Interface-Flattening Transform for EM Field Modeling in Tilted, Cylindrically-Stratified Geophysical Media
📝 Abstract
We propose and investigate an “interface-flattening” transformation, hinging upon Transformation Optics (T.O.) techniques, to facilitate the rigorous analysis of electromagnetic (EM) fields radiated by sources embedded in tilted, cylindrically-layered geophysical media. Our method addresses the major challenge in such problems of appropriately approximating the domain boundaries in the computational model while, in a full-wave manner, predicting the effects of tilting in the layers. When incorporated into standard pseudo-analytical algorithms, moreover, the proposed method is quite robust, as it is not limited by absorption, anisotropy, and/or eccentering profile of the cylindrical geophysical formations, nor is it limited by the radiation frequency. These attributes of the proposed method are in contrast to past analysis methods for tilted-layer media that often place limitations on the source and medium characteristics. Through analytical derivations as well as a preliminary numerical investigation, we analyze and discuss the method’s strengths and limitations.
💡 Analysis
We propose and investigate an “interface-flattening” transformation, hinging upon Transformation Optics (T.O.) techniques, to facilitate the rigorous analysis of electromagnetic (EM) fields radiated by sources embedded in tilted, cylindrically-layered geophysical media. Our method addresses the major challenge in such problems of appropriately approximating the domain boundaries in the computational model while, in a full-wave manner, predicting the effects of tilting in the layers. When incorporated into standard pseudo-analytical algorithms, moreover, the proposed method is quite robust, as it is not limited by absorption, anisotropy, and/or eccentering profile of the cylindrical geophysical formations, nor is it limited by the radiation frequency. These attributes of the proposed method are in contrast to past analysis methods for tilted-layer media that often place limitations on the source and medium characteristics. Through analytical derivations as well as a preliminary numerical investigation, we analyze and discuss the method’s strengths and limitations.
📄 Content
1 Interface-Flattening Transform for EM Field Modeling in Tilted, Cylindrically-Stratified Geophysical Media Kamalesh Sainath and Fernando L. Teixeira Abstract—We propose and investigate an “interface-flattening” transformation, hinging upon Transformation Optics (T.O.) tech- niques, to facilitate the rigorous analysis of electromagnetic (EM) fields radiated by sources embedded in tilted, cylindrically- layered geophysical media. Our method addresses the major challenge in such problems of appropriately approximating the domain boundaries in the computational model while, in a full- wave manner, predicting the effects of tilting in the layers. When incorporated into standard pseudo-analytical algorithms, moreover, the proposed method is quite robust, as it is not limited by absorption, anisotropy, and/or eccentering profile of the cylindrical geophysical formations, nor is it limited by the radiation frequency. These attributes of the proposed method are in contrast to past analysis methods for tilted-layer media that often place limitations on the source and medium characteristics. Through analytical derivations as well as a preliminary numerical investigation, we analyze and discuss the method’s strengths and limitations. Index Terms—Borehole geophysics, electromagnetic analysis, well-logging, stratified media, transformation optics. I. INTRODUCTION Many applications demand the numerical evaluation of electromagnetic (EM) fields produced by radiators embedded in complex, inhomogeneous environments [1]. For example, robust (i.e., with respect to environment and source charac- teristics) computational modeling of EM sensors operating in complex geological formations is necessary both to better understand the effects of the nearby Earth formation inhomo- geneity profile and constitutive properties [2] on the sensor’s response (“forward modeling”) and to facilitate resistivity profile inversion (e.g., to assess a formation’s hydrocarbon productivity potential) [2]. The repeated use of the numerical modeler as the forward engine in many inversion methods, combined with the electrically large size of many geophysi- cal problems, requires a fast computational scheme that can rigorously model EM phenomena arising from the subsurface environment’s dominant geophysical features. Bridging the gap between robustness and solution speed, a layered-medium geophysical model is commonly employed. In particular, cylindrically-layered environments characterized by parallel, vertically-oriented (i.e., along z) interfaces arise quite frequently [1], [3]. This layered-medium approximation The authors are with the ElectroScience Laboratory (ESL), Department of Electrical and Computer Engineering, The Ohio State University (OSU), Columbus, Ohio, USA 43212 (e-mail: {sainath.1@,teixeira@ece.}osu.edu). This work was supported by the NASA-NSTRF program and by OSC under Grant PAS-0061. is oftentimes motivated (when this domain approximation, near the sensor at least, is approximately valid) because in such media one can employ pseudo-analytical methods based upon cylindrical eigenfunctions expansions. These expansions are particularly attractive since they can rigorously account for eccentered, anisotropic, and/or absorptive geophysical me- dia, as well as a broad range of radiation frequencies, with ease [1], [3]–[5]. However, it is worthwhile asking whether (and how) one can extend upon the types of media (with respect to, for example, spatial inhomogeneity structure) that can be rigorously modeled using these cylinder wave ex- pansions. In particular, removal of the vertical, parallel-layer modeling constraint, which would facilitate modeling of the presence and effects of tilted layered media, offers the distinct opportunity to investigate novel EM wave propagation and scattering phenomena that can aid interpretation of collected EM subsurface sensor data. The presence of relative vertical layer tilt can arise, for example, due to mechanical effects or gravitational pull effects either on the tool itself or on the drilling fluid (mud filtrate) invasion zone in the course of deviated drilling [5]. However, rigorous modeling of wave propagation and scattering within these types of media, without having to place undesirable constraints on the frequency range, source distribution, and material properties,1 is encumbered by a challenge on appropriately approximating the boundaries comprising tilted layers. To overcome this challenge and enable pseudo-analytical- based modeling of tilted, cylindrically-layered media, we propose and explore a strategy based on Transformation Optics (T.O.) [8]–[12] to obtain material blueprints of annular slabs to coat each of the cylindrical interfaces in a transformed, standard problem2 which contains only vertically-oriented, parallel interfaces. This interface flattening is carried out by defining a coordinate mesh/metric deformation followed by incorporating the metric deformation into the ambient material prop
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