both the reciprocity theorem and energy conservation.

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

“…Combining the above results when we take the projection of equation (3)(4)(5)(6)(7)(8)(9)(10)(11) ontoẑ allow us to find the LHS of (3-10):…”

“…Logging-while-drilling (LWD) tools, as that illustrated in Fig. 2.1, are commonly modeled using a mode-matching-based formulation [10,31,43], whereby a finite-element (FE) discretization is used along each horizontal layer of a cylindrical stratified medium. In the vertical direction, an analytical modematching procedure is then used to match the fields between each horizontal layer.…”

“…3.1. This geometry can be well approximated as a cylindrically stratified medium [10,14]. In this case, pseudo-analytical methods [10,14,29,[47][48][49][50] are a good alternative to brute-force methods as the former provide more efficient algorithms and a more practical forward engine for integration into inversion algorithms designed to estimate the Earth formation properties given the tool response [26,48,49].…”

“…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

“…Unlike the NMM approach in [28] and [40], the number of transversal eigenmodes required to handle the problem is typically not larger than 45, but for problems requiring about these many modes, the ratio of the maximum and minimum eigenvalues can reach 10 6 -10 7 , leading to ill-conditioning. For the local reflection/transmission coefficients in (44)- (47), we factor out the diagonal matrices J andH (which can be inverted easily) from the inverse matrices.…”