TWO-DIMENSIONAL HIGH-ACCURACY SIMULATION OF … · merically) a self-adaptive hp goal-oriented...

SIAM J. APPL. MATH. c© 2006 Society for Industrial and Applied MathematicsVol. 66, No. 6, pp. 2085–2106

TWO-DIMENSIONAL HIGH-ACCURACY SIMULATION OFRESISTIVITY LOGGING-WHILE-DRILLING (LWD)

MEASUREMENTS USING A SELF-ADAPTIVE GOAL-ORIENTEDhp FINITE ELEMENT METHOD∗

D. PARDO† , L. DEMKOWICZ‡ , C. TORRES-VERDIN§ , 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 elec-trical properties of rock formations. Logging instruments as well as rock formation properties areassumed to exhibit axial symmetry around the axis of a vertical borehole. The simulations areperformed with a self-adaptive goal-oriented hp-finite element method that delivers exponential con-vergence rates in terms of the quantity of interest (for example, the difference in the electrical currentmeasured at two receiver antennas) against the CPU time. Goal-oriented adaptivity allows for ac-curate approximations of the quantity of interest without the need to obtain an accurate solutionin the entire computational domain. In particular, goal-oriented hp-adaptivity becomes essential tosimulating LWD instruments, since it reduces the computational cost by several orders of magni-tude with respect to the global energy-norm-based hp-adaptivity. Numerical results illustrate theefficiency and high accuracy of the method, and provide physical interpretation of resistivity mea-surements obtained with LWD instruments. These results also describe the advantages of usingmagnetic buffers in combination with solenoidal antennas for strengthening the measured EM signalso that the “signal-to-noise” ratio is minimized.

Key words. hp-finite elements, exponential convergence, goal-oriented adaptivity, computa-tional electromagnetics, Maxwell’s equations, through casing resistivity tools (TCRT)

AMS subject classifications. 78A25, 78A55, 78M10, 65N50

DOI. 10.1137/050631732

1. Introduction. A plethora of energy-norm-based algorithms intended to gen-erate optimal grids have been developed throughout recent decades (see, for example,[10, 23] and references therein) to accurately solve a large class of engineering prob-lems. However, the energy-norm is a quantity of limited relevance for most engineeringapplications, especially when a particular objective is pursued, such as simulating theelectromagnetic response of geophysical resistivity logging instruments in a boreholeenvironment. In these instruments, the amplitude of the measurement (for example,the electric field) is typically several orders of magnitude smaller at the receiver an-tennas than at the transmitter antennas. Thus, small relative errors of the solutionin the energy-norm do not imply small relative errors of the solution at the receiver

∗Received by the editors May 17, 2005; accepted for publication (in revised form) February 14,2006; published electronically October 16, 2006. This work was financially supported by Baker-Atlasand the Joint Industry Research Consortium on Formation Evaluation supervised by Prof. C. Torres-Verdin.

http://www.siam.org/journals/siap/66-6/63173.html†Institute for Computational Engineering and Sciences (ICES) and Department of Petroleum and

Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712 ([email protected]).

‡Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin,Austin, TX 78712 ([email protected]).

§Department of Petroleum and Geosystems Engineering, The University of Texas at Austin,Austin, TX 78712 ([email protected]).

¶Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin,Austin, TX 78712 ([email protected]). On leave from Department of Computer Methods inMetallurgy, AGH University of Science and Technology, Cracow, Poland.

2085

2086 PARDO, DEMKOWICZ, TORRES-VERDIN, AND PASZYNSKI

antennas. Indeed, it is not uncommon to construct adaptive grids delivering a relativeerror in the energy-norm below 1% while the solution at the receiver antennas stillexhibits a relative error above 1000% (see [18]).

Consequently, in order to accurately simulate logging-while-drilling (LWD) resis-tivity measurements in this paper, we develop a self-adaptive strategy to approximatea specific feature of the solution. Refinement strategies of this type are called goal-oriented adaptive algorithms [16, 22], and are based on minimizing the error of aprescribed quantity of interest mathematically expressed in terms of a linear func-tional (see [5, 12, 17, 16, 22, 24] for details).

In this paper, we formulate, implement, and study (both theoretically and nu-merically) a self-adaptive hp goal-oriented algorithm intended to solve electrodynamicproblems. This algorithm is an extension of the fully automatic (energy-norm-based)hp-adaptive strategy described in [10, 23], and a continuation of concepts presentedin [19, 25] for elliptic problems.

We apply the self-adaptive hp goal-oriented algorithm to accurately simulate in-duction LWD instruments in a borehole environment with axial symmetry. Theseinstruments are widely used by the geophysical logging industry, and their simula-tion requires resolution of electromagnetic (EM) singularities generated by the LWDgeometry and rock formation materials [28], as well as resolution of high materialcontrasts that occur between the mandrel and the borehole.

Other methods for simulation of LWD measurements include the transmissionline matrix method [14], fast Fourier transform [29], and finite differences [26, 13]. Incontrast to previous contributions, here we consider a detailed geometry of the logginginstrument, which requires the resolution of strong singularities in the EM fields, weaccount for the finite conductivity of the mandrel, we incorporate magnetic buffers inboth transmitter and receiver antennas, we consider the effect of the magnetic per-meability of the mandrel, and we provide extremely accurate results with guaranteedrelative error bounds below 0.1% (0.001% if desired). We also consider a high contrastin conductivity among different layers in the formation, and we present a comparisonbetween using two and three receiver antennas.

The organization of this paper is as follows. In section 2, we describe the maincharacteristics of induction logging instruments. We also describe our problem ofinterest, composed of an induction LWD instrument in a borehole environment, andused for the assessment of the rock formation electrical properties. In section 3,we introduce Maxwell’s equations, governing the EM phenomena and explaining thephysics of resistivity measurements. We also derive the corresponding variationalformulation for axisymmetric problems. A self-adaptive goal-oriented hp algorithmfor electrodynamic problems is described in section 4. The corresponding details ofimplementation are discussed in the same section. Simulations and numerical resultsconcerning the response of LWD instruments in a borehole environment are shown insection 5. Section 6 draws the main conclusions and outlines future lines of research.Finally, in the appendix, we compare numerical results with a semianalytical solutionobtained using Bessel functions for a simplified LWD model problem. The comparisonis intended to verify the code as well as to illustrate the high-accuracy results obtainedwith the self-adaptive goal-oriented hp-finite element method (FEM).

hp-FEM: ELECTROMAGNETIC APPLICATIONS 2087

2. Alternate current (AC) logging applications. In this article, we consideran induction1 LWD instrument operating at 2 MHz. The instrument makes use ofone of the following two types of source antennas/coils:

• solenoidal coils (Figure 1, left panel), and• toroidal coils (Figure 1, right panel).

Fig. 1. Two coil antennas: a solenoid antenna (left panel) composed of a wire wrapped arounda cylinder, and a toroid antenna (right panel) composed of a wire wrapped around a toroid.

2.1. Induction LWD instruments based on solenoidal coils. For axisym-metric problems, these logging instruments generate a TMφ field; i.e., the only non-zero components of the EM fields are Eφ, Hρ, and Hz, where (ρ, φ, z) denote thecylindrical system of coordinates.

A solenoidal coil (Figure 1) produces an impressed current Jimp that we mathe-matically describe as

Jimp(r) = φIδ(ρ− a)δ(z),(2.1)

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 functionδ(ρ− a)δ(z) with an approximate function UF that considers the finite dimensions ofthe coil, and such that

∫UF dρdz = 1.

The analytical electric far-field solution excited by a solenoidal coil of radius aradiating in homogeneous media is given in terms of the electric field by (see [15])

E = φωμkIπa2 e−jkd

4πd

[1 − j

kd

]ρ

d,(2.2)

where k =√ω2ε− jωσ is the wave number; j =

√−1 is the imaginary unit; ω is

angular frequency; ε, μ, and σ stand for dielectric permittivity, magnetic permeability,and electrical conductivity of the medium, respectively; and d is the distance betweenthe source coil and the receiver coil.

In order to avoid the dependence upon the dimensions of the solenoid, we imposea current on the solenoidal coil equal to 1/(πa2)A, i.e., equivalent to that of 1A with a

1Induction logging instruments are characterized by the fact that impressed current Jimp isdivergence-free (i.e., ∇ · Jimp = 0).


vertical magnetic dipole (VMD). The corresponding far-field solution in homogeneousmedia is given by (see [15])

E = φωμkIe−jkd

4πd

[1 − j

kd

]ρ

d.(2.3)

Thus, solution (2.3) is independent of the dimensions of the coil.2

2.2. Induction LWD instruments based on toroidal coils. For axisymmet-ric problems, these logging instruments generate a TEφ field; i.e., the only nonzerocomponents of the EM fields are Hφ, Eρ, and Ez.

A toroidal coil induces a magnetic current IM in the azimuthal direction. If weplace a toroid of radius a radiating in homogeneous media, the resulting magneticfar-field is given by (see [15])

H = φ(σ + jωε)πa2IM jke−jkd

4πd

[1 − j

kd

]ρ

d.(2.4)

In order to avoid the dependence upon the dimensions of the toroid, we impose amagnetic current on the toroidal coil equal to that induced by a (σ + jωε)A electriccurrent excitation with a vertical electrical dipole (VED), also known as a Hertziandipole. The corresponding magnetic far-field solution in homogeneous media is givenby (see [15])

H = φ(σ + jωε)Ijke−jkd

4πd

[1 − j

kd

]ρ

d.(2.5)

In this case, IM = I/(πa2).

2.2.1. Goal of the computations. We are interested in simulating the EMresponse of an induction LWD instrument in a borehole environment.

For a solenoidal coil, the main objective of our simulation is to compute the firstdifference of the voltage between the two receiving coils of radius a divided by the(vertical) distance Δz between them, i.e.,

V1 − V2

Δz=

(∮l1

E(l) dl −∮l2

E(l) dl

)/(Δz) =

2πa

Δz(E(l1) − E(l2)) ,(2.6)

where l1 and l2 are the first and second receiving coils, respectively, and l1 ∈ l1,l2 ∈ l2 are two arbitrary points located at the receiving coils. Notice that, due to theaxisymmetry of the electric field, E(lji ) = E(lki ) for all lji , l

ki ∈ li.

This quantity of interest (first difference of voltage) is widely used in resistivitylogging applications. Indeed, a first-order asymptotic approximation of the electricfield response at low frequencies (Born’s approximation) shows that the voltage at areceiver coil is proportional to the rock formation resistivity in the proximity of sucha coil (see [15] for details). At higher frequencies (> 20 kHz), asymptotic approx-imations (see [3] for details) also indicate the dependence of the voltage upon therock formation conductivity. Thus, an adequate approximation of the rock formation

2In resistivity logging applications, it is customary to consider solutions that have been divided bythe 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 instrumentscan be readily compared.


conductivity (which is unknown a priori in practical applications) can be estimatedfrom the voltage measured at the receiving coils. Computing the first difference ofthe voltage between two receivers (rather than the voltage at one receiver) is conve-nient for improving the vertical resolution of the measurements. This well-known factamong well-logging practitioners will be illustrated here with numerical experiments.

For a toroidal coil, the main objective of these simulations is to compute the firstdifference of the electric current at the two receiving coils of radius a divided by the(vertical) distance Δz between them, i.e.,

I1 − I2Δz

=

(∮l1

H(l) dl −∮l2

H(l) dl

)/(Δz) =

2πa

Δz(H(l1) − H(l2)) .(2.7)

Notice that the main difference between a toroidal and a solenoidal coil is thatthe former generates an impressed magnetic current, while the latter produces animpressed electric current. This fact leads to the physical consideration that, if thevoltage due to a solenoidal coil is proportional to the rock formation conductivity, thenthe electric current enforced by a toroidal coil is also proportional to the rock formationresistivity. Thus, the selection of the quantity of interest for toroidal coils (firstdifference of electric current) is dictated by the physical relation between solenoidaland toroidal coils and by the previous choice of a quantity of interest for solenoidalcoils (first difference of voltage).

2.3. Description of an LWD instrument in a borehole environment. Weconsider an LWD instrument composed of the following axisymmetric materials (alldimensions are given in cm):

• one transmitter and two receiver coils defined on1. ΩC1 = {(ρ, φ, z) : 7.1 < ρ < 7.3, −2.5 < z < 2.5},2. ΩC

2= {(ρ, φ, z) : 7.1 < ρ < 7.3, 98.75 < z < 101.25}, and,

3. ΩC3= {(ρ, φ, z) : 7.1 < ρ < 7.3, 113.75 < z < 116.25}, respectively;

• three magnetic buffers with resistivity 104 Ω·m and relative permeability 104,defined on

1. ΩB1= {(ρ, φ, z) : 6.675 < ρ < 6.985, −5 < z < 5},

2. ΩB2= {(ρ, φ, z) : 6.675 < ρ < 6.985, 97.5 < z < 102.5}, and,

3. ΩB3= {(ρ, φ, z) : 6.675 < ρ < 6.985, 112.5 < z < 117.5}, respectively;

and• a metallic mandrel with resistivity 10−6 Ω·m defined on ΩM = {(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 < 102.5} ∪ {(ρ, φ, z) : 6.675 < ρ < 7.6, 112.5 < z < 117.5}).

This LWD instrument moves along the vertical direction (z-axis) in a subsurfaceborehole environment composed of

• a borehole mud with resistivity 0.1 Ω · m defined on1. ΩBH = {(ρ, φ, z) : ρ < 10.795} − (∪iΩBi ∪ ΩM ), and

• three formation materials of resistivities 100 Ω · m, 10000 Ω · m, and 1 Ω · m,defined on

1. ΩM1= {(ρ, φ, z) : ρ ≥ 10.795, (z < −50 or z > 100)},

2. ΩM2 = {(ρ, φ, z) : ρ ≥ 10.795, −50 ≤ z < 0}, and,3. ΩM3 = {(ρ, φ, z) : ρ ≥ 10.795, 0 ≤ z ≤ 100}, respectively.

Figure 2 shows the geometry of the described logging instrument and borehole envi-ronment.

3. Maxwell’s equations. In this section, we first introduce the time-harmonicMaxwell equations in the frequency domain. They form a set of first-order partial


100 Ohm − m

10000 Ohm − m

100 Ohm − m

1 Ohm − m

50 c

m

cm15

0.00

0001

Ohm

− m

100

cm

100

cm

0.1

Ohm

− m

0.00

0001

Ohm

m

5 cm

Man

drel

Borehole0.1 Ohm − mRadius = 10.795 cm

Radius 7.6 cm

10000 Relative Permeability

Magnetic Buffer10000 Ohm − m

10 c

m

6.675 cm

Fig. 2. 2D cross section of the geometry of an induction LWD problem composed of a metallicmandrel, one transmitter and two receiver coils equipped with magnetic buffers, a borehole, and fourlayers in the rock formation (with different resistivities). The right panel is an enlarged view of thegeometry (left panel) in the vicinity of the transmitter antenna.

differential equations (PDEs). Then, we describe boundary conditions needed for thesimulation of our logging applications of interest. Finally, we derive a variationalformulation in terms of either the electric or the magnetic field, and we reduce thedimension of the computational problem by considering axial symmetry.

3.1. Time-harmonic Maxwell equations. Assuming a time-harmonic depen-dence of the form ejωt, where t denotes time and ω �= 0 is angular frequency, Maxwell’sequations can be written as⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

∇×H = (σ + jωε)E + Jimp Ampere’s law,

∇×E = −jωμ H − Mimp Faraday’s law,

∇ · (εE) = ρ Gauss’ law of electricity, and

∇ · (μH) = 0 Gauss’ law of magnetism.

(3.1)

Here H and E denote the magnetic and electric fields, respectively; Jimp is a pre-scribed, impressed electric current density; Mimp is a prescribed, impressed magneticcurrent density; ε, μ, and σ stand for dielectric permittivity, magnetic permeability,and electrical conductivity of the medium, respectively; and ρ denotes the electriccharge distribution. We assume μ �= 0.

The equations described in (3.1) 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 bothelectric field E and magnetic field H must be square integrable. According to (3.1)2and (3.1)4, Mimp is divergence-free.

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, andby utilizing Gauss’ electric law, we arrive at the so-called continuity equation,

∇ · (σE) + jωρ + ∇ · Jimp = 0.(3.2)

3.2. Boundary conditions (BCs). There exist a variety of BCs that can beincorporated into Maxwell’s equations. In the following, we describe those BCs thatare of interest for the logging applications discussed in this paper. At this point, weare considering general 3D domains. A discussion on boundary terms correspondingto the axisymmetry condition is postponed to section 3.4.

3.2.1. Perfect electric conductor (PEC). Maxwell’s equations are to be sat-isfied in the whole space minus domains occupied by a PEC. A PEC is an idealizationof a highly conductive media. Inside a region where σ → ∞, the corresponding elec-tric field converges to zero3 by applying Ampere’s law. Faraday’s law implies that thetangential component of the electric field E must remain continuous across materialinterfaces in the absence of impressed magnetic surface currents. Consequently, thetangential component of the electric field must vanish along the PEC boundary, i.e.,

n×E = 0,(3.3)

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 fieldpremultiplied by the permeability must remain continuous across material interfaces.Therefore, the normal component of the magnetic field must vanish along the PECboundary, i.e.,

n · H = 0.(3.4)

The tangential component of magnetic field (surface current) and normal com-ponent of the electric field (surface charge density) need not be zero and may bedetermined a posteriori.

3.2.2. Source antennas. Antennas are modeled by prescribing an impressedvolume current Jimp. Using the equivalence principle (see, for example, [11]), we canreplace the original impressed electric volume current Jimp with an equivalent electricsurface current

JimpS = [n×H]S ,(3.5)

defined on an arbitrary surface S enclosing the support of Jimp, 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 Mimp can be replaced by the equiv-alent magnetic surface current

MimpS = −[n×E]S,(3.6)

defined on an arbitrary surface S enclosing the support of Mimp.

3This result is true under the physical consideration that impressed volume current Jimp andσE should remain finite, i.e., 〈Jimp, ψ〉, 〈σE, ψ〉 < ∞ for every test function ψ. See [21] for details.


3.2.3. Closure of the domain. We consider a bounded computational domainΩ. A variety of BCs can be imposed on the boundary ∂Ω such that the difference be-tween 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 describedin [7]) or an absorption-type BC such as a perfect matched layer (PML) [6, 26, 13].Also, since the EM fields and their derivatives decay exponentially in the presence oflossy media (nonzero conductivity), we may simply impose a homogeneous Dirichletor Neumann BC on the boundary of a sufficiently large computational domain.

In the field of geophysical logging applications, it is customary to impose a homo-geneous 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 aresistive media). We will follow the same approach.

According to the BCs discussed above, we will divide boundary Γ = ∂Ω into thedisjoint union of

• ΓE , where MimpΓE

= −[n×E]ΓE(with Mimp

ΓEpossibly zero), with

• ΓH , where JimpΓH

= [n×H]ΓH, (with Jimp

ΓHpossibly zero).

3.3. Variational formulation. From Maxwell’s equations and the BCs de-scribed above, we derive the corresponding standard variational formulation in termsof the electric or magnetic field as follows.

First, we notice from Faraday’s law that ∇×E ∈ (L2(Ω))3 if and only if Mimp ∈(L2(Ω))3. Since our objective is to find a solution E ∈ H(curl; Ω) = {F ∈ (L2(Ω))3 :∇×F ∈ (L2(Ω))3}, we shall assume in the case of the electric field formulation(E-formulation) derived below that Mimp ∈ (L2(Ω))3. If the prescribed Mimp /∈(L2(Ω))3, we may still solve Maxwell’s equations with H(curl)-conforming finite el-ements for the magnetic field by using the H-formulation (3.3.2), or simply by pre-scribing an equivalent source Mimp such that Mimp − Mimp does not radiate outsidethe antenna [27].

Similarly, for the H-formulation, we will assume that Jimp ∈ (L2(Ω))3.

3.3.1. E-formulation. By dividing Faraday’s law by magnetic permeability μ,multiplying the resulting equation by ∇×F, where F ∈ HΓE

(curl; Ω) = {F ∈H(curl; Ω) : (n×F)|ΓE

= 0 } is an arbitrary test function, and integrating overthe domain Ω, we arrive at the identity

∫Ω

1

μ(∇×E) · (∇×F)dV = −jω

∫Ω

H · (∇×F)dV −∫

Ω

1

μMimp · (∇×F)dV.(3.7)

Integrating∫Ω

H · (∇×F) dV by parts, and applying Ampere’s law, we obtain

∫Ω

H · (∇×F) dV =

∫Ω

(∇×H) · F dV −∫

ΓH

n×H · Ft dS

=

∫Ω

(σ + jωε)E · F dV +

∫Ω

Jimp · F dV −∫

ΓH

n×H · Ft dS.(3.8)

Ft = F− (F ·n) ·n is the tangential component of vector F on ΓH , and n is the unitnormal outward (with respect to Ω if ΓH ⊂ ∂Ω) vector. Substitution of (3.8) into


(3.7) and use of (3.5) yields the following variational formulation:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Find E ∈ EΓE+ HΓE

(curl; Ω) such that∫Ω

1

μ(∇×E) · (∇×F) dV −

∫Ω

k2E · F dV = −jω

∫Ω

Jimp · F dV

+ jω

∫ΓH

JimpΓH

· Ft dS −∫

Ω

1

μMimp · (∇×F) dV ∀ F ∈ HΓE

(curl; Ω),

(3.9)

where k2 = ω2ε − jωσ is the wave number and EΓEis a lift (typically EΓE

= 0) ofthe essential BC data EΓE

= −M impΓE

(denoted with the same symbol).Conversely, we can derive (3.1), (3.3), and (3.5) from variational problem (3.9).

3.3.2. H-formulation. By dividing Ampere’s law by σ+jωε, multiplying the re-sulting equation by ∇×F, where F ∈ HΓH

(curl; Ω) = {F ∈ H(curl; Ω) : (n×F)|ΓH=

0 } is an arbitrary test function, and integrating over the domain Ω, we arrive at theidentity

−jω

∫Ω

1

k2(∇×H) · (∇×F)dV =

∫Ω

E · (∇×F) dV

− jω

∫Ω

1

k2Jimp · (∇×F) dV.

(3.10)

Integrating∫Ω

E · (∇×F) dV by parts and applying Faraday’s law, we obtain∫Ω

E · (∇×F) dV =

∫Ω

(∇×E) · F dV −∫

ΓE

n×E · Ft dS

= − jω

∫Ω

μH · F dV −∫

Ω

Mimp · F dV −∫

ΓE

n×E · Ft dS.(3.11)

Substitution of (3.11) into (3.10) and use of (3.6) yields the following variationalformulation:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Find H ∈ HΓH+ HΓH

(curl; Ω) such that∫Ω

1

σ + jωε(∇×H) · (∇×F)dV + jω

∫Ω

μH · FdV = −∫

Ω

Mimp · FdV

+

∫ΓE

MimpΓE

· Ft dS +

∫Ω

1

σ + jωεJimp · (∇×F)dV ∀ F ∈ HΓH

(curl; Ω),

(3.12)

where HΓHis a lift (typically HΓH

= 0) of the essential BC data HΓH= J imp

ΓH

(denoted with the same symbol).

3.4. Cylindrical coordinates and axisymmetric problems. We considercylindrical coordinates (ρ, φ, z). For the geophysical logging applications consideredin this article, we assume that both the logging instrument and the rock formationproperties are axisymmetric (invariant with respect to the azimuthal coordinate φ)around the axis of the borehole. Under this assumption, we obtain that for any vectorfield A = ρAρ + φAφ + zAz,

∇×A = −ρ∂Aφ

∂z+ φ

(∂Aρ

∂z− ∂Az

∂ρ

)+ z

1

ρ

∂(ρAφ)

∂ρ.(3.13)


3.4.1. E-formulation. Next, we consider the space of all test functions F ∈HD(curl; Ω) such that F = (0, Fφ, 0). According to (3.13),

∇×F = −ρ∂Fφ

∂z+ z

1

ρ

∂(ρFφ)

∂ρ.(3.14)

Variational formulation (3.9) reduces to a formulation in terms of the scalar fieldEφ only, namely,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find Eφ ∈ Eφ,D + H1D(Ω) such that∫

Ω

1

μ

(∂Eφ

∂z

∂Fφ

∂z+

1

ρ2

∂(ρEφ)

∂ρ

∂(ρFφ)

∂ρ

)dV −

∫Ω

k2Eφ Fφ dV

= − jω

∫Ω

J impφ Fφ dV + jω

∫ΓN

J impφ,ΓN

Fφ dS

−∫

Ω

1

μ

[−M imp

ρ

∂Fφ

∂z+ M imp

z

1

ρ

∂(ρFφ)

∂ρ

]dV ∀ Fφ ∈ H1

D(Ω),

(3.15)

where H1D(Ω) = {Eφ : (0, Eφ, 0) ∈ HD(curl; Ω)} = {Eφ ∈ L2(Ω) : 1

ρEφ +∂Eφ

∂ρ ∈L2(Ω),

∂Eφ

∂z ∈ L2(Ω), Eφ|ΓD= 0}. Similarly, for a test function F = (Fρ, 0, Fz),

variational problem (3.9) simplifies to⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find E = (Eρ, 0, Ez) ∈ ED + HD(curl; Ω) such that∫Ω

1

μ

(∂Eρ

∂z− ∂Ez

∂ρ

)(∂Fρ

∂z− ∂Fz

∂ρ

)dV −

∫Ω

k2(EρFρ + EzFz) dV

= − jω

∫Ω

J impρ Fρ + J imp

z Fz dV + jω

∫ΓN

J impρ,ΓN

Fρ + J impz,ΓN

Fz dS

−∫

Ω

1

μM imp

φ

[∂Fρ

∂z− ∂Fz

∂ρ

]dV ∀ F = (Fρ, 0, Fz) ∈ HD(curl; Ω),

(3.16)

where HD(curl; Ω) = {(Eρ, Ez) : E = (Eρ, 0, Ez) ∈ L2(Ω), (∇×E)|φ =∂Eρ

∂z − ∂Ez

∂ρ ∈L2(Ω), (n×E)|ΓD

= 0}.In summary, problem (3.9) decouples into a system of two simpler problems de-

scribed by (3.15) and (3.16).Remark 1. It has been shown in [4, Lemma 4.9] that space H1

D(Ω) can also beexpressed as H1

D(Ω) = {Eφ ∈ L2(Ω) : 1ρEφ ∈ L2(Ω), ∇(ρ,z)Eφ ∈ L2(Ω)}.

3.4.2. H-formulation. Using the same decomposition of test functions (i.e.,F = (0, Fφ, 0), and F = (Fρ, 0, Fz)) for variational problem (3.12), we arrive atthe following two decoupled variational problems in terms of (0, Hφ, 0) (3.17) and(Hρ, 0, Hz) (3.18), respectively:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find Hφ ∈ Hφ,D + H1D(Ω) such that∫

Ω

1

σ + jωε

(∂Hφ

∂z

∂Fφ

∂z+

1

ρ2

∂(ρHφ)

∂ρ

∂(ρFφ)

∂ρ

)dV

+ jω

∫Ω

μHφ FφdV = −∫

Ω

M impφ Fφ dV +

∫ΓN

M impφ,ΓN

Fφ dS

+

∫Ω

1

σ + jωε

[−J imp

ρ

∂Fφ

∂z+ J imp

z

1

ρ

∂(ρFφ)

∂ρ

]dV ∀ Fφ ∈ H1

D(Ω) .

(3.17)

hp-FEM: ELECTROMAGNETIC APPLICATIONS 2095⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find H = (Hρ, 0, Hz) ∈ HD + HD(curl; Ω) such that∫Ω

1

σ + jωε

(∂Hρ

∂z− ∂Hz

∂ρ

) (∂Fρ

∂z− ∂Fz

∂ρ

)dV

+ jω

∫Ω

μ(HρFρ + HzFz) dV = −∫

Ω

M impρ Fρ + M imp

z Fz dV

+

∫ΓN

M impρ,ΓN

Fρ + M impz,ΓN

Fz dS +

∫Ω

1

σ + jωεJ impφ

[∂Fρ

∂z− ∂Fz

∂ρ

]dV

∀ F = (Fρ, 0, Fz) ∈ HD(curl; Ω) .

(3.18)

From the formulation of problems (3.15) through (3.18), we remark the following:• Physically, solutions of problems (3.16) and (3.17) correspond to the TEφ-

mode (i.e., Eφ = 0), and solutions of problems (3.15) and (3.18) correspondto the TMφ-mode (i.e., Hφ = 0).

• The axis of symmetry is not a boundary of the original 3D problem, andtherefore, a BC should not be needed to solve this problem. Nevertheless, for-mulations of problems (3.15) through (3.18) require the use of spaces H1

D(Ω)and HD(curl; Ω) described above. The former space involves the singularweight 1

ρ , which implicitly requires a homogeneous Dirichlet BC along the

axis of symmetry. The latter space can be considered as it is (by using 2Dedge elements), and no BC is necessary4 to solve the problem.

4. Self-adaptive goal-oriented hp-FEM. We are interested in solving varia-tional problems (3.9) and (3.12) (or alternatively, (3.15), (3.16), (3.17), and (3.18)),which we state here in terms of sesquilinear form b and antilinear form f :

{Find E ∈ ED + V,

b(E,F) = f(F) ∀F ∈ V ,(4.1)

where• ED 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

b(E,F) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∫Ω

1

μ(∇×E) · (∇×F) dV︸︷︷︸

aE(E,F)

−∫

Ω

k2E · F dV︸︷︷︸cE(E,F)

E-Form,

∫Ω

1

k2(∇×E) · (∇×F) dV︸︷︷︸

aH(E,F)

−∫

Ω

μE · F dV︸︷︷︸cH(E,F)

H-Form,(4.2)

where sesquilinear forms aE , aH , cE , and cH are Hermitian, continuous, and

4From the computational point of view, this effect can be achieved by artificially adding a ho-mogeneous natural (Neumann) BC.


semipositive definite. We define an “energy” inner product on V as

(E,F) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∫Ω

1

μ(∇×E) · (∇×F) dV︸︷︷︸

aE(E,F)

+

∫Ω

|k2|E · F dV︸︷︷︸cE(E,F)

E-Form,

∫Ω

1

|k2| (∇×E) · (∇×F) dV︸︷︷︸aH(E,F)

+

∫Ω

μE · F dV︸︷︷︸cH(E,F)

H-Form,

(4.3)

with the corresponding (energy) norm denoted by ‖E‖. Notice the inclusionof the material properties in the definition of the norm.

4.1. Representation of the error in the quantity of interest. Given anhp-FE subspace Vhp ⊂ V, we discretize (4.1) as follows:{

Find Ehp ∈ ED + Vhp,

b(Ehp,Fhp) = f(Fhp) ∀Fhp ∈ Vhp .(4.4)

The objective of goal-oriented adaptivity is to construct an optimal hp-grid, in thesense that it minimizes the problem size needed to achieve a given tolerance error fora given quantity of interest L, with L denoting a linear and continuous functional. Byrecalling the linearity of L, we have

Error of interest = L(E) − L(Ehp) = L(E − Ehp) = L(e),(4.5)

where e = E − Ehp denotes the error function. By defining the residual rhp ∈ V′ asrhp(F) = f(F) − b(Ehp,F) = b(E − Ehp,F) = b(e,F), we look for the solution of thedual problem: {

Find W ∈ V,

b(F,W) = L(F) ∀F ∈ V.(4.6)

Problem (4.6) has a unique solution in V. The solution W is usually referred to asthe influence function.

By discretizing (4.6) via, for example, Vhp ⊂ V, we obtain{Find Whp ∈ Vhp,

b(Fhp,Whp) = L(Fhp) ∀Fhp ∈ Vhp .(4.7)

Definition of the dual problem plus the Galerkin orthogonality for the originalproblem imply the final representation formula for the error in the quantity of interest,namely,

L(e) = b(e,W) = b(e,W − Fhp︸︷︷︸ε

) = b(e, ε) .

At this point, Fhp ∈ Vhp is arbitrary, and b(e, ε) = b(e, ε) denotes the bilinearform corresponding to the original sesquilinear form.

Notice that, in practice, the dual problem is solved not for W but for its complexconjugate 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) reducesto 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 bilinearform b, we wish to obtain a sharp upper bound for |L(e)| that depends upon the meshparameters (element size h and order of approximation p) only locally. Then, a self-adaptive algorithm intended to minimize this bound will be defined.

First, using a procedure similar to the one described in [10], we approximate E andW with fine grid functions Eh

2 , p+1, Wh2 , p+1, which have been obtained by solving the

corresponding linear system of equations associated with the finite element subspaceVh

2 , p+1. In the remainder of this article, E and W will denote the fine grid solutions

of the direct and dual problems (E = Eh2 , p+1, and W = Wh

2 , p+1, 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 contri-

butions. Let bK denote a contribution from element K to sesquilinear form b. It thenfollows that

|L(e)| = |b(e, ε)| ≤∑K

|bK(e, ε)| ,(4.8)

where summation over K indicates summation over elements.

4.2. Projection-based interpolation operator. Once we have a representa-tion formula for the error in the quantity of interest in terms of the sum of elementcontributions given by (4.8), we wish to express this upper bound in terms of localquantities, i.e. in terms of quantities that do not vary globally when we modify thegrid locally. For this purpose, we introduce the idea of projection-based interpolationoperators.

First, in order to simplify the notation, we define the following three spaces ofadmissible solutions:

• V = HD(curl; Ω),• V2D = HD(curl; Ω), and• V 1D = H1

D(Ω).The corresponding hp-finite element spaces will be denoted by Vhp, V2D

hp , and V 1Dhp ,

respectively.At this point, we introduce three projection-based interpolation operators that

have been defined in [9, 8], and used in [10, 23] for the construction of the fullyautomatic energy-norm-based hp-adaptive algorithm:

• Πcurl,3Dhp : V −→ Vhp,

• Πcurl,2Dhp : V2D −→ V2D

hp , and

• Π1Dhp : V 1D −→ V 1D

hp .We shall also consider three Galerkin projection operators:

• Pcurl,3Dhp : V −→ Vhp,

• Pcurl,2Dhp : V2D −→ V2D

hp , and

• P 1Dhp : V 1D −→ V 1D

hp .

To further simplify the notation, we will utilize the unique symbol Πcurlhp to denote

all projection-based interpolation operators mentioned above. Depending upon theproblem formulation (and corresponding space of admissible solutions), Πcurl

hp should

be understood as Πcurl,3Dhp for problems (3.9) and (3.12), Πcurl,2D

hp for problems (3.16)

and (3.18), or Π1Dhp for problems (3.15) and (3.17). Similarly, we will use the unique


symbol Pcurlhp to denote either Pcurl,3D

hp , Pcurl,2Dhp , or P 1D

hp .

We define Ehp = Pcurlhp E. Equation (4.8) then becomes

|L(e)| ≤∑K

|bK(E, ε)| =∑K

|bK(E − Πcurlhp E, ε) + bK(Πcurl

hp E − Pcurlhp E, ε)| .(4.9)

Given an element K, we conjecture that |bK(Πcurlhp E − PhpE, ε)| will be negligible

compared to |bK(E − Πcurlhp E, ε)|. Under this assumption, we conclude that

|L(e)| �∑K

|bK(E − Πcurlhp E, ε)| .(4.10)

In particular, for ε = W − Πcurlhp W, we have

|L(e)| �∑K

|bK(E − Πcurlhp E,W − Πcurl

hp W)| .(4.11)

By applying the Cauchy–Schwarz inequality, we obtain the next upper bound for|L(e)|:

|L(e)| �∑K

‖e‖K‖ε‖K ,(4.12)

where e = E − Πcurlhp E, ε = W − Πcurl

hp W, and ‖ · ‖K denotes energy-norm ‖ · ‖restricted to element K.

4.3. Fully automatic goal-oriented hp-refinement algorithm. We de-scribe an hp self-adaptive algorithm that utilizes the main ideas of the fully auto-matic (energy-norm-based) hp-adaptive algorithm described in [10, 23]. We start byrecalling the main objective of the self-adaptive (energy-norm-based) hp-refinementstrategy, which consists of solving the following maximization problem:⎧⎪⎪⎨⎪⎪⎩

Find an optimal hp-grid in the following sense:

hp = arg maxhp

∑K

‖E − Πcurlhp E‖2

K − ‖E − Πcurlhp

E‖2K

ΔN,

(4.13)

where• E = Eh

2 , p+1 is the fine grid solution, and

• ΔN > 0 is the increment in the number of unknowns from grid hp to grid hp.Similarly, for goal-oriented hp-adaptivity, we propose the following algorithm

based on estimate (4.12):⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find an optimal hp-grid in the following sense:

hp = arg maxhp

∑K

[‖E − Πcurl

hp E‖K · ‖W − Πcurlhp W‖K

ΔN

−‖E − Πcurl

hpE‖K · ‖W − Πcurl

hpW‖K

ΔN

],

(4.14)

where


• E = Eh2 , p+1 and W = Wh

2 , p+1 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 hp to grid hp.Implementation of the goal-oriented hp-adaptive algorithm is based on the op-

timization procedure used for energy-norm hp-adaptivity [10, 23], which utilizes amultistep approach (first optimization of edges, and then optimization of interior de-grees of freedom). The subspace associated to an optimal finite element grid is alwayscontained in the subspace associated with the finite element fine grid computed duringthe previous step.

4.4. Implementation details. In what follows, we discuss the main implemen-tation details needed to extend the fully automatic (energy-norm-based) hp-adaptivealgorithm [10, 23] to a fully automatic goal-oriented hp-adaptive algorithm.

1. First, the solution W of the dual problem on the fine grid is necessary. Thisgoal can be attained either by using a direct (frontal) solver or an iterative(two-grid) solver (see [18]).

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 errorevaluation of a 2D function is replaced by the product ‖ E − Πcurl

hp E ‖ · ‖W − Πcurl

hp W ‖.4. After these simple modifications, the energy-norm-based self-adaptive algo-

rithm may now be utilized as a self-adaptive goal-oriented hp algorithm.

5. Numerical results. In this section, we apply the goal-oriented hp self-adap-tive strategy described in section 4 to simulate the response of the induction LWDinstrument operating at 2 MHz considered in section 2.3, using formulation (3.15)for solenoidal coils and (3.17) for toroidal coils. Exactly the same results are ob-tained with formulations (3.18) and (3.16), respectively, as predicted by the theory.Thus, formulations (3.18) and (3.16) have been used as an extra verification of thesimulations, and the corresponding results have been omitted in this article to avoidduplicity.

Figure 3 displays the first vertical difference of the electric field (divided by thedistance between the two receivers) for the described LWD instrument equipped withsolenoidal coils (left and center panel). The right panel corresponds to the computa-tion of the normalized second vertical difference of the electric field when consideringan extra receiving antenna 15 cm above the second receiving antenna. The threecurves (two for the second vertical difference of the electric field) correspond to

1. the rock formation with no mud-filtrate invasion,2. the rock formation with a 2 Ω·m 40 cm horizontal mud layer invading the

1 Ω·m rock formation layer, and a 5Ω·m 90cm horizontal mud layer invadingthe 10000 Ω·m rock formation layer, and

3. the previous (mud-invaded) rock formation, using a mandrel with relativemagnetic permeability of 100.

For toroidal antennas, we display in Figure 4 the first vertical difference of the mag-netic field (divided by the distance between the two receivers). The three displayedcurves correspond to the three situations discussed above.

These results illustrate the strong dependence of the LWD response on the rockformation resistivity. We observe that solenoidal antennas are more sensitive to highlyconductive formations as well as to the electrical permeability of the mandrel, while


0.04 0.08 0.12 0.16 0.2−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Electric Field (V/m2)

Vert

ical P

ositio

n o

f R

eceiv

ing A

nte

nna (

m)

Invasion Study

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Vert

ical P

ositio

n o

f R

eceiv

ing A

nte

nna (

m)

Solenoid

No Invasion0.4/0.9 m Invasion0.4/0.9m Invasion, Perm. Mandrel=100


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

2 Ohm−m (0.4m) 2 Ohm−m (0.4m)

5 Ohm−m (0.9m) 5 Ohm−m (0.9m)

0.2 0.3 0.4 0.5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude Second Vert. Diff. Electric Field (V/m^3)

Ve

rtic

al P

osi

tion

of

Se

con

d R

ece

ivin

g A

nte

nn

a (

m)

Invasion Study

No Invasion0.4/0.9m Invasion

2 Ohm−m (0.4m) 1 Ohm−m

5 Ohm−m (0.9m) 1000 Ohm−m

100 Ohm−m

100 Ohm−m

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 betweenreceivers) at the receiving coils. The normalized amplitude of the second vertical difference of theelectric field is displayed in the right panel. Results obtained with the self-adaptive goal-orientedhp-FEM. The spatial distribution of electrical resistivity is also displayed to facilitate the physicalinterpretation of results.

10−5

10−3

10−1

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Magnetic Field (A/m2)

Vert

ical P

ositio

n o

f R

eceiv

ing A

nte

nna (

m)

Invasion Study

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Vert

ical P

ositio

n o

f R

eceiv

ing A

nte

nna (

m)

Toroid



100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

2 Ohm−m (0.4m) 2 Ohm−m (0.4m)

5 Ohm−m (0.9m) 5 Ohm−m (0.9m)

Fig. 4. LWD problem equipped with a toroidal source. Amplitude (left panel) and phase (rightpanel) 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 hp-FEM. The spatialdistribution of electrical resistivity is also displayed to facilitate the physical interpretation of results.


10−4

10−2

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. of Electric Field (V/m2)

Vert

ical P

ositio

n o

f R

eceiv

ing A

nte

nna (

m)

Solenoid

With Magnetic BuffersWithout Magnetic Buffers

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

10−5

100

105

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. of Magnetic Field (A/m2)

Vert

ical P

ositio

n o

f R

eceiv

ing A

nte

nna (

m)

Toroid

With Magnetic BuffersWithout Magnetic Buffers

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

Fig. 5. LWD problem equipped with a solenoidal source. Results obtained with the self-adaptivegoal-oriented hp-FEM correspond to the use of solenoidal antennas (left panel), and toroidal anten-nas (right panel). The spatial distribution of electrical resistivity is also displayed to facilitate thephysical interpretation of results.

toroidal antennas are more sensitive to highly resistive formations. We also observethat the second vertical difference of the electric field is more sensitive to water in-vasion than the first vertical difference of the electric field (in both conductive andresistive formations).

Figure 5 illustrates the effect of the magnetic buffers. By removing the magneticbuffers from the logging instrument’s design, the amplitude of the received signaldecreases by a factor of up to 200 in the case of a solenoidal source. For practicalapplications, a strong signal on the receivers is desired to minimize the noise-to-signalratio. Thus, it is appropriate to use magnetic buffers in combination with solenoidalantennas. In contrast, the use of magnetic buffers with toroidal antennas is notadvisable since they weaken the received signal. In both cases, the phase and shapeof the solution is not sensitive to the presence (or not) of magnetic buffers, and thecorresponding results have been omitted.

The solver of linear equations utilized for these simulations is a multifrontal mas-sively parallel sparse direct solver (MUMPS) [2, 1] running in a single-processor ma-chine equipped with a Pentium IV 3.0 GHz processor. The total amount of timeutilized by our FEM depends upon the choice of initial grid and the quantity of in-terest to be computed. Twelve minutes were needed to compute each curve—log—ofFigure 5, composed of 80 points.

The exponential convergence obtained using the self-adaptive goal-oriented hp-FEM is shown in Figure 6 (left panel), by considering an arbitrary fixed position ofthe logging instrument for a solenoid antenna. The final grid delivers a relative errorin the quantity of interest below 0.00001%; i.e., the first 7 significant digits of thequantity of interest are exact. In Figure 6 (right panel), we display the exponential


0 1000 8000 27000 6400010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Number of Unknowns N (scale N1/3)

Rela

tive E

rror

in %

Goal−Oriented hp−Adaptivity

0 1000 8000 27000 6400010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Number of Unknowns N (scale N1/3)

Rela

tive E

rror

in %

Energy−norm hp−Adaptivity

Upper bound for |L(e)|/|L(u)||L(e)|/|L(u)|

Energy−norm error|L(e)|/|L(u)|

Fig. 6. LWD problem equipped with a solenoidal source. Left panel: convergence behaviorobtained with the self-adaptive goal-oriented hp-FEM shows exponential convergence rates for esti-mate (4.8) (solid curve) used for optimization. The dashed curve describes the relative error in thequantity of interest. Right panel: convergence behavior obtained with the self-adaptive energy-normhp-FEM shows exponential convergence rates for the energy-norm. The dashed curve describes therelative error in the quantity of interest.

convergence of the energy-norm-based hp-FEM. The final hp-grid delivers an energy-norm error below 0.01%. Nevertheless, the quantity of interest still contains a relativeerror above 15%.

A final goal-oriented hp-grid delivering a relative error in the quantity of interestof 0.1% is displayed in Figure 7.

6. Summary and conclusions. We have successfully applied a self-adaptivegoal-oriented hp-FEM algorithm to simulate the axisymmetric response of an induc-tion LWD instrument in a borehole environment. These simulations would not bepossible with energy-norm adaptive algorithms. Also, the use of hp-FEM providesthe flexibility needed to accurately approximate the solution within the formation(using the p method) as well as the strong singularities caused by the abrupt geome-try of the mandrel (using the h method).

Numerical results illustrate the exponential convergence of the method (allow-ing for high accuracy simulations), the suitability of the presented formulations foraxisymmetric electrodynamic problems, and the main physical characteristics of thepresented induction LWD instrument. These results suggest the use of solenoidal an-tennas for the assessment of highly conductive rock formation materials, and toroidalantennas for the assessment of highly resistive materials. Solenoidal antennas shouldbe used in combination with magnetic buffers to strengthen the measured EM signal,while the use of magnetic buffers with toroidal antennas should be avoided. Bothtypes of antennas can be used to study mud-filtrate invasion. Second vertical differ-ences of electromagnetic fields are more sensitive to mud-filtrate invasion than first


Transmitter −→

Receiver I −→Receiver II −→

p = 8

p = 7

p = 6

p = 5

p = 4

p = 3

p = 2

p = 1

Fig. 7. LWD instrument equipped with a solenoidal source. Portion (120 cm× 200 cm) of thefinal hp-grid. Different shades indicate different polynomial orders of approximation, ranging from1 (light grey) to 8 (white).

vertical differences.Since the influence function used by the self-adaptive goal-oriented hp-adaptive

algorithm is approximated via finite elements, the numerical method presented in thisarticle is problem independent, and it can be applied to 1D, 2D, and 3D finite elementdiscretizations of H1-, H(curl)-, and H(div)-spaces.

Appendix. A loop-antenna radiating in a homogeneous lossy mediumin 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 anadditional mechanism to verify the code, as well as to provide comparative resultsbetween analytical and numerical solutions.

We consider a solenoid (or a toroid) of radius a radiating at a frequency of 2 MHzin a homogeneous lossy medium (with resistivity equal to 1 Ω ·m), in the presence ofan infinitely large cylindrical mandrel (with resistivity equal to 10−6 Ω · m) of radiusb < 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 givenby [15, 20]

Eφ(ρ, z) =−ωμ

4πa

∫ ∞

−∞[J1(kρa) + ΓH

(1)1 (kρa)]H

(1)1 (kρρ)e

ikzzdkρ,(A.1)

where Γ = −J1(kρb)/H(1)1 (kρb), Jp and H

(1)p are the Bessel and Hankel functions,

respectively, of the first type of order p, and kz =√k2 − k2

ρ.

For a toroidal coil located at z = 0, the resulting solution for a ≤ ρ ≤ b is givenby

Hφ(ρ, z) =−i

4πa

∫ ∞

−∞[J1(kρa) + ΓH

(1)1 (kρa)]H

(1)1 (kρρ)e

ikzzdkρ,(A.2)


z

COIL (Toroid/Solenoid)

MANDREL

b a

Fig. 8. Geometry of a loop-antenna radiating in a homogeneous lossy medium in the presenceof a highly conductive metallic mandrel.

10−8

10−6

10−4

10−2

100

0.5

1

1.5

2

2.5

3

3.5

Amplitude (V/m)

Dis

tan

ce in

z−

axi

s fr

om

tra

nsm

itte

r to

re

ceiv

er

(in

m)

−180 −90 0 90 1800.5

1

1.5

2

2.5

3

3.5

Phase (degrees)

Dis

tan

ce in

z−

axi

s fr

om

tra

nsm

itte

r to

re

ceiv

er

(in

m)

Analytical solution

Mandrel Resisitivity: 10−7



Mandrel Resisitivity: 1

Analytical solution





Fig. 9. Solution (electric field) along the vertical axis passing through a solenoid radiating in ahomogeneous medium in the presence of a metallic mandrel. Analytical solution (mandrel is a PEC)against the numerical solution for different mandrel resistivities (10−7, 10−5, 10−3, and 1 Ω · m)obtained with the self-adaptive goal-oriented hp-FEM.

where Γ = −J0(kρb)/H(1)0 (kρb).

In Figures 9 and 10, we display a comparison between analytical and numericalresults (obtained using the self-adaptive hp goal-oriented algorithm) for the solenoidaland toroidal coils, respectively. We selected b = 0.0254 cm and a = 0.03048 cm. Thenumerical results accurately reproduce the analytical ones, in terms of both amplitude


10−10

10−5

100

105

0.5

1

1.5

2

2.5

3

3.5

Amplitude (A/m)

Dis

tance

in z

−axi

s fr

om

tra

nsm

itter

to r

ece

iver

(in m

)

Analytical solution





−180 −90 0 90 1800.5

1

1.5

2

2.5

3

3.5

Phase (degrees)

Dis

tance

in z

−axi

s fr

om

tra

nsm

itter

to r

ece

iver

(in m

)

Analytical solution





Fig. 10. Solution (magnetic field) along a vertical axis passing through a toroid radiating in ahomogeneous medium in the presence of a metallic mandrel. Analytical solution (mandrel is a PEC)against the numerical solution for different mandrel resistivities (10−7, 10−5, 10−3, and 1 Ω · m)obtained with the self-adaptive goal-oriented hp-FEM.

and phase.When considering a solenoid, the logging instrument response using a mandrel of

resistivity 10−5 Ω · m or a PEC mandrel are indistinguishable in terms of amplitude.A similar situation occurs for a toroid. In terms of phase, induction instrumentsequipped with solenoidal coils appear to be more sensitive to the mandrel resistivitythan those equipped with toroidal coils.

Acknowledgments. We would like to acknowledge the expertise and technicaladvice received from L. Tabarovsky, A. Bespalov, T. Wang, and other members of theScience Department of Baker-Atlas.

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