ELECTRICAL logging-while-drilling (LWD) tools are

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1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 2, FEBRUARY Cylindrical FDTD Analysis of LWD Tools Through Anisotropic Dipping-Layered Earth Media Hwa Ok Lee and Fernando L. Teixeira, Senior Member, IEEE Abstract Electrical logging-while-drilling (LWD) tools are commonly used in oil and gas exploration to estimate the conductivity (resistivity) of adjacent earth media. In general, earth media exhibit anisotropic conductivities. This implies that when LWD tools are used for deviated and horizontal drilling, the resulting borehole problem may include dipping-layered media with dipping beds having full 3 3 conductivity tensors. To model this problem, we describe a 3-D cylindrical finite-difference timedomain (FDTD) algorithm extended to fully anisotropic conductive media and implemented with cylindrical perfectly matched layers to mimic open-domain problems. The 3-D FDTD algorithm is validated against analytical results in simple formations, showing good agreement, and used to simulate the response of LWD tools through anisotropic dipping beds for various values of anisotropic conductivities and dipping angles. Index Terms Anisotropy, borehole problems, finite-difference time-domain (FDTD) method, logging-while-drilling (LWD) tools, well logging. I. INTRODUCTION ELECTRICAL logging-while-drilling (LWD) tools are commonly used in borehole exploration for oil and gas prospection. The parameter of interest in this case is the conductivity (resistivity) of earth formations adjacent to the borehole. In general, earth formation conductivities are anisotropic, exhibiting different values along vertical and horizontal directions [1]. This can be due to geological factors such as the presence of clay and sand laminates with directionally dependent resistivities that produce macroscopic anisotropy [2], [3], or, as another example, this can be due to salt water penetrating porous fractured formations and increasing the conductivity in the direction parallel to the fracture [3], [4]. The study of the impact of anisotropy on the tool response is important for correct interpretation of measurements [5]. In vertical drilling, anisotropic formations often lead to borehole problems with layered media having uniaxial conductivity tensors in a coordinate system aligned to the borehole (tool) axis. However, in deviated and horizontal drilling, they can lead to borehole problems with dipping-layered media having full 3 3 conductivity tensors [3]. In recent years, various numerical and seminumerical methods have been developed to study the response of electrical Manuscript received July 28, 2005; revised May 18, This work was supported in part by the National Science Foundation under Grant ECS and in part by the Ohio Supercomputer Center under Grant PAS-0061 and Grant PAS The authors are with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, Ohio State University, Columbus, OH USA ( lee.@osu.edu; teixeira@ece.osu.edu). Digital Object Identifier /TGRS logging tools in complex earth media. Among them, one can cite stabilized biconjugate gradient fast Fourier transform (FFT) [6], numerical mode matching (NMM) [7], transmission line matrix [8], finite-element [9], finite-difference frequencydomain (FDFD) [2] [10], and finite-difference time-domain (FDTD) [11] [13]. FDTD is particularly attractive since it is easily implemented in inhomogeneous formations and is matrix-free, having a (low) computational complexity that scales only linearly with the number of degrees of freedom. However, standard FDTD [14] relies on a Cartesian grid and is restricted to diagonal conductivity tensors. An extension of FDTD to arbitrary anisotropic media in Cartesian grids is discussed in [15]. Extensions of FDTD to cylindrical grids are discussed, e.g., in [13], [16], and [17]. The use of cylindrical 3-D FDTD to simulate LWD tool response in eccentric borehole and isotropic dipping beds has been recently discussed in [12], [13], and [18]. In this paper, we combine these latter developments to derive a 3-D FDTD scheme in nonuniform cylindrical grids capable of modeling borehole problems in dipping layered formations withfull (3 3) anisotropic conductive beds. A 3-D cylindrical grid is used to conform to the LWD tool geometry and reduce discretization errors. Nonuniform discretization is adopted in the radial direction to minimize memory requirements. An unsplit perfectly matched layer (PML) in cylindrical coordinates [17], [19], [20] is incorporated to truncate the computational domain. This new 3-D FDTD algorithm is validated against Sommerfeld integral results in homogeneous formations [21] and NMM [22] results in layered formations and used to simulate LWD tool response through anisotropic dipping beds for different dipping angles. II. FORMULATION A. Anisotropy Conductivity The anisotropic earth conductivity [5] can be expressed in the principal (Cartesian) axis system as σ = σ h σ h 0 (1) 0 0 σ v where σ h is the horizontal conductivity and σ v is the vertical conductivity. The anisotropy ratio is defined as k = (σ h /σ v ) 1/2. During deviated or horizontal drilling, the tool axis does not coincide with the principal axis anymore, as illustrated in Fig. 1. We denote θ as the dipping angle between /$ IEEE

2 384 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 2, FEBRUARY 2007 This is a full 3 3 tensor when the principal axis and the borehole axis do not coincide, i.e., θ 0. Moreover, when θ 0, the tensor elements become functions of position even within a homogeneous layer (i.e., a layer having uniform σ v and σ h ). B. Cylindrical 3-D FDTD in Anisotropic Conductive Media A staggered 3-D FDTD cylindrical grid is employed with central finite differences (uniform grid case) for partial derivatives along each cylindrical coordinate and leap-frog update for the time derivative [17], [23]. Since nonmagnetic media are assumed, the FDTD update equations for the magnetic field components (based on Faraday s law) retain the standard form [16], [17], [23], [24]. Assuming a nonuniform grid, the general update equations of the electric field components (Ampere s law) incorporating the off-diagonal terms of the conductivity and permittivity tensors can be written as Fig. 1. Illustration of the LWD tool in a three-layer formation with an anisotropic dipping bed. (a) Geometry of the LWD tool. (b) Anisotropic dipping bed with actual (apparent) thickness. (c) Anisotropic dipping bed with true vertical thickness. the principal z axis and the tool z axis and choose a cylindrical coordinate system (ρ, φ, z) aligned to the tool axis to represent a point in space. The conductivity tensor σ in the preceding principal axis coordinates is transformed to another tensor σ in the tool s cylindrical coordinates by using a rotation matrix R(θ, φ) defined as cos θ cos φ cos θ sin φ sin θ R(θ, φ) = sin φ cos φ 0. (2) sin θ cos φ sin θ sin φ cos θ The anisotropic conductivity tensor σ in the cylindrical coordinates of the LWD tool becomes σ = R 1 (θ, φ) σ R(θ, φ) = σ ρρ σ ρφ σ ρz σ φρ σ φφ σ φz (3) σ zρ σ zφ σ zz where and σ ρρ = σ ρ cos 2 φ + σ φ sin 2 φ σ ρφ = σ ρ sin φ cos φ + σ φ sin φ cos φ σ ρz =(σ v σ h )cos φ sin θ cos θ σ φρ = σ ρ cos φ sin φ + σ φ sin φ cos φ σ φφ = σ ρ sin 2 φ + σ φ cos 2 φ σ φz = (σ v σ h )sin φ sin θ cos θ σ zρ =(σ v σ h )cos φ sin θ cos θ σ zφ = (σ v σ h )sin φ sin θ cos θ σ zz = σ h sin 2 θ + σ v cos 2 θ (4) σ ρ = σ h cos 2 θ + σ v sin 2 θ σ φ = σ h. (5) E n+1 ρ(i+1/2,j,k) E n+1 φ(i,j+1/2,k) E n+1 z(i,j,k+1/2) ( 1 H ρ i+1/2 = ( 1 t ɛ + 1 ) 1 2 σ n+1/2 z(i+1/2,j+1/2,k) Hn+1/2 z(i+1/2,j 1/2,k) φ φ(i+1/2,j,k+1/2) Hn+1/2 φ(i+1/2,j,k 1/2) z i ρ(i,j+1/2,k+1/2) Hn+1/2 ρ(i,j+1/2,k 1/2) z i z(i+1/2,j+1/2,k) Hn+1/2 z(i 1/2,j+1/2,k) ρ i+1/2 φ(i+1/2,j,k+1/2) Hn+1/2 φ(i 1/2,j,k+1/2) ρ i+1/2 + 1 ρ i [( H n+1/2 φ(i+1/2,j,k+1/2) +Hn+1/2 φ(i 1/2,j,k+1/2) 2 ( ρ(i,j+1/2,k+1/2) Hn+1/2 ρ(i,j 1/2,k+1/2) φ ( 1 + t ɛ 1 2 σ ) Eρ(i+1/2,j,k) n E n φ(i,j+1/2,k) E n z(i,j,k+1/2) ) ) )] where indexes i, j, and k denote the integer grid points and index n denote the time step. ρ i, φ, and z i represent the (nonuniform) radial, (uniform) azimuthal, and (nonuniform) vertical discretization cell sizes, respectively. In the preceding equation, a semi-implicit approximation has been used for the electric field components in conduction current term. Typically, small spatial cells are used close to the borehole, and larger grid cells are used in the outer regions. To accurately capture the skin effect, the discretization criterion for the maximum grid cell size (for the range of operation frequencies and conductivities considered here) is given by δ/6, where δ is the smallest skin depth corresponding to the (largest) conductivity in the (6)

3 LEE AND TEIXEIRA: CYLINDRICAL FDTD ANALYSIS OF LWD TOOLS 385 formation. In a cylindrical grid with nonuniform discretization along ρ and z, the time step is given by [23] t = 2 ( 1 v ρ min + c f 1 ρ min φ ) 2 + ( ) (7) 2 1 z min where ρ min, ρ min φ, and z min are the smallest space grid increments in the cylindrical coordinates, v is the maximum phase speed, and c f (Courant factor) is a parameter less than one to ensure a numerically stable update in (6). In practice, LWD tools operate in the continuous-wave mode (time-harmonic excitation at frequency f c ). For the present FDTD simulations, a ramp sinusoidal function of the form v s (t) =r(t)sin(ωt), where r(t) is a raised cosine ramp function given by { 0, t < 0 r(t) = 0.5[1 cos(ωt/2α)], 0 t αt 1, t > αt (8) with frequency f c =1/T =2MHz and ramp factor α = 0.5, is used as time-domain excitation. This excitation minimizes dc offset and high-frequency contamination [25] (due to the excitation discontinuity at t =0) that are otherwise present in a conventional sinusoidal excitation with turn-on at t =0 to accelerate convergence [13]. The FDTD yields time-domain voltages at the two receiver coil antennas that are converted to frequency domain based on the time-to-frequency transformation approach described in [13] and [26], which is more efficient than a direct FFT in this case. Due to the low-q (lossy) nature of the problem, the phase difference and amplitude ratio extracted from the time-domain results typically converge after 1.5T for the examples considered here. III. NUMERICAL RESULTS Unless mentioned otherwise, we consider an LWD tool with 4-in-radius steel mandrel inside a 5-in-radius borehole, as illustrated in Fig. 1. The LWD tool has one transmitter and two receivers consisting of 4.5-in-radius loop antennas positioned around the steel mandrel. The receivers are located at 30 and 24 in away from the transmitter along the tool axis. The frequency of operation is 2 MHz. The ratio of the frequencydomain voltage phasors at the two receivers yields a voltage amplitude ratio and phase difference, which characterize the tool response. We assume a Dirichlet boundary condition at the steel mandrel surface (assumed as perfect electric conductor). A. Apparent Resistivity: Homogeneous Anisotropic Formation Figs. 2 and 3 show the apparent resistivities R aph and R aam based on the phase difference and amplitude ratio of the receiver voltages, respectively, versus the anisotropy ratio k. Seven different dipping angles θ between the LWD tool axis and the anisotropy principal axis are considered. A homogeneous formation with no borehole is assumed in this case to allow comparison against analytical results. Both the relative permittivity and permeability are set equal to 1. Figs. 2 and 3 depict the FDTD results for σ h =0.1 S/m and σ h = Fig. 2. Apparent resistivities R aph and R aam for σ h =0.1S/m in a homogeneous anisotropic formation versus the anisotropy ratio. 0.5 S/m, respectively. The computation domain is discretized using a cylindrical grid with (N ρ,n φ,n z ) = (30, 125, 180) grid points. A nonuniform discretization is employed along the radial direction, with ρ varying from cm close to the mandrel to cm at the outer edge. Note that the maximum outermost grid cell size along the radial directtion is chosen as ( ρ max )=δ min /6, where the smallest skin depth in the formation δ min = 2/ωµσ max is associated with the largest conductivity in the formation [13]. In this example, a uniform cell size z =2.54 cm is used along the vertical direction, and a five-layer PML is inserted at the outer grid cells along the ρ and z directions. The five-layer PML in ρ direction employs a cubic profile in the real part of the PML stretching variables only [17] because imaginary streching was found to cause instabilities in the fully anisotropic case. We observe that the apparent resistivities are sensitive to both k and θ. These results show good agreement against the analytical results presented in [21], with a relative error of below 5%. The discrepancies can be attributed to FDTD discretization errors and to the fact that the analytical model used in [21] considers

4 386 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 2, FEBRUARY 2007 Fig. 4. Tilted magnetic dipole equivalent model used to construct reference NMM solutions for dipping anisotropic formations. The tilt angle of the magnetic dipole corresponds to the dipping angle of the formation. Fig. 5. Comparison of the FDTD and NMM results for an LWD tool (phase difference) crossing a three-layer formation with anisotropic bed and tilt angle equal to 15. Fig. 3. Apparent resistivities R aph and R aam for σ h =0.5 S/m in a homogeneous anisotropic formation versus the anisotropy ratio. magnetic dipole excitations, which do not include mandrel or finite-size antenna effects. B. Dipping Anisotropic Bed: FDTD Versus NMM Results We next validate the results in a three-layer formation with an anisotropic dipping bed. The FDTD results are validated against NMM [22] results. We simulate the phase difference in an LWD tool, as illustrated in Fig. 1. In this particular case, no borehole is present, and the radii of the mandrel and coil antennas is reduced to 0.5 and 1 in, respectively. This is done in order to better conform to the equivalent (infinitesimal) magnetic dipole source and formation considered in the NMM formulation. The top and bottom layers are isotropic with σ iso =10 S/m. The anisotropic bed has an actual thickness equal to 60 in, with vertical conductivity of σ v =0.5 S/m and horizontal conductivity of σ h =2.5S/m. Two different dipping angles are considered, viz., θ =15 and θ =45.TheNMM implements a tilted magnetic dipole source in a three-layer horizontal anisotropic formation, with an tilt angle equal to the dipping angle of the original formation, as illustrated in Fig. 4. The magnetic field is sampled in a direction aligned to the dipoles so as to mimic the response of the LWD tool across a dipping anisotropic formation. The FDTD and NMM results are compared in Figs. 5 and 6, showing with very good agreement. The small discrepancies can be attributed to the infinitesimal dipole approximation and the absence of mandrel effects in the NMM model. C. Anisotropic Dipping Bed: Actual Thickness We next simulate the phase difference for an LWD tool penetrating a three-layer formation, as illustrated in Fig. 1. We consider various dipping angles and assume a geometry with an actual bed thickness (or apparent thickness). In this case, the vertical bed thickness is equal to 60 in for θ =0 and increases by a factor of 1/ cos θ for larger dipping angles. Both top and bottom layers have an isotropic conductivity of σ iso = 10 S/m. The midlayer is anisotropic with horizontal conductivity of σ h =2.5 S/m and vertical conductivity of σ v =0.5 S/m. The borehole is filled with a mud fluid with conductivity equal to σ mud = S/m for an oil-based mud and σ mud =2S/m for a water-based mud. Fig. 7 shows the FDTD simulation of

5 LEE AND TEIXEIRA: CYLINDRICAL FDTD ANALYSIS OF LWD TOOLS 387 Fig. 6. Comparison of the FDTD and NMM results for an LWD tool (phase difference) crossing a three-layer formation with anisotropic bed and tilt angle equal to 45. Fig. 8. Simulation results of the LWD tool with σ mud = S/m crossing an anisotropic dipping bed (actual thickness) with σ h =2.5, σ v =0.5, and σ iso =10S/m. σh σ v. This helps explain the observed behavior as the dipping angle increases. Fig. 8 depicts the simulation results of the same formation now with oil-based mud. As expected, the overall characteristics for σ mud = are very similar to those for σ mud = 2 S/m, except for very large dipping angles, where the results become more sensitive to the mud properties and the horn effect is more pronounced. Fig. 7. Simulation results of the LWD tool with σ mud =2 S/m crossing an anisotropic dipping bed (actual thickness) with σ h =2.5, σ v =0.5, and σ iso =10S/m. the logging response when σ h, σ v, and σ iso are chosen to be equal to 2.5, 0.5, and 10 S/m, respectively. The conductivity of the mud is σ mud =2 S/m. The computation domain is discretized using (N ρ,n φ,n z ) = (50, 127, 280) grid points. In the ρ direction, ρ varies from to cm. From Fig. 7, we note that as the dipping angle increases, the apparent thickness of the anisotropic dipping bed is widened due to both the outward slower transition to a larger σ iso at the bed interfaces and the larger actual thickness. Moreover, the horn effect at the interfaces increases for larger dipping angles. Within the anisotropic bed depth, the phase difference is reduced for larger dipping angles. Note that in a vertical well, the apparent resistivity depends only on σ h, while in a horizontal well, the apparent resistivity depends on the geometric average D. Anisotropic Dipping Bed: True Vertical Thickness We consider the same tool geometry as in the previous case, but we now fix the vertical thickness (with respect to the tool axis) of the anisotropic bed at 60 in for the various dipping angles. In this way, the geometric effect (secant factor) that increases the actual thickness for larger dipping angles is artificially supressed. Any increase on the apparent thickness in this case can be solely attributed to bed boundary effects. Note that this does not corresponds to scaling the previous logging results (apparent thickness) by a secant factor because the tool length is kept invariant here. The conductivities in the formation are the same as before, and the mud conductivity is σ mud = S/m. The results are shown in Fig. 9. Compared with Fig. 8, this results shows a smaller apparent thickness and larger horn effects for large dipping angles, as expected. IV. SUMMARY We have described a 3-D cylindrical FDTD simulation of LWD logging tools through dipping formations with anisotropic conductivites. For this end, the FDTD algorithm has been extended to model full 3 3 anisotropic conductive media in cylindrical coordinates, incorporating a cylindrical PML that absorbs boundary conditions. This new FDTD scheme was validated against analytical results in homogeneous formations and NMM results in layered formations, showing very good

6 388 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 2, FEBRUARY 2007 Fig. 9. Simulation results of the LWD tool with σ mud = S/m crossing an inhomogeneous anisotropic dipping bed (true vertical thickness) with σ h =2.5, σ v =0.5, andσ iso =10S/m. agreement. The FDTD model was further used to verify the impact of different dipping angles on the response of LWD tools through anisotropic dipping beds. ACKNOWLEDGMENT The authors would like to thank Y.-K. Hue (Ohio State University) for providing the NMM results, L. San Martin (Halliburton Energy Services) for suggestions, and the anonymous reviewers for useful comments. REFERENCES [1] J. D. Klein, P. R. Martin, and D. F. Allen, The petrophysics of electrically anisotropic reservoirs, Log Anal., vol. 38, no. 3, pp , [2] T. Wang and S. Fang, 3-D electromagnetic anisotropy modeling using finite differences, Geophysics, vol. 66, no. 6, pp , [3] C. J. Weiss and G. A. Newman, Electromagnetic induction in a fully 3-D anisotropic earth, Geophysics, vol. 67, no. 4, pp , [4] B. Anderson, S. Bonner, M. G. Luling, and R. Rosthal, Response of 2-MHz LWD resistivity and wireline induction tools in dipping beds and laminated formations, in Proc. 31st Annu. SPWLA Log Symp. Dig.,1990, pp. A1 A25. [5] J. H. Moran and S. Gianzero, Effects of formation anisotropy on resistivity-logging measurements, Geophysics, vol. 44, no. 7, pp , Jul [6] Z. Q. Zhang and Q. H. Liu, Applications of the BCGS-FFT method to 3-D induction well logging problems, IEEE Trans. Geosci. Remote Sens., vol. 41, no. 5, pp , May [7] G.-X. Fan, Q. H. Liu, and S. P. Blanchard, 3-D numerical mode-matching (NMM) method for resistivity well-logging tools, IEEE Trans. Antennas Propag., vol. 48, no. 10, pp , Oct [8] Y. Zhang, C. C. Liu, and L. C. Shen, The performance evaluation of LWD logging tools using magnetic and electric dipoles by numerical simulations, IEEE Trans. Geosci. Remote Sens.,vol.34,no.4, pp , Jul [9] B. Anderson, Simulation of induction logging by the finite-element method, Geophysics, vol. 49, no. 11, pp , [10] S. Davydycheva, V. Druskin, and T. Habashy, An efficient finitedifference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media, Geophysics, vol. 68, no. 5, pp , [11] S. Liu and M. Sato, Electromagnetic logging technique based on borehole radar, IEEE Trans. Geosci. Remote Sens., vol. 40, no. 9, pp , Sep [12] Y.-K. Hue, F. L. Teixeira, L. E. San Martin, and M. Bittar, Modeling of EM logging tools in arbitrary 3-D borehole geometries using PML-FDTD, IEEE Geosci. Remote Sens. Lett., vol. 2, no. 1, pp , Jan [13], Three-dimensional simulation of eccentric LWD tool reponse in boreholes through dipping formations, IEEE Trans. Geosci. Remote Sens., vol. 43, no. 2, pp , Feb [14] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media, IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp , May [15] J. Schneider and S. Hudson, The finite-difference time-domain method applied to anisotropic material, IEEE Trans. Antennas Propag., vol. 41, no. 7, pp , Jul [16] J.-Q. He and Q. H. Liu, A nonuniform cylindrical FDTD algorithm with improved PML and quasi-pml absorbing boundary conditions, IEEE Trans. Geosci. Remote Sens., vol. 37, no. 2, pp , Mar [17] F. L. Teixeira and W. C. Chew, Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media, IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp , Jul [18] T. Wang and J. Sigorelli, Finite difference modeling of electromagnetic tool response for logging drilling, Geophysics, vol. 69, no. 1, pp , [19] F. L. Teixeira and W. C. Chew, Systematic derivation of PML absorbing media in cylindrical and spherical coordinates, IEEE Microw. Guided Wave Lett., vol. 7, no. 11, pp , Nov [20], Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves, Microw. Opt. Technol. Lett., vol. 17, no. 4, pp , [21] A. Q. Howard, Jr., Petrophysics of magnetic dipole fields in an anisotropic earth, IEEE Trans. Antennas Propag., vol. 48, no. 9, pp , Sep [22] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, [23] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method. Boston, MA: Artech House, [24] F. L. Teixeira and W. C. Chew, Lattice electromagnetic theory from a topological viewpoint, J. Math. Phys., vol. 40, no. 1, pp , [25] C. M. Furse, D. H. Roper, and D. N. Buechler, The problem and treatment of DC offsets in FDTD simulations, IEEE Trans. Antennas Propag., vol. 48, no. 8, pp , Aug [26] C. M. Furse, Faster than Fourier: Ultra-efficient time to frequencydomain conversions for FDTD simulations, IEEE Antennas Propag. Mag., vol. 42, no. 6, pp , Dec Hwa Ok Lee received the M.S. degree in electrical engineering from The Ohio State University, Columbus, in She is currently working toward the Ph.D. degree at The Ohio State University. Her current research interests include computational electromagnetics and numerical methods. Fernando L. Teixeira (S 89 M 93 SM 04) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, Urbana, in From 1999 to 2000, he was a Postdoctoral Research Associate with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge. Since 2000, he has been with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, Ohio State University Columbus, where he is currently an Associate Professor. He has coauthored more than 70 journal articles and book chapters. His current research interests include analytical and numerical techniques for wave propagation and scattering problems in communication, sensing, and devices applications. Dr. Teixeira is a member of Phi Kappa Phi and Sigma Xi, and a Full Member of the International Union of Radio Science (URSI) Commission B. He was a recipient of many prizes for his research including the NSF CAREER Award in 2004 and the triennial USNC/URSI Booker Fellowship in 2005.