SIMULATION OF RESISTIVITY LOGGING-WHILE-DRILLING (LWD) MEASUREMENTS USING A SELF-ADAPTIVE GOAL-ORIENTED HP FINITE ELEMENT METHOD.

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1 SIMULATION OF RESISTIVITY LOGGING-WHILE-DRILLING (LWD) MEASUREMENTS USING A SELF-ADAPTIVE GOAL-ORIENTED HP FINITE ELEMENT METHOD. D. PARDO,, L. DEMKOWICZ, C. TORRES-VERDÍN, AND M. PASZYNSKI, Abstract. We simulate electromagnetic (EM) measurements acquired with a Logging-While- Drilling (LWD) instrument in a borehole environment. The measurements are used to assess electrical properties of rock formations. Logging instruments as well as rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. The simulations are performed with a self-adaptive goal-oriented -Finite Element Method (FEM) that delivers exponential convergence rates in terms of the quantity of interest (for example, the difference in the electrical current measured at two receiver antennas) against the CPU time. Goal-oriented adaptivity allows for accurate approximations of the quantity of interest without the need of obtaining an accurate solution in the entire computational domain. In particular, goal-oriented -adaptivity becomes essential to simulate LWD instruments, since it reduces the computational cost by several orders of magnitude with respect to the global energy-norm based -adaptivity. Numerical results illustrate the efficiency and high-accuracy of the method, and provide physical interpretation of resistivity measurements obtained with LWD instruments. These results also describe the advantages of using magnetic buffers in combination with solenoidal antennas for strengthening the measured EM signal so that the signal-to-noise ratio is minimized. Key words. -finite elements, exponential convergence, goal-oriented adaptivity, computational electromagnetics, Maxwell s equations, through casing resistivity tools (TCRT). AMS subject classifications. 78A25, 78A55, 78M0, 65N50. Introduction. A plethora of energy-norm based algorithms intended to generate optimal grids have been developed throughout the last decades (see, for example, [7, 8] and references therein) to accurately solve a large class of engineering problems. However, the energy-norm is a quantity of limited relevance for most engineering applications, especially when a particular objective is pursued, for instance, to simulate the electromagnetic response of geophysical resistivity logging instruments in a borehole environment. In these instruments, the amplitude of the measurement (for example, the electric field) is typically several orders of magnitude smaller at the receiver antennas than at the transmitter antennas. Thus, small relative errors of the solution in the energy-norm do not imply small relative errors of the solution at the receiver antennas. Indeed, it is not uncommon to construct adaptive grids delivering a relative error in the energy-norm below % while the solution at the receiver antennas still exhibits a relative error above 000% (see []). Consequently, in order to accurately simulate LWD resistivity measurements in this paper, we develop a self-adaptive strategy to approximate a specific feature of the solution. Refinement strategies of this type are called goal-oriented adaptive algorithms [, 7], and are based on minimizing the error of a prescribed quantity of interest mathematically expressed in terms of a linear functional (see [, 9, 2,, 7, 9] for details). Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin TX 7872 Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin TX 7872 On leave from AGH University of Science and Technology, Department of Computer Methods in Metallurgy, Cracow, Poland

2 2 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI In this paper, we formulate, implement, and study (both theoretically and numerically) a self-adaptive goal-oriented algorithm intended to solve electrodynamic problems. This algorithm is an extension of the fully automatic (energy-norm based) -adaptive strategy described in [7, 8], and a continuation of concepts presented in [4, 20] for elliptic problems. We apply the self-adaptive goal-oriented algorithm to accurately simulate induction LWD instruments in a borehole environment with axial symmetry. These instruments are widely used by the geophysical logging industry, and their simulation requires resolution of EM singularities generated by the LWD geometry and rock formation materials [22], as well as resolution of high material constrasts that occur between the mandrel and the borehole. The organization of this document is as follows: In Section 2, we describe the main characteristics of induction logging instruments. We also describe our problem of interest, composed of an induction LWD instrument in a borehole environment, and used for the assessment of the rock formation electrical properties. In Section, we introduce Maxwell s equations, governing the electromagnetic phenomena and explaining the physics of resistivity measurements. We also derive the corresponding variational formulation for axisymmetric problems. A self-adaptive goal-oriented algorithm for electrodynamic problems is described in Section 4. The corresponding details of implementation are discussed in the same section. Simulations and numerical results concerning the response of LWD instruments in a borehole environment are shown in Section 5. Section 6 draws the main conclusions, and outline future lines of research. Finally, in the Appendix, we compare numerical results with a semi-analytical solution obtained using Bessel functions for a simplified LWD model problem. The comparison is intended to verify the code as well as to illustrate the high accuracy results obtained with the self-adaptive goal-oriented -FEM. 2. Alternate Current (AC) Logging Applications. In this article, we consider an induction LWD instrument operating at 2 Mhz. The instrument makes use of one of the following two types of source antennas/coils: solenoidal coils (Fig. 2., left panel), and toroidal coils (Fig. 2., right panel). 2.. Induction LWD Instruments Based on Solenoidal Coils. For axisymmetric problems, these logging instruments generate a T M φ field, i.e., the only non-zero components of the electromagnetic (EM) fields are E φ, H ρ, and H z, where (ρ, φ, z) denote the cylindrical system of coordinates. A solenoidal coil (Fig. 2.) produces an impressed current J imp that we mathematically describe as (2.) J imp (r) = ˆφIδ(ρ a)δ(z), where I is the electric current measured in Amperes (A), δ is the Dirac s delta function, and a is the radius of the solenoid. In the numerical computations, we replace function δ(ρ a)δ(z) with an approximate function U F that considers the finite dimensions of the coil, and such that U F dρdz =. The analytical electric far-field solution excited by a solenoidal coil of radius a radiating in homogeneous media is given in terms of the electric field by (see [0]) Induction logging instruments are characterized by the fact that impressed current J imp is divergence free (i.e., J imp = 0).

3 HP -FEM: ELECTROMAGNETIC APPLICATIONS Fig. 2.. Two coil antennas: a solenoid antenna (left panel) composed of a wire wrapped around a cylinder, and a toroid antenna (right panel) composed of a wire wrapped around a toroid. (2.2) E = ˆφωµkIπa 2 e jkd 4πd [ j kd ]ρ d, where k = ω 2 ɛ jωσ is the wave number, j = is the imaginary unit, ω is angular frequency, ɛ, µ, and σ stand for dielectric permittivity, magnetic permeability, and electrical conductivity of the medium, respectively, and d is the distance between the source coil and the receiver coil. In order to avoid the dependence upon the dimensions of the solenoid, we impose a current on the solenoidal coil equal to /(πa 2 ) A, i.e., equivalent to that of A with a Vertical Magnetic Dipole (VMD). The corresponding far-field solution in homogeneous media is given by (see [0]) (2.) E = ˆφωµkI e jkd 4πd [ j kd ]ρ d. Thus, solution (2.) is independent of the dimensions of the coil Induction LWD Instruments Based on Toroidal Coils. For axisymmetric problems, these logging instruments generate a T E φ field, i.e., the only nonzero components of the EM fields are H φ, E ρ, and E z. A toroidal coil induces a magnetic current I M in the azimuthal direction. If we place a toroid of radius a radiating in homogeneous media, the resulting magnetic far-field is given by (see [0]) (2.4) H = ˆφ(σ + jωɛ)πa 2 I M jk e jkd 4πd [ j kd ]ρ d. In order to avoid the dependence upon the dimensions of the toroid, we impose a magnetic current on the toroidal coil equal to that induced by a (σ + jωɛ) A electric current excitation with a Vertical Electrical Dipole (VED), also known as Hertzian 2 In resistivity logging applications, it is customary to consider solutions that have been divided by the geometrical factor (also called K-factor) [], so that results are independent (as much as possible) of the logging instrument s geometry. Thus, solutions obtained from different logging instruments can be readily compared.

4 4 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI dipole. The corresponding magnetic far-field solution in homogeneous media is given by (see [0]) () H = ˆφ(σ + jωɛ)ijk e jkd 4πd [ j kd ]ρ d. In this case, I M = I/(πa 2 ) Goal of the Computations. We are interested in simulating the EM response of an induction LWD instrument in a borehole environment. For a solenoidal coil, the main objective of our simulation is to compute the first difference of the voltage between the two receiving coils of radius a divided by the (vertical) distance z between them, i.e., (2.6) V V 2 z ( ) = E(l) dl E(l) dl l l 2 /( z) = 2πa z (E(l ) E(l 2 )), where l and l 2 are the first and second receiving coils, respectively, and l l, l 2 l 2 are two arbitrary points located at the receiving coils. Notice that due to the axisymmetry of the electric field, E(l j i ) = E(lk i ) for all lj i, lk i l i. This quantity of interest (first difference of voltage) is widely used in resistivity logging applications. Indeed, a first-order asymptotic approximation of the electric field response at low frequencies (Born s approximation) shows that the voltage at a receiver coil is proportional to the rock formation resistivity in the proximity of such a coil (see [0] for details). At higher frequencies (> 20 Khz), asymptotic approximations (see [] for details) also indicate the dependence of the voltage upon the rock formation conductivity. Thus, an adequate approximation of the rock formation conductivity (which is unknown a priori in practical applications) can be estimated from the voltage measured at the receiving coils. Computing the first difference of the voltage between two receivers (rather than the voltage at one receiver) is convenient for improving the vertical resolution of the measurements. This well-known fact among well-logging practitioners will be illustrated here with numerical experiments. For a toroidal coil, the main objective of these simulations is to compute the first difference of the electric current at the two receiving coils of radius a divided by the (vertical) distance z between them, i.e., (2.7) I I 2 z ( ) = H(l) dl H(l) dl l l 2 /( z) = 2πa z (H(l ) H(l 2 )). Notice that the main difference between a toroidal and a solenoidal coil is that the former generates an impressed magnetic current, while the latter produces an impressed electric current. This fact leads to the physical consideration that, if the voltage due to a solenoidal coil is proportional to the rock formation conductivity, then the electric current enforced by a toroidal coil is also proportional to the rock formation resistivity. Thus, the selection of the quantity of interest for toroidal coils (first difference of electric current) is dictated by the physical relation between solenoidal and toroidal coils, and the previous choice of a quantity of interest for solenoidal coils (first difference of voltage). 2.. Description of a LWD Instrument in a Borehole Environment. We consider a LWD instrument composed of the following axisymmetric materials (all dimensions are given in cm): one transmitter and two receiver coils defined on

5 HP -FEM: ELECTROMAGNETIC APPLICATIONS 5. C = {(ρ, φ, z) : 7. < ρ < 7., < z < }, 2. C2 = {(ρ, φ, z) : 7. < ρ < 7., < z < 0.25}, and,. C = {(ρ, φ, z) : 7. < ρ < 7.,.75 < z < 6.25}, respectively, three magnetic buffers with resistivity 0 4 m and relative permeability 0 4, defined on. B = {(ρ, φ, z) : < ρ < 6.985, 5 < z < 5}, 2. B2 = {(ρ, φ, z) : < ρ < 6.985, 97.5 < z < 0}, and,. B = {(ρ, φ, z) : < ρ < 6.985, < z < 7.5}, respectively, and a metallic mandrel with resistivity 0 6 m defined on M = {(ρ, φ, z) : ρ < 7.6} ({(ρ, φ, z) : < ρ < 7.6, 5 < z < 5} {(ρ, φ, z) : < ρ < 7.6, 97.5 < z < 0} {(ρ, φ, z) : < ρ < 7.6, < z < 7.5}). This LWD instrument is moves along the vertical direction (z-axis) in a subsurface borehole environment composed of: a borehole mud with resistivity 0. m defined on. BH = {(ρ, φ, z) : ρ < 0.795} ( i Bi M ), and, three formation materials of resistivities 00 m, 0000 m, and m, defined on. M = {(ρ, φ, z) : ρ 0.795, (z < 50 or z > 00)}, 2. M2 = {(ρ, φ, z) : ρ 0.795, 50 z < 0}, and,. M = {(ρ, φ, z) : ρ 0.795, 0 z 00}, respectively. Fig. 2.2 shows the geometry of the described logging instrument and borehole environment cm 00 Ohm m 5 cm Magnetic Buffer 0000 Ohm m 0000 Relative Permeability 00 cm Ohm m 0. Ohm m Ohm m 0000 Ohm m 00 Ohm m 00 cm 50 cm 0 cm Mandrel Ohm m 5 cm Borehole 0. Ohm m Radius = cm Radius 7.6 cm Fig D cross-section of the geometry of an induction LWD problem composed of a metallic mandrel, one transmitter and two receiver coils equipped with magnetic buffers, a borehole, and four layers in the rock formation (with different resistivities). The right panel is an enlarged view of the geometry (left panel) in the vicinity of the transmitter antenna.. Maxwell s Equations. In this section, we first introduce the time-harmonic Maxwell s equations in the frequency domain. They form a set of first-order Partial Differential Equations (PDE s). Then, we describe boundary conditions needed for

6 6 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI the simulation of our logging applications of interest. Finally, we derive a variational formulation in terms of either the electric or the magnetic field, and we reduce the dimension of the computational problem by considering axial symmetry... Time-harmonic Maxwell s equations. Assuming a time-harmonic dependence of the form e jωt, where t denotes time, and ω 0 is angular frequency, Maxwell s equations can be written as H = (σ + jωɛ)e + J imp Ampere s Law, E = jωµ H M imp Faraday s Law, (.) (ɛe) = ρ Gauss Law of Electricity, and (µh) = 0 Gauss Law of Magnetism. Here H and E denote the magnetic and electric field, respectively, J imp is a prescribed, impressed electric current density, M imp is a prescribed, impressed magnetic current density, ɛ, µ, and σ stand for dielectric permittivity, magnetic permeability, and electrical conductivity of the medium, respectively, and ρ denotes the electric charge distribution. We assume µ 0. The equations described in (.) are to be understood in the distributional sense, i.e. they are satisfied in the classical sense in subdomains of regular material data, and they also imply appropriate interface conditions across material interfaces. Energy considerations lead to the assumption that the absolute value of both electric field E and magnetic field H must be square integrable. M imp is assumed to be divergence free due to physical considerations. Maxwell s equations are not independent. Taking the divergence of Faraday s Law yields the Gauss Law of magnetism. By taking the divergence of Ampere s Law, and by utilizing Gauss Electric Law we arrive at the so called continuity equation, (.2) (σe) + jωρ + J imp = Boundary Conditions (BC s). There exist a variety of BC s that can be incorporated into Maxwell s equations. In the following, we describe those BC s that are of interest for the logging applications discussed in this paper. At this point, we are considering general D domains. A discussion on boundary terms corresponding to the axisymmetry condition is postponed to Section Perfect Electric Conductor (PEC). Maxwell s equations are to be satisfied in the whole space minus domains occupied by a PEC. A PEC is an idealization of a highly conductive media. Inside a region where σ, the corresponding electric field converges to zero by applying Ampere s law. Faraday s law implies that the tangential component of the electric field E must remain continuous across material interfaces in the absence of impressed magnetic surface currents. Consequently, the tangential component of the electric field must vanish along the PEC boundary, i.e., (.) n E = 0, where n is the unit normal (outward) vector. This result is true under the physical consideration that impressed volume current J imp and σe should remain finite, i.e., J imp, ψ, σe, ψ < for every test function ψ. See [6] for details.

7 HP -FEM: ELECTROMAGNETIC APPLICATIONS 7 Since the electric field vanishes inside a PEC, Faraday s law implies that the magnetic field should also vanish inside a PEC in the absence of magnetic currents. The same Faraday s law implies that the normal component of the magnetic field premultiplied by the permeability must remain continuous across material interfaces. Therefore, the normal component of the magnetic field must vanish along the PEC boundary, i.e., (.4) n H = 0. The tangential component of magnetic field (surface current) and normal component of the electric field (surface charge density) need not be zero, and may be determined a-posteriori Source Antennas. Antennas are modeled by prescribing an impressed volume current J imp. Using the equivalence principle (see, for example, [8]), we can substitute the original impressed electric volume current J imp by the equivalent electric surface current (.5) J imp S = [n H] S, defined on an arbitrary surface S enclosing the support of J imp, where [n H] S denotes the jump of n H accross S. Similarly, an impressed magnetic volume current M imp can be replaced by the equivalent magnetic surface current (.6) M imp S = [n E] S, defined on an arbitrary surface S enclosing the support of M imp..2.. Closure of the Domain. We consider a bounded computational domain. A variety of BC s can be imposed on the boundary such that the difference between solution of such a problem and solution of the original problem defined over R is small. For example, it is possible to use an infinite element technique (as described in [4]). Also, since the electromagnetic fields and their derivatives decay exponentially in the presence of lossy media (non-zero conductivity), we may simply impose a homogeneous Dirichlet or Neumann BC on the boundary of a sufficiently large computational domain. In the field of geophysical logging applications, it is customary to impose a homogeneous Dirichlet BC on the boundary of a large computational domain (for example, 2-20 meters in each direction from a 2 Mhz source antenna in the presence of a resistive media). We will follow the same approach... Variational Formulation. From Maxwell s equations and the BC s described above, we derive the corresponding standard variational formulation in terms of the electric or magnetic field as follows. First, we notice from Faraday s law that E (L 2 ()) if and only if M imp (L 2 ()). Since our objective is to find a solution E H(curl; ) = {F (L 2 ()) : F (L 2 ()) }, we shall assume in the case of the electric field formulation (E-formulation) derived below that M imp (L 2 ()). If the prescribed M imp / (L 2 ()), we may still solve Maxwell s equations with H(curl)-conforming finite elements for the magnetic field by using the H-formulation (..2), or simply by prescribing an equivalent source M imp such that M imp M imp does not radiate outside the antenna [2]. Similarly, for the H-formulation, we will assume that J imp (L 2 ()).

8 8 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI... E-Formulation. By dividing Faraday s law by magnetic permeability µ, multiplying the resulting equation by F, where F H(curl; ) is an arbitrary test function, and integrating over the domain, we arrive at the identity µ ( E) ( F)dV (.7) = jω H ( F)dV µ Mimp ( F)dV. Integrating H ( F) dv by parts, and applying Ampere s law, we obtain (.8) H ( F) dv = ( H) F dv [n H] ΓN F t ds = (σ + jωɛ)e F dv + J imp F dv [n H] ΓN F t ds, with denoting a surface contained in the closure of where an impressed electric surface current J imp may be prescribed. F t = F (F n) n is the tangential component of vector F on, and n is the unit normal outward (with respect if ) vector. Substitution of (.8) into (.7), and use of equation (.5) yields to the following variational identity, valid for any test function F H(curl; ): µ ( E) ( F)dV k 2 E F dv = jω J imp F dv (.9) +jω F t ds µ Mimp ( F) dv, J imp where k 2 = ω 2 ɛ jωσ. Finally, in order to obtain a unique solution E H(curl; ) for problem (.9), we introduce a Dirichlet boundary condition on a part Γ D of the boundary of the computational domain. Thus, we obtain the following variational formulation: Find E E D + H D (curl; ) such that: µ ( E) ( F) dv k 2 E F dv = jω J imp F dv (.0) +jω J imp F t ds µ Mimp ( F) dv F H D (curl; ), where E D is a lift (typically E D = 0) of the essential boundary condition data E D (denoted with the same symbol), and H D (curl; ) = {F H(curl; ) : (n F) ΓD = 0} is the space of admissible test functions associated with problem (.0). Conversely, we can derive (.), (.), and (.5) from variational problem (.0). Remark. At this point, an impressed magnetic surface current M imp S defined on a subset of Γ D may be introduced into the formulation by using equation (.6). It follows that E D = n E ΓD = M imp Γ D...2. H-Formulation. By dividing Ampere s law by σ + jωɛ, multiplying the resulting equation by F, where F H(curl; ) is an arbitrary test function, and integrating over the domain, we arrive at the identity

9 (.) Integrating (.2) HP -FEM: ELECTROMAGNETIC APPLICATIONS 9 jω k 2 ( H) ( F)dV = jω E ( F) dv k 2 Jimp ( F) dv. E ( F) dv by parts, and applying Faraday s law, we obtain E ( F) dv = ( E) F dv [n E] ΓN F t ds = jω µh F dv M imp F dv [n E] ΓN F t ds, with denoting a surface contained in the closure of where an impressed magnetic surface current M imp may be prescribed. Substitution of (.2) into (.), and use of equation (.6) yields the following variational identity, valid for any test function F H(curl; ): jω k 2 ( H) ( F)dV + jω µh F dv = (.) M imp F dv + F t ds jω k 2 Jimp ( F) dv. M imp Finally, in order to obtain a unique solution H H(curl; ) for problem (.) we introduce a Dirichlet boundary condition on a part Γ D of the boundary of the computational domain. Thus, we obtain the following variational formulation: (.4) Find H H D + H D (curl; ) such that: σ + jωɛ ( H) ( F)dV + jω + M imp F t ds + µh FdV = M imp FdV σ + jωɛ Jimp ( F)dV F H D (curl; ), where H D is a lift (typically H D = 0) of the essential boundary condition data H D (denoted with the same symbol). At this point, an impressed electric surface current J imp S defined on a subset of Γ D may be introduced into the formulation by using equation (.5). It follows that H D = n H ΓD = J imp.4. Cylindrical Coordinates and Axisymmetric Problems. We consider cylindrical coordinates (ρ, φ, z). For the geophysical logging applications considered in this article, we assume that both the logging instrument and the rock formation properties are axisymmetric (invariant with respect to the azimuthal component φ) around the axis of the borehole. Under this assumption, we obtain that for any vector field A = ˆρA ρ + ˆφA φ + ẑa z, (.5) A = ˆρ A φ z Γ D. + ˆφ( A ρ z A z ρ ) + ẑ (ρa φ ). ρ ρ.4.. E-Formulation. Next, we consider the space of all test functions F H D (curl; ) such that F = (0, F φ, 0). According to (.5), (.6) F = ˆρ F φ z + ẑ (ρf φ ). ρ ρ

10 0 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI Variational formulation (.0) reduces to a formulation in terms of the scalar field E φ, namely, (.7) Find E φ E φ,d + H D () such that: ( Eφ F φ µ z z + (ρe φ ) (ρ F ) φ ) ρ 2 ρ ρ jω J imp φ F φ dv + jω [ M imp F φ ρ µ z + M z imp ρ J imp φ, dv k 2 E φ Fφ dv = (ρ F φ ) ρ Fφ ds ] dv F φ H D(), where H D() = {E φ : (0, E φ, 0) H D (curl; )} = {E φ L 2 () : ρ E φ + E φ ρ L 2 E φ (), L 2 (), E φ ΓD = 0}. Similarly, for a test function F = (F ρ, 0, F z ), z variational problem (.0) simplifies to: (.8) Find E = (E ρ, 0, E z ) E D + H D (curl; ) such that: ( Eρ µ z E ) ( z Fρ ρ z F ) z dv k 2 (E ρ Fρ + E z Fz ) dv = ρ jω Jρ imp F ρ + Jz imp F z dv + jω J imp ρ,γ Fρ N + J imp z,γ Fz N ds [ µ M imp Fρ φ z F ] z dv F = (F ρ, 0, F z ) ρ H D (curl; ), where H D (curl; ) = {(E ρ, E z ) : E = (E ρ, 0, E z ) L 2 (), ( E) φ = E ρ z E z ρ L2 (), (n E) ΓD = 0}. In summary, problem (.0) decouples into a system of two simpler problems described by (.7) and (.8). Remark 2. It has been shown in [2] (Lemma 4.9) that space H D() can also be expressed as H D() = {E φ L 2 () : ρ E φ L 2 (), (ρ,z) E φ L 2 ()} H-Formulation. Using the same decomposition of test functions (i.e., F = (0, F φ, 0), and F = (F ρ, 0, F z )) for variational problem (.4), we arrive at the following two decoupled variational problems in terms of (0, H φ, 0) (.9), and (H ρ, 0, H z ) (.20), respectively: (.9) Find H φ H φ,d + H D () such that: ( Hφ F φ σ + jωɛ z z + (ρh φ ) (ρ F ) φ ) ρ 2 dv ρ ρ +jω µh φ Fφ dv = M imp φ F φ dv + M imp φ,γ Fφ N ds Γ [ N + J imp F φ ρ σ + jωɛ z + J z imp (ρ F ] φ ) dv F φ ρ ρ H D().

11 (.20) HP -FEM: ELECTROMAGNETIC APPLICATIONS Find H = (H ρ, 0, H z ) H D + H D (curl; ) such that: ( Hρ σ + jωɛ z H ) ( z Fρ ρ z F ) z dv ρ +jω µ(h ρ Fρ + H z Fz ) dv = Mρ imp F ρ + Mz imp F z dv [ + M imp ρ,γ Fρ N + M imp z,γ Fz N ds + σ + jωɛ J imp Fρ φ z F ] z dv ρ F = (F ρ, 0, F z ) H D (curl; ). From the formulation of problems (.7) trough (.20), we remark the following: Physically, solution of problems (.8), and (.9) correspond to the T E φ - mode (i.e. E φ = 0), and solution of problems (.7), and (.20) correspond to the T M φ -mode (i.e. H φ = 0). The axis of symmetry is not a boundary of the original D problem, and therefore, a boundary condition should not be needed to solve this problem. Nevertheless, formulations of problems (.7) through (.20) require the use of spaces H D () and H D (curl; ) described above. The former space involves the singular weight ρ, which implicitly requires a homogeneous Dirichlet boundary condition along the axis of symmetry. The latter space can be considered as it is (by using 2D edge elements), and no BC is necessary 4 to solve the problem. 4. Self-Adaptive Goal-Oriented -FEM. We are interested in solving variational problems (.0) and (.4) (or alternatively, (.7), (.8), (.9), and (.20)), that we state here in terms of sesquilinear form b, and antilinear form f: { Find E ED + V (4.) b(e, F) = f(f) F V, where E D is a lift of the essential (Dirichlet) BC. V is a Hilbert space. f V is an antilinear and continuous functional on V. b is a sesquilinear form. More precisely, we have: (4.2) b(e, F) = µ a(e, F) k2 c(e, F) E-Formulation, k 2 a(e, F) µc(e, F) H-Formulation where sesquilinear forms a and c are assumed to be Hermitian, continuous and V-coercive. We define an energy inner product on V as: (E, F) := µ a(e, F) + k2 c(e, F) E-Formulation (4.), k 2 a(e, F) + µc(e, F) H-Formulation with the corresponding (energy) norm denoted by E. 4 From the computational point of view, this effect can be achieved by artificially adding a homogeneous natural (Neumann) BC.

12 2 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI 4.. Representation of the Error in the Quantity of Interest. Given an -FE subspace V V, we discretize (4.) as follows: { Find E E D + V (4.4) b(e, F ) = f(f ) F V. The objective of goal-oriented adaptivity is to construct an optimal -grid, in the sense that it minimizes the problem size needed to achieve a given tolerance error for a given quantity of interest L, with L denoting a linear and continuous functional. By recalling the linearity of L, we have: (4.5) Error of interest = L(E) L(E ) = L(E E ) = L(e), where e = E E denotes the error function. By defining the residual r V as r (F) = f(f) b(e, F) = b(e E, F) = b(e, F), we look for the solution of the dual problem: { Find W V (4.6) b(f, W) = L(F) F V. Using the Lax-Milgram theorem we conclude that problem (4.6) has a unique solution in V. The solution W, is usually referred to as the influence function. By discretizing (4.6) via, for example, V V, we obtain: { Find W V (4.7) b(f, W ) = L(F ) F V. Definition of the dual problem plus the Galerkin orthogonality for the original problem imply the final representation formula for the error in the quantity of interest, namely, L(e) = b(e, W) = b(e, W F ) = }{{} b(e, ɛ). ɛ At this point, F V is arbitrary, and b(e, ɛ) = b(e, ɛ) denotes the bilinear form corresponding to the original sesquilinear form. Notice that, in practice, the dual problem is solved not for W but for its complex conjugate W utilizing the bilinear form and not the sesquilinear form. The linear system of equations is factorized only once, and the extra cost of solving (4.7) reduces to only one backward and one forward substitution (if a direct solver is used). Once the error in the quantity of interest has been determined in terms of bilinear form b, we wish to obtain a sharp upper bound for L(e) that depends upon the mesh parameters (element size h and order of approximation p) only locally. Then, a selfadaptive algorithm intended to minimize this bound will be defined. First, using a procedure similar to the one described in [7], we approximate E and W with fine grid functions E h 2, p+, W h 2, p+, which have been obtained by solving the corresponding linear system of equations associated with the FE subspace V h 2, p+. In the remainder of this article, E and W will denote the fine grid solutions of the direct and dual problems (E = E h 2, p+, and W = W h 2, p+, respectively), and we will restrict ourselves to discrete FE spaces only. Next, we bound the error in the quantity of interest by a sum of element contributions. Let b K denote a contribution from element K to sesquilinear form b. It then follows that

13 HP -FEM: ELECTROMAGNETIC APPLICATIONS (4.8) L(e) = b(e, ɛ) K b K (e, ɛ), where summation over K indicates summation over elements Projection based interpolation operator. Once we have a representation formula for the error in the quantity of interest in terms of the sum of element contributions given by (4.8), we wish to express this upper bound in terms of local quantities, i.e. in terms of quantities that do not vary globally when we modify the grid locally. For this purpose, we introduce the idea of projection-based interpolation operators. First, in order to simplify the notation, we define the following three spaces of admissible solutions: V = H D (curl; ), V 2D = H D (curl; ), and, V D = H D (). The corresponding -Finite Element spaces will be denoted by V, V 2D D, and V, respectively. At this point, we introduce three projection-based interpolation operators that have been defined in [6, 5], and used in [7, 8] for the construction of the fully automatic energy-norm based -adaptive algorithm: Π curl,d : V V, Π curl,2d : V 2D V 2D, and, Π D : V D V D. We shall also consider three projection operators in the energy-norm: P curl,d : V V, P curl,2d : V 2D V 2D, and, P D : V D V D. To further simplify the notation, we will utilize the unique symbol Π curl to denote all projection based interpolation operators mentioned above. Depending upon the problem formulation (and corresponding space of admissible solutions), Π curl should be understood as Π curl,d for problems (.0) and (.4), Π curl,2d for problems (.8) and (.20), or Π D for problems (.7) and (.9). Similarly, we will use the unique symbol P curl to denote either P curl,d, P curl,2d, or P D. We denote E = P curl E. Equation (4.8) then becomes (4.9) L(e) K b K (E, ɛ) = K b K (E Π curl E, ɛ) + b K (Π curl E P curl E, ɛ). Given an element K, it is expected that b K (Π curl E P E, ɛ) will be negligible compared to b K (E Π curl E, ɛ). Under this assumption, we conclude that: (4.0) L(e) K b K (E Π curl E, ɛ). In particular, for ɛ = W Π curl W, we have: (4.) L(e) K b K (E Π curl E, W Π curl W).

14 4 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI By applying Cauchy-Schwartz inequality, we obtain the next upper bound for L(e) : (4.2) L(e) K ẽ K ɛ K, where ẽ = E Π curl E, ɛ = W Πcurl W, and K denotes energy-norm restricted to element K. 4.. Fully Automatic Goal-Oriented -Refinement Algorithm. We describe an self-adaptive algorithm that utilizes the main ideas of the fully automatic (energy-norm based) -adaptive algorithm described in [7, 8]. We start by recalling the main objective of the self-adaptive (energy-norm based) -refinement strategy, which consists of solving the following maximization problem: (4.) Find an optimal -grid in the following sense: = arg max c K E Π curl E 2 K E Πcurl c E 2 K N where E = E h 2, p+ is the fine grid solution, and N > 0 is the increment in the number of unknowns from grid to grid ĥp. Similarly, for goal-oriented -adaptivity, we propose the following algorithm based on estimate (4.2): Find an optimal -grid in the following sense: [ E Π curl = arg max E K W Π curl W K (4.4) c N K E Π curl c E K W Π curl c W ] K, N where: E = E h 2, p+ and W = W h 2, p+ are the fine grid solutions corresponding to the direct and dual problems, and N > 0 is the increment in the number of unknowns from grid to grid ĥp. Implementation of the goal-oriented -adaptive algorithm is based on the optimization procedure used for energy-norm -adaptivity [7, 8] Implementation details. In what follows, we discuss the main implementation details needed to extend the fully automatic (energy-norm based) -adaptive algorithm [7, 8] to a fully automatic goal-oriented -adaptive algorithm.. First, the solution W of the dual problem on the fine grid is necessary. This goal can be attained either by using a direct (frontal) solver or an iterative (two-grid) solver (see []). 2. Subsequently, we should treat both solutions as satisfying two different partial differential equations (PDE s). We select functions E and W as the solutions of the system of two PDE s.. We proceed to redefine the evaluation of the error. The energy-norm error evaluation of a two dimensional function is replaced by the product E Π curl E W Πcurl W.,

15 HP -FEM: ELECTROMAGNETIC APPLICATIONS 5 4. After these simple modifications, the energy-norm based self-adaptive algorithm may now be utilized as a self-adaptive goal-oriented algorithm. 5. Numerical Results. In this Section, we apply the goal-oriented selfadaptive strategy described in Section 4 to simulate the response of the induction LWD instrument operating at 2 Mhz considered in Section 2., using formulation (.7) for solenoidal coils, and (.9) for toroidal coils. Exactly the same results are obtained with formulations (.20) and (.8), respectively, as predicted by the theory. Thus, formulations (.20) and (.8) have been used as an extra verification of the simulations, and the corresponding results have been omitted in this article to avoid duplicity. Fig. 5. displays the first vertical difference of the electric field (divided by the distance between the two receivers) for the described LWD instrument equipped with solenoidal coils. The three curves correspond to:. the rock formation with no mud-filtrate invasion, 2. the rock formation with a 2 m 40 cm. horizontal mud layer invading the m rock formation layer, and a 5 m 90cm. horizontal mud layer invading the 0000 m rock formation layer, and. the previous (mud invaded) rock formation, using a mandrel with relative magnetic permeability of 00. For toroidal antennas, we display in Fig. 5.2 the first vertical difference of the magnetic field (divided by the distance between the two receivers). The three displayed curves correspond to the three situations discussed above. Invasion Study No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel=00 Solenoid No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel=00 Vertical Position of Receiving Antenna (m) 2 00 Ohm m Ohm m Ohm m (0.4m) Ohm m 0.52 Ohm m (0.4m) Ohm m (0.9m) 0000 Ohm m 5 Ohm m (0.9m) Vertical Position of Receiving Antenna (m) Ohm m 0000 Ohm m 00 Ohm m 00 Ohm m Amplitude First Vert. Diff. Electric Field (V/m 2 ) Phase (degrees) Fig. 5.. LWD problem equipped with a solenoidal source. Amplitude (left panel) and phase (right panel) of the first vertical difference of the electric field (divided by the distance between receivers) at the receiving coils. Results obtained with the self-adaptive goal-oriented -FEM. The spatial distribution of electrical resistivity is also displayed to facilitate the physical interpretation of results.

16 6 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI Invasion Study No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel=00 Toroid No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel=00 Vertical Position of Receiving Antenna (m) 2 00 Ohm m Ohm m Ohm m (0.4m) Ohm m 0.52 Ohm m (0.4m) Ohm m (0.9m) 0000 Ohm m 5 Ohm m (0.9m) Vertical Position of Receiving Antenna (m) Ohm m 0000 Ohm m 00 Ohm m 00 Ohm m Amplitude First Vert. Diff. Magnetic Field (A/m 2 ) Phase (degrees) Fig LWD problem equipped with a toroidal source. Amplitude (left panel) and phase (right panel) of the first vertical difference of the magnetic field (divided by the distance between receivers) at the receiving coils. Results obtained with the self-adaptive goal-oriented -FEM. The spatial distribution of electrical resistivity is also displayed to facilitate the physical interpretation of results. These results illustrate the strong dependence of the LWD response on the rock formation resistivity. We observe that solenoidal antennas are more sensitive to highly conductive formations as well as to the electrical permeability of the mandrel, while toroidal antennas are more sensitive to highly resistive formations. Fig. 5. illustrates the effect of the magnetic buffers. By removing the magnetic buffers from the logging instrument s design, the amplitude of the received signal decreases by a factor of up to 200 in the case of a solenoidal source. For practical applications, a strong signal on the receivers is desired to minimize the noise-to-signal ratio. Thus, it is appropriate to use magnetic buffers in combination with solenoidal antennas. On the contrary, the use of magnetic buffers with toroidal antennas is not advisable since they weaken the received signal. In both cases, the phase and shape of the solution is not sensitive to the presence (or not) of magnetic buffers, and the corresponding results have been omitted. The exponential convergence obtained using the self-adaptive goal-oriented - FEM is shown in Fig. 5.4 (left panel), by considering an arbitrary fixed position of the logging instrument for a solenoid antenna. The final grid delivers a relative error in the quantity of interest below %, i.e. the first 7 significant digits of the quantity of interest are exact. In Fig. 5.4 (right panel), we display the exponential convergence of the energy-norm based -FEM. The final -grid delivers an energynorm error below 0.0%. Nevertheless, the quantity of interest still contains a relative error above 5%. A final goal-oriented -grid delivering a relative error in the quantity of interest of 0.% is displayed in Fig. 5.5.

17 HP -FEM: ELECTROMAGNETIC APPLICATIONS 7 Solenoid With Magnetic Buffers Without Magnetic Buffers Toroid With Magnetic Buffers Without Magnetic Buffers With Magnetic Buffers Without Magnetic Buffers Vertical Position of Receiving Antenna (m) Ohm m Ohm m 0000 Ohm m Vertical Position of Receiving Antenna (m) Ohm m Ohm m 0000 Ohm m Vertical Position of Receiving Antenna (m) Ohm m Ohm m 0000 Ohm m 00 Ohm m 00 Ohm m 00 Ohm m Amplitude First Vert. Diff. of Electric Field (V/m 2 ) Amplitude First Vert. Diff. of Magnetic Field (A/m 2 ) Phase (degrees) Fig. 5.. LWD problem equipped with a solenoidal source. Results obtained with the selfadaptive goal-oriented -FEM correspond to the use of solenoidal antennas (left panel), and toroidal antennas (right panel), respectively. The spatial distribution of electrical resistivity is also displayed to facilitate the physical interpretation of results. 6. Summary and Conclusions. We have successfully applied a self-adaptive goal-oriented -FE algorithm to simulate the axisymmetric response of an induction LWD instrument in a borehole environment. These simulations would not be possible with energy-norm adaptive algorithms. Numerical results illustrate the exponential convergence of the method (allowing for high accuracy simulations), the suitability of the presented formulations for axisymmetric electrodynamic problems, and the main physical characteristics of the presented induction LWD instrument. These results suggest the use of solenoidal antennas for the assessment of highly conductive rock formation materials, and toroidal antennas for the assessment of highly resistive materials. Solenoidal antennas should be used in combination with magnetic buffers to strengthen the measured EM signal, while the use of magnetic buffers with toroidal antennas should be avoided. Both types of antennas can be used to study mud-filtrate invasion. Since the influence function used by the self-adaptive goal-oriented -adaptive algorithm is approximated via Finite Elements, the numerical method presented in this article is problem independent, and it can be applied to D, 2D, and D FE discretizations of H -, H(curl)-, and H(div)-spaces. Appendix. A Loop-Antenna Radiating in a Homogeneous Lossy Medium in the Presence of a Highly Conductive Metallic Mandrel. In this appendix, we consider a problem with known analytical solution. We use this problem as an additional mechanism to verify the code, as well as to provide comparative results between analytical and numerical solutions. We consider a solenoid (or a toroid) of radius a radiating at a frequency of 2 Mhz

18 8 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI Goal Oriented Adaptivity Upper bound for L(e) / L(u) L(e) / L(u) Energy norm Adaptivity Energy norm error L(e) / L(u) 0 0 Relative Error in % Relative Error in % Number of Unknowns N (scale N / ) Number of Unknowns N (scale N / ) Fig LWD problem equipped with a solenoidal source. Left panel: convergence behavior obtained with the self-adaptive goal-oriented -FEM shows exponential convergence rates for estimate (4.8) (solid curve) used for optimization. The dashed curve describes the relative error in the quantity of interest. Right panel: convergence behavior obtained with the self-adaptive energy-norm -FEM shows exponential convergence rates for the energy-norm. The dashed curve describes the relative error in the quantity of interest. p = 8 p = 7 Receiver II Receiver I p = 6 p = 5 p = 4 p = Transmitter p = 2 p = Fig LWD instrument equipped with a solenoidal source. Portion (20 cm x 200 cm) of the final -grid. Different colors indicate different polynomial orders of approximation, ranging from (light grey) to 8 (white).

19 HP -FEM: ELECTROMAGNETIC APPLICATIONS 9 in a homogeneous lossy medium (with resistivity equal to m), in the presence of an infinitely large cylindrical mandrel (with resistivity equal to 0 6 m) of radius b < a. The coil and the mandrel exhibit axial symmetry (see Fig. A.). z COIL (Toroid/Solenoid) b a MANDREL Fig. A.. Geometry of a loop-antenna radiating in a homogeneous lossy medium in the presence of a highly conductive metallic mandrel. For a solenoidal coil located at z = 0, the resulting solution for a ρ b is given by [0, 5]: (A.) E φ (ρ, z) = ωµ 4πa [J (k ρ a) + ΓH () (k ρa)]h () (k ρρ)e ikzz dk ρ, where Γ = J (k ρ b)/h () (k ρb), J p and H p () are the Bessel and Hankel functions of the first type of order p, respectively, and k z = k 2 kρ. 2 by: (A.2) For a toroidal coil located at z = 0, the resulting solution for a ρ b is given H φ (ρ, z) = i 4πa [J (k ρ a) + ΓH () (k ρa)]h () (k ρρ)e ikzz dk ρ, where Γ = J 0 (k ρ b)/h () 0 (k ρb). In figures A.2 and A., we display a comparison between analytical and numerical results (obtained using the self-adaptive goal-oriented algorithm) for the solenoidal and toroidal coils, respectively. We selected b = cm, and a = cm. The numerical results accurately reproduce the analytical ones, both in terms of amplitude and phase. When considering a solenoid, the logging instrument response using a mandrel of resistivity 0 5 m or a PEC mandrel are indistinguishable in terms of amplitude. A similar situation occurs for a toroid. In terms of phase, induction instruments equipped with solenoidal coils appear to be more sensitive to the mandrel resistivity than those equipped with toroidal coils. Acknowledgments. This work was financially supported by Baker-Atlas and the Joint Industry Research Consortium on Formation Evaluation supervised by Prof. C. Torres-Verdin. We would also like to acknowledge the expertise and technical advise

20 20 D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, M. PASZYNSKI Distance in z axis from transmitter to receiver (in m) Analytical solution Mandrel Resisitivity: 0 7 Mandrel Resisitivity: 0 5 Mandrel Resisitivity: 0 Mandrel Resisitivity: Distance in z axis from transmitter to receiver (in m) Analytical solution Mandrel Resisitivity: 0 7 Mandrel Resisitivity: 0 5 Mandrel Resisitivity: 0 Mandrel Resisitivity: Amplitude (V/m) Phase (degrees) Fig. A.2. Solution (electric field) along the vertical axis passing through a solenoid radiating in a homogeneous medium in the presence of a metallic mandrel. Analytical solution (mandrel is a PEC) against the numerical solution for different mandrel resistivities (0 7, 0 5, 0, and m) obtained with the self-adaptive goal oriented -FEM. received from L. Tabarovsky, A. Bespalov, T. Wang, and other members of the Science Department of Baker-Atlas. REFERENCES [] B. I. Anderson, Modeling and Inversion Methods for the Interpretation of Resistivity Logging Tool Response, PhD thesis, Delft University of Technology, 200. [2] F. Assous, C.(Jr.) Ciarlet, and S Labrunie, Theoretical tools to solve the axisymmetric Maxwell equations., Math. Meth. Appl. Sci., 25 (2002), pp [] R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods., in ENU- MATH 97. Proceedings of the 2nd European conference on numerical mathematics and advanced applications held in Heidelberg, Germany, September 28-October, 997. Including a selection of papers from the st conference (ENUMATH 95) held in Paris, France, September 995. Singapore: World Scientific , Bock, Hans Georg (ed.) et al., 998. [4] W. Cecot, W. Rachowicz, and L. Demkowicz, An -adaptive finite element method for electromagnetics. III: A three-dimensional infinite element for Maxwell s equations., Int. J. Numer. Methods Eng., 57 (200), pp [5] L. Demkowicz, Finite element methods for Maxwell equations., Encyclopedia of Computational Mechanics, (eds. E. Stein, R. de Borst, T.J.R. Hughes), Wiley and Sons, 200, in review, (200). [6] L. Demkowicz and A. Buffa, H, H(curl), and H(div) conforming projection-based interpolation in three dimensions: quasi optimal p-interpolation estimates., Comput. Methods Appl. Mech. Eng., 94 (2005), pp [7] L. Demkowicz, W. Rachowicz, and Ph. Devloo, A fully automatic -adaptivity., J. Sci. Comput., 7 (2002), pp [8] R. F. Harrington, Time-harmonic electromagnetic fields., McGraw-Hill, New York, 96. [9] V. Heuveline and R. Rannacher, Duality-based adaptivity in the -finite element method., J. Numer. Math., (200), pp. 95.

21 HP -FEM: ELECTROMAGNETIC APPLICATIONS 2 Distance in z axis from transmitter to receiver (in m) Analytical solution Mandrel Resisitivity: 0 7 Mandrel Resisitivity: 0 5 Mandrel Resisitivity: 0 Mandrel Resisitivity: Distance in z axis from transmitter to receiver (in m) Analytical solution Mandrel Resisitivity: 0 7 Mandrel Resisitivity: 0 5 Mandrel Resisitivity: 0 Mandrel Resisitivity: Amplitude (A/m) Phase (degrees) Fig. A.. Solution (magnetic field) along a vertical axis passing through a toroid radiating in a homogeneous medium in the presence of a metallic mandrel. Analytical solution (mandrel is a PEC) against the numerical solution for different mandrel resistivities (0 7, 0 5, 0, and m) obtained with the self-adaptive goal oriented -FEM. [0] J. R. Lovell, Finite Element Methods in Resistivity Logging, PhD thesis, Delft University of Technology, 99. [] J.T. Oden and S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method., Comput. Math. Appl., 4 (200), pp [2] M. Paraschivoiu and A. T. Patera, A hierarchical duality approach to bounds for the outputs of partial differential equations., Comput. Methods Appl. Mech. Eng., 58 (998), pp [] D. Pardo, Integration of -adaptivity with a two grid solver: applications to electromagnetics., PhD thesis, The University of Texas at Austin, April [4] D. Pardo, L. Demkowicz, and C. Torres-Verdin, A Goal Oriented -Adaptive Finite Element Method with Electromagnetic Applications. Part I: Electrostatics., ICES Report Submitted to Int. J. Numer. Methods Eng., 0 (2005), p. 0. [5] M. Paszynski, L. Demkowicz, and D. Pardo, Verification of Goal-Oriented -Adaptivity., ICES Report 05-06, (2005). [6] C. R. Paul and S. A. Nasar, Introduction to Electromagnetic Fields, McGraw-Hill, New York, 982. [7] S. Prudhomme and J.T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors., Comput. Methods Appl. Mech. Eng., 76 (999), pp.. [8] W. Rachowicz, D. Pardo, and L. Demkowicz, Fully automatic -adaptivity in three dimensions., Tech. Report 04-22, ICES Report, [9] R. Rannacher and F.T. Suttmeier, A posteriori error control in finite element methods via duality techniques: application to perfect plasticity., Comput. Mech., 2 (998), pp. 2. [20] P. Solin and L. Demkowicz, Goal-oriented -adaptivity for elliptic problems., Comput. Methods Appl. Mech. Eng., 9 (2004), pp [2] J. Van Bladel, Singular Electromagnetic Fields and Sources., Oxford University Press., New York, 99. [22] T. Wang and J. Signorelli, Finite-difference Modeling of Electromagnetic Tool Response for Logging While Drilling., Geophysics, 69 (2004), pp