c 2006 Society for Industrial and Applied Mathematics

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1 SIAM J. APPL. MATH. Vol. 66, No. 6, pp c 2006 Society for Industrial and Applied Mathematics TWO-DIMENSIONAL HIGH-ACCURACY SIMULATION OF RESISTIVITY LOGGING-WHILE-DRILLING (LWD) MEASUREMENTS USING A SELF-ADAPTIVE GOAL-ORIENTED FINITE ELEMENT METHOD D. PARDO, L. DEMKOWICZ, C. TORRES-VERDÍN, AND M. PASZYNSKI Abstract. We simulate electromagnetic (EM) measurements acquired with a logging-whiledrilling (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 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 to obtain an accurate solution in the entire computational domain. In particular, goal-oriented -adaptivity becomes essential to simulating 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 DOI. 0.37/ Introduction. A plethora of energy-norm-based algorithms intended to generate optimal grids have been developed throughout recent decades (see, for example, [0, 23] 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, such as simulating 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 Received by the editors May 7, 2005; accepted for publication (in revised form) February 4, 2006; published electronically October 6, This work was financially supported by Baker-Atlas and the Joint Industry Research Consortium on Formation Evaluation supervised by Prof. C. Torres- Verdin. Institute for Computational Engineering and Sciences (ICES) and Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 7872 (dzubiaur@yahoo. es). Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 7872 (leszek@ices.utexas.edu). Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 7872 (cverdin@uts.cc.utexas.edu). Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 7872 (maciek@ices.utexas.edu). On leave from Department of Computer Methods in Metallurgy, AGH University of Science and Technology, Cracow, Poland. 2085

2 2086 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI 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 [8]). Consequently, in order to accurately simulate logging-while-drilling (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 goaloriented adaptive algorithms [6, 22], and are based on minimizing the error of a prescribed quantity of interest mathematically expressed in terms of a linear functional (see [5, 2, 7, 6, 22, 24] for details). 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 [0, 23], and a continuation of concepts presented in [9, 25] 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 electromagnetic (EM) singularities generated by the LWD geometry and rock formation materials [28], as well as resolution of high material contrasts that occur between the mandrel and the borehole. Other methods for simulation of LWD measurements include the transmission line matrix method [4], fast Fourier transform [29], and finite differences [26, 3]. In contrast to previous contributions, here we consider a detailed geometry of the logging instrument, which requires the resolution of strong singularities in the EM fields, we account for the finite conductivity of the mandrel, we incorporate magnetic buffers in both transmitter and receiver antennas, we consider the effect of the magnetic permeability of the mandrel, and we provide extremely accurate results with guaranteed relative error bounds below 0.% (0.00% if desired). We also consider a high contrast in conductivity among different layers in the formation, and we present a comparison between using two and three receiver antennas. The organization of this paper 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 3, we introduce Maxwell s equations, governing the EM 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 outlines future lines of research. Finally, in the appendix, we compare numerical results with a semianalytical 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 -finite element method (FEM).

3 -FEM: ELECTROMAGNETIC APPLICATIONS 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 (Figure, left panel), and toroidal coils (Figure, right panel). Fig.. 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.. Induction LWD instruments based on solenoidal coils. For axisymmetric problems, these logging instruments generate a TM φ field; i.e., the only nonzero components of the EM fields are E φ, H ρ, and H z, where (ρ, φ, z) denote the cylindrical system of coordinates. A solenoidal coil (Figure ) 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 [5]) (2.2) E = ˆφωμkIπa 2 e jkd 4πd [ j ] ρ kd 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 Induction logging instruments are characterized by the fact that impressed current J imp is divergence-free (i.e., J imp = 0).

4 2088 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI vertical magnetic dipole (VMD). The corresponding far-field solution in homogeneous media is given by (see [5]) (2.3) E = ˆφωμkI e jkd 4πd [ j ] ρ kd d. Thus, solution (2.3) is independent of the dimensions of the coil Induction LWD instruments based on toroidal coils. For axisymmetric problems, these logging instruments generate a TE 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 [5]) (2.4) H = ˆφ(σ + jωɛ)πa 2 I M jk e jkd 4πd [ j ] ρ kd 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 a Hertzian dipole. The corresponding magnetic far-field solution in homogeneous media is given by (see [5]) (2.5) H = ˆφ(σ + jωɛ)ijk e jkd 4πd [ j ] ρ kd 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 Δz between them, i.e., ( ) V V 2 (2.6) = E(l) dl E(l) dl /(Δz) = 2πa Δz l l 2 Δz (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 [5] for details). At higher frequencies (> 20 khz), asymptotic approximations (see [3] for details) also indicate the dependence of the voltage upon the rock formation conductivity. Thus, an adequate approximation of the rock formation 2 In resistivity logging applications, it is customary to consider solutions that have been divided by the geometrical factor (also called K-factor) [3], 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.

5 -FEM: ELECTROMAGNETIC APPLICATIONS 2089 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 Δz between them, i.e., (2.7) I I 2 Δz ( ) = H(l) dl H(l) dl l l 2 /(Δz) = 2πa Δz (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 by the previous choice of a quantity of interest for solenoidal coils (first difference of voltage) Description of an LWD instrument in a borehole environment. We consider an LWD instrument composed of the following axisymmetric materials (all dimensions are given in cm): one transmitter and two receiver coils defined on. C = {(ρ, φ, z) :7. <ρ<7.3, 2.5 <z<2.5}, 2. C2 = {(ρ, φ, z) :7. <ρ<7.3, <z<0.25}, and, 3. C3 = {(ρ, φ, z) :7. <ρ<7.3, 3.75 <z<6.25}, respectively; three magnetic buffers with resistivity 0 4 m and relative permeability 0 4, defined on. B = {(ρ, φ, z) :6.675 <ρ<6.985, 5 <z<5}, 2. B2 = {(ρ, φ, z) :6.675 <ρ<6.985, 97.5 <z<02.5}, and, 3. B3 = {(ρ, φ, z) :6.675 <ρ<6.985, 2.5 <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) :6.675 <ρ<7.6, 5 <z<5} {(ρ, φ, z) :6.675 <ρ<7.6, 97.5 <z<02.5} {(ρ, φ, z) : <ρ<7.6, 2.5 <z<7.5}). This LWD instrument 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, 3. M3 = {(ρ, φ, z) :ρ 0.795, 0 z 00}, respectively. Figure 2 shows the geometry of the described logging instrument and borehole environment. 3. Maxwell s equations. In this section, we first introduce the time-harmonic Maxwell equations in the frequency domain. They form a set of first-order partial

6 2090 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI 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. 2. 2D 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. differential equations (PDEs). Then, we describe boundary conditions needed for 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. 3.. Time-harmonic Maxwell 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, (3.) 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 fields, 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 (3.) 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. According to (3.) 2 and (3.) 4, M imp is divergence-free. Maxwell s equations are not independent. Taking the divergence of Faraday s law

7 -FEM: ELECTROMAGNETIC APPLICATIONS 209 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, (3.2) (σe)+jωρ + J imp = Boundary conditions (BCs). There exist a variety of BCs that can be incorporated into Maxwell s equations. In the following, we describe those BCs that are of interest for the logging applications discussed in this paper. At this point, we are considering general 3D 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 3 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., (3.3) n E = 0, where n is the unit normal (outward) vector. 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., (3.4) 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, []), we can replace the original impressed electric volume current J imp with an equivalent electric surface current (3.5) 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 across S in the case of an interface condition, or simply n H on S in the case of a boundary condition. Similarly, an impressed magnetic volume current M imp can be replaced by the equivalent magnetic surface current (3.6) M imp S = [n E] S, defined on an arbitrary surface S enclosing the support of M imp. 3 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 [2] for details.

8 2092 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI Closure of the domain. We consider a bounded computational domain. A variety of BCs 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 3 is small. For example, it is possible to use an infinite element technique (as described in [7]) or an absorption-type BC such as a perfect matched layer (PML) [6, 26, 3]. Also, since the EM fields and their derivatives decay exponentially in the presence of lossy media (nonzero 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. According to the BCs discussed above, we will divide boundary Γ = intothe disjoint union of Γ E, where M imp Γ E = [n E] ΓE (with M imp Γ E possibly zero), with Γ H, where J imp Γ H =[n H] ΓH, (with J imp Γ H possibly zero) Variational formulation. From Maxwell s equations and the BCs 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 ()) 3 if and only if M imp (L 2 ()) 3. Since our objective is to find a solution E H(curl;)={F (L 2 ()) 3 : F (L 2 ()) 3 }, we shall assume in the case of the electric field formulation (E-formulation) derived below that M imp (L 2 ()) 3. If the prescribed M imp / (L 2 ()) 3, we may still solve Maxwell s equations with H(curl)-conforming finite elements for the magnetic field by using the H-formulation (3.3.2), or simply by prescribing an equivalent source M imp such that M imp M imp does not radiate outside the antenna [27]. Similarly, for the H-formulation, we will assume that J imp (L 2 ()) E-formulation. By dividing Faraday s law by magnetic permeability μ, multiplying the resulting equation by F, where F H ΓE (curl;) = {F H(curl;) : (n F) ΓE =0} is an arbitrary test function, and integrating over the domain, we arrive at the identity (3.7) μ ( E) ( F)dV = jω H ( F)dV μ Mimp ( F)dV. Integrating H ( F) dv by parts, and applying Ampere s law, we obtain (3.8) H ( F) dv = ( H) F dv n H F t ds Γ H = (σ + jωɛ)e F dv + J imp F dv n H F t ds. Γ H F t = F (F n) n is the tangential component of vector F on Γ H, and n is the unit normal outward (with respect to if Γ H ) vector. Substitution of (3.8) into

9 -FEM: ELECTROMAGNETIC APPLICATIONS 2093 (3.7) and use of (3.5) yields the following variational formulation: Find E E ΓE + H ΓE (curl; ) such that μ ( E) ( F) dv k 2 E F dv = jω J imp F dv (3.9) + jω F t ds μ Mimp ( F) dv F H ΓE (curl;), J imp Γ H Γ H where k 2 = ω 2 ɛ jωσ is the wave number and E ΓE is a lift (typically E ΓE =0)of the essential BC data E ΓE = M imp Γ E (denoted with the same symbol). Conversely, we can derive (3.), (3.3), and (3.5) from variational problem (3.9) H-formulation. By dividing Ampere s law by σ+jωɛ, multiplying the resulting equation by F, where F H ΓH (curl;)={f H(curl;) :(n F) ΓH = 0 } is an arbitrary test function, and integrating over the domain, we arrive at the identity jω k 2 ( H) ( F)dV = E ( F) dv (3.0) jω k 2 Jimp ( F) dv. Integrating E ( F) dv by parts and applying Faraday s law, we obtain E ( F) dv = ( E) F dv n E F t ds Γ (3.) E = jω μh F dv M imp F dv n E F t ds. Γ E Substitution of (3.) into (3.0) and use of (3.6) yields the following variational formulation: (3.2) Find H H ΓH + H ΓH (curl; ) such that σ + jωɛ ( H) ( F)dV + jω μh FdV = + M imp Γ E F t ds + Γ E M imp FdV σ + jωɛ Jimp ( F)dV F H ΓH (curl;), where H ΓH is a lift (typically H ΓH = 0) of the essential BC data H ΓH = J imp Γ H (denoted with the same symbol) 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 coordinate φ) around the axis of the borehole. Under this assumption, we obtain that for any vector field A =ˆρA ρ + ˆφA φ +ẑa z, (3.3) A = ˆρ A φ z ( + ˆφ Aρ z A ) z +ẑ (ρa φ ). ρ ρ ρ

10 2094 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI E-formulation. Next, we consider the space of all test functions F H D (curl; ) such that F =(0,F φ, 0). According to (3.3), F = ˆρ F φ z +ẑ (ρf φ ) (3.4). ρ ρ Variational formulation (3.9) reduces to a formulation in terms of the scalar field E φ only, namely, (3.5) 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 Γ N J imp φ,γ N 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 z L 2 (), E φ ΓD =0}. Similarly, for a test function F =(F ρ, 0,F z ), variational problem (3.9) simplifies to 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 (3.6) = jω Jρ imp F ρ + Jz imp F z dv + jω Fz ds μ M imp φ [ Fρ z F z ρ ] dv J imp ρ,γ Fρ N + J imp z,γ N Γ N F =(F ρ, 0,F z ) H D (curl;), where H D (curl;)={(e ρ,e z ):E =(E ρ, 0,E z ) L 2 (), ( E) φ = Eρ z Ez ρ L 2 (), (n E) ΓD =0}. In summary, problem (3.9) decouples into a system of two simpler problems described by (3.5) and (3.6). Remark. It has been shown in [4, 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 (3.2), we arrive at the following two decoupled variational problems in terms of (0,H φ, 0) (3.7) and (H ρ, 0,H z ) (3.8), respectively: (3.7) Find H φ H φ,d + H D () such that ( Hφ F φ σ + jωɛ z z + (ρh φ ) ρ 2 ρ + jω μh φ Fφ dv = F φ dv + + σ + jωɛ [ Jρ imp F φ M imp φ z + J z imp ρ (ρ F ) φ ) dv ρ (ρ F φ ) ρ Γ N M imp φ,γ N Fφ ds ] dv F φ H D().

11 (3.8) -FEM: ELECTROMAGNETIC APPLICATIONS 2095 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 + Γ N 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 (3.5) through (3.8), we remark the following: Physically, solutions of problems (3.6) and (3.7) correspond to the TE φ - mode (i.e., E φ = 0), and solutions of problems (3.5) and (3.8) correspond to the TM φ -mode (i.e., H φ = 0). The axis of symmetry is not a boundary of the original 3D problem, and therefore, a BC should not be needed to solve this problem. Nevertheless, formulations of problems (3.5) through (3.8) 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 BC 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 (3.9) and (3.2) (or alternatively, (3.5), (3.6), (3.7), and (3.8)), which we state here in terms of sesquilinear form b and antilinear form f: (4.) { Find E ED + V, 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. We have (4.2) b(e, F) = μ ( E) ( F) dv }{{} a E (E,F) k 2 ( E) ( F) dv } {{ } a H (E,F) k 2 E F dv } {{ } c E (E,F) μe F dv } {{ } c H (E,F) E-Form, H-Form, where sesquilinear forms a E, a H, c E, and c H are Hermitian, continuous, and 4 From the computational point of view, this effect can be achieved by artificially adding a homogeneous natural (Neumann) BC.

12 2096 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI semipositive definite. We define an energy inner product on V as (4.3) (E, F) := μ ( E) ( F) dv }{{} a E (E,F) k 2 ( E) ( F) dv } {{ } a H (E,F) + k 2 E F dv } {{ } c E (E,F) + μe F dv } {{ } c H (E,F) E-Form, H-Form, with the corresponding (energy) norm denoted by E. Notice the inclusion of the material properties in the definition of the norm. 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. 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

13 -FEM: ELECTROMAGNETIC APPLICATIONS 2097 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 [0], we approximate E and W with fine grid functions E h,p+, W h 2 2,p+, which have been obtained by solving the corresponding linear system of equations associated with the finite element subspace,p+. In the remainder of this article, E and W will denote the fine grid solutions V h 2 of the direct and dual problems (E = E h,p+, and W = W h 2 2,p+, respectively), and we will restrict ourselves to discrete finite element 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 (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 [9, 8], and used in [0, 23] for the construction of the fully automatic energy-norm-based -adaptive algorithm: Π curl,3d : V V, Π curl,2d : V 2D V 2D, and Π D : V D V D. We shall also consider three Galerkin projection operators: P curl,3d : 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,3d for problems (3.9) and (3.2), Π curl,2d for problems (3.6) and (3.8), or Π D for problems (3.5) and (3.7). Similarly, we will use the unique

14 2098 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI symbol P curl to denote either P curl,3d We define E = P curl, P curl,2d,orp D. 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, we conjecture 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,wehave (4.) L(e) K b K (E Π curl E, W Π curl W). By applying the Cauchy Schwarz 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 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 [0, 23]. We start by recalling the main objective of the self-adaptive (energy-norm-based) -refinement strategy, which consists of solving the following maximization problem: Find an optimal -grid in the following sense: (4.3) = arg max ĥp K E Π curl E 2 K E Πcurl ĥp E 2 K ΔN where E = E h 2,p+ is the fine grid solution, and ΔN >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) ĥp ΔN K E Π curl ĥp E K W Π curlw ] K ĥp, ΔN where,

15 -FEM: ELECTROMAGNETIC APPLICATIONS 2099 E = E h 2,p+ and W = W h 2,p+ are the fine grid solutions corresponding to the direct and dual problems, and ΔN >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 [0, 23], which utilizes a multistep approach (first optimization of edges, and then optimization of interior degrees of freedom). The subspace associated to an optimal finite element grid is always contained in the subspace associated with the finite element fine grid computed during the previous step Implementation details. In what follows, we discuss the main implementation details needed to extend the fully automatic (energy-norm-based) -adaptive algorithm [0, 23] 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 [8]). 2. Subsequently, we should treat both solutions as satisfying two different PDEs. We select functions E and W as the solutions of the system of two PDEs. 3. We proceed to redefine the evaluation of the error. The energy-norm error evaluation of a 2D function is replaced by the product E Π curl E W Π curl W. 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 self-adaptive strategy described in section 4 to simulate the response of the induction LWD instrument operating at 2 MHz considered in section 2.3, using formulation (3.5) for solenoidal coils and (3.7) for toroidal coils. Exactly the same results are obtained with formulations (3.8) and (3.6), respectively, as predicted by the theory. Thus, formulations (3.8) and (3.6) have been used as an extra verification of the simulations, and the corresponding results have been omitted in this article to avoid duplicity. Figure 3 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 (left and center panel). The right panel corresponds to the computation of the normalized second vertical difference of the electric field when considering an extra receiving antenna 5 cm above the second receiving antenna. The three curves (two for the second vertical difference of the electric field) correspond to. the rock formation with no mud-filtrate invasion, 2. the rock formation with a2 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 3. the previous (mud-invaded) rock formation, using a mandrel with relative magnetic permeability of 00. For toroidal antennas, we display in Figure 4 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. 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

16 200 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI Invasion Study No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel= Solenoid No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel= Invasion Study No Invasion 0.4/0.9m Invasion 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 Vertical Position of Second Receiving Antenna (m) 2 00 Ohm m Ohm m (0.4m) Ohm m 0 5 Ohm m (0.9m) 000 Ohm m Ohm m 00 Ohm m 00 Ohm m Amplitude First Vert. Diff. Electric Field (V/m 2 ) Phase (degrees) Amplitude Second Vert. Diff. Electric Field (V/m^3) Fig. 3. LWD problem equipped with a solenoidal source. Amplitude (left panel) and phase (center panel) of the first vertical difference of the electric field (divided by the distance between receivers) at the receiving coils. The normalized amplitude of the second vertical difference of the electric field is displayed in the right panel. 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 Invasion Study No Invasion 0.4/0.9 m Invasion 0.4/0.9m Invasion, Perm. Mandrel= 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. 4. 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.

17 -FEM: ELECTROMAGNETIC APPLICATIONS Solenoid With Magnetic Buffers Without Magnetic Buffers Toroid 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 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 ) Fig. 5. LWD problem equipped with a solenoidal source. Results obtained with the self-adaptive goal-oriented -FEM correspond to the use of solenoidal antennas (left panel), and toroidal antennas (right panel). The spatial distribution of electrical resistivity is also displayed to facilitate the physical interpretation of results. toroidal antennas are more sensitive to highly resistive formations. We also observe that the second vertical difference of the electric field is more sensitive to water invasion than the first vertical difference of the electric field (in both conductive and resistive formations). Figure 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. In contrast, 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 solver of linear equations utilized for these simulations is a multifrontal massively parallel sparse direct solver (MUMPS) [2, ] running in a single-processor machine equipped with a Pentium IV 3.0 GHz processor. The total amount of time utilized by our FEM depends upon the choice of initial grid and the quantity of interest to be computed. Twelve minutes were needed to compute each curve log of Figure 5, composed of 80 points. The exponential convergence obtained using the self-adaptive goal-oriented - FEM is shown in Figure 6 (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 Figure 6 (right panel), we display the exponential

18 202 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND 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 /3 ) Number of Unknowns N (scale N /3 ) Fig. 6. 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. 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 Figure Summary and conclusions. We have successfully applied a self-adaptive goal-oriented -FEM 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. Also, the use of -FEM provides the flexibility needed to accurately approximate the solution within the formation (using the p method) as well as the strong singularities caused by the abrupt geometry of the mandrel (using the h method). 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. Second vertical differences of electromagnetic fields are more sensitive to mud-filtrate invasion than first

19 -FEM: ELECTROMAGNETIC APPLICATIONS 203 p =8 p =7 Receiver II Receiver I p =6 p =5 p =4 p =3 Transmitter p =2 p = Fig. 7. LWD instrument equipped with a solenoidal source. Portion ( 20 cm 200 cm) of the final -grid. Different shades indicate different polynomial orders of approximation, ranging from (light grey) to 8 (white). vertical differences. 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 3D finite element 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 a 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 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 Figure 8). For a solenoidal coil located at z = 0, the resulting solution for a ρ b is given by [5, 20] E φ (ρ, z) = ωμ (A.) [J (k ρ a)+γh () 4πa (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, respectively, of the first type of order p, and k z = k 2 kρ. 2 For a toroidal coil located at z = 0, the resulting solution for a ρ b is given by H φ (ρ, z) = i (A.2) [J (k ρ a)+γh () 4πa (k ρa)]h () (k ρρ)e ikzz dk ρ,

20 204 PARDO, DEMKOWICZ, TORRES-VERDÍN, AND PASZYNSKI z COIL (Toroid/Solenoid) b a MANDREL Fig. 8. Geometry of a loop-antenna radiating in a homogeneous lossy medium in the presence of a highly conductive metallic mandrel. Distance in z axis from transmitter to receiver (in m) Analytical solution Mandrel Resisitivity: 0 7 Mandrel Resisitivity: 0 5 Mandrel Resisitivity: 0 3 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 3 Mandrel Resisitivity: Amplitude (V/m) Phase (degrees) Fig. 9. 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 3, and m) obtained with the self-adaptive goal-oriented -FEM. where Γ = J 0 (k ρ b)/h () 0 (k ρb). In Figures 9 and 0, 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, in terms of both amplitude

21 -FEM: ELECTROMAGNETIC APPLICATIONS 205 Distance in z axis from transmitter to receiver (in m) Analytical solution Mandrel Resisitivity: 0 7 Mandrel Resisitivity: 0 5 Mandrel Resisitivity: 0 3 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 3 Mandrel Resisitivity: Amplitude (A/m) Phase (degrees) Fig. 0. 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 3, and m) obtained with the self-adaptive goal-oriented -FEM. 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. We would like to acknowledge the expertise and technical advice received from L. Tabarovsky, A. Bespalov, T. Wang, and other members of the Science Department of Baker-Atlas. REFERENCES [] P. R. Amestoy, I. S. Duff, J.-Y. L Excellent, and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (200), pp [2] P. R. Amestoy, I. S. Duff, and J.-Y. L Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Engrg., 84 (2000), pp [3] B. I. Anderson, Modeling and Inversion Methods for the Interpretation of Resistivity Logging Tool Response, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, 200; available online at [4] F. Assous, C. Ciarlet, Jr., and S. Labrunie, Theoretical tools to solve the axisymmetric Maxwell equations, Math. Methods Appl. Sci., 25 (2002), pp [5] R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, in ENUMATH 97, Proceedings of the 2nd European Conference on Numerical Mathematics and Advanced Applications, Heidelberg, Germany, 997, H. G. Bock, F. Brezzi, R.