Simulation of borehole-eccentered triaxial induction measurementsusing a Fourier hp finite-element method

Myung Jin Nam1, David Pardo2, and Carlos Torres-Verdín3

ABSTRACT

We developed a numerical method for accurate simulationof triaxial induction measurements in the presence of bore-hole-eccentered logging instruments. This 3D simulationmethod combines a Fourier series expansion in a new systemof coordinates with a 2D, goal-oriented, high-order, and self-adaptive hp finite-element refinement strategy. The resultingnumerical scheme provides accurate and reliable simulationsof triaxial induction measurements while reducing thecomputational complexity of conventional 3D problems.We employed this method to quantify effects of borehole-eccentered triaxial tools fabricated with conductive andresistive mandrels. Numerical experiments indicate that aconductive mandrel can mitigate borehole-eccentered effectson horizontal magnetic field measurements to a level below25%, which contrast those observed in the presence of re-sistive mandrels, in which measurements can differ by morethan 100% depending solely upon the borehole-eccentereddistance. When the mandrel and borehole were conductive,Hxy measurements experienced significant (up to 100%) tool-eccentered effects across layers whose electrical conductivitywas different from that of the borehole.

INTRODUCTION

Ever since Doll (1951) proposed the use of electrical boreholelogging for formation evaluation, this technique has been exten-sively used for hydrocarbon reservoir characterization and surveil-lance. Induction logging instruments are the only choice forthe evaluation of electrical conductivity in a borehole filled with

oil-based mud (OBM) because galvanic devices (e.g., the laterolog)exhibit poor performance in resistive mud. Traditional inductionlogging devices incorporating one transmitter coil and one or sev-eral receiver coils are sensitive only to the horizontal resistivity invertical wells that penetrate horizontal beds (Barber et al., 2004;Abubakar et al., 2006). As a result, these devices exhibit low sen-sitivity to the presence of hydrocarbon saturation in thinly lami-nated sand-shale sequences; in such cases, vertical conductivityis a better indicator of hydrocarbon pay zones (Anderson et al.,2001). Vertical conductivity is higher than horizontal conductivityin transversely isotropic hydrocarbon-bearing formations becausevertical conductivities are mainly determined by the high resistivityof hydrocarbon-bearing layers, whereas horizontal conductivitymainly reflects the low resistivity of shale (Luling et al., 1994).As a remedy to the above problems, multicomponent triaxial

electromagnetic (EM) induction tools were designed to measurehorizontal and vertical conductivity (Krieghauser et al., 2000; Zhda-nov et al., 2001; Rosthal et al., 2003; Davydycheva, 2011). Thesetools include three mutually orthogonal transmitter coils located atthe same vertical position and three collocated, mutually orthogonalreceiver coils to measure the three possible vector-field componentsassociated with each source. Horizontal and vertical conductivitiesare estimated from the nine magnetic-field measurements corre-sponding to all possible couplings between each of the three trans-mitters and three receivers.The main limitation of triaxial induction measurements is that

borehole effects on transverse induction measurements are muchlarger than those on conventional coaxial tools. These effects aremost noticeable when the tool is decentralized, specifically in a con-ductive borehole. Nowadays, some service companies employ aconductive mandrel rather than an insulating sleeve to reduce bore-hole effects on triaxial induction measurements. This paper studiesborehole effects on triaxial measurements in the presence of a

Manuscript received by the Editor 28 December 2011; revised manuscript received 25 June 2012; published online 20 December 2012.1Sejong University, Energy and Mineral Resources Engineering, Seoul, Republic of Korea. Formerly the University of Texas at Austin, Department of

Petroleum and Geosystems Engineering, Austin, Texas USA. E-mail: nmj1203@gmail.com.2University of the Basque Country (UPV/EHU), Department of Applied Mathematics, Statistics, and Operational Research, Leioa, Spain, and IKERBAS-

QUE, Basque Foundation for Science, Bilbao, Spain. Formerly the University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin,Texas USA. E-mail: dzubiaur@gmail.com.

3The University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas USA. E-mail: cverdin@mail.utexas.edu.© 2012 Society of Exploration Geophysicists. All rights reserved.

D41

GEOPHYSICS, VOL. 78, NO. 1 (JANUARY-FEBRUARY 2013); P. D41–D52, 15 FIGS.10.1190/GEO2011-0524.1

borehole-eccentered conductive or resistive mandrel. Our mainobjective is to accurately assess borehole-eccentered effects viareliable numerical simulations.When considering realistic tool properties and dimensions,

simulations of triaxial measurements in the presence of borehole-eccentered tools is rather challenging due to the 3D nature of theproblem. Previous attempts to solve similar problems employ 3Dsoftware (e.g., Druskin et al., 1999; Newman and Alumbaugh,2002; Davydycheva et al., 2003; Hou et al., 2006; Mallan andTorres-Verdín, 2007). However, the complexity of arbitrary 3D geo-metries increases the computational requirements, and, as a result, a3D simulation algorithm is in general computationally expensive.To overcome this problem and achieve faster computations, wedevelop a new formulation that reduces the cost, hence the burden,of 3D computations.The computational complexity of 3D algorithms can be reduced

by employing a new system of coordinates ζ ¼ ðζ1; ζ2; ζ3Þ de-signed in such a way that material properties become invariant withrespect to the quasi-azimuthal direction ζ2 (Pardo et al., 2008).Using a Fourier series expansion in the new system of coordinateswith respect to ζ2 yields to a 3D formulation consisting of a se-quence of weakly coupled 2D problems. Specifically, this couplingtakes place only in the borehole part of the domain (and neither inthe formation nor in the logging instrument), thereby producing asparsity pattern in the resulting stiffness matrix that significantlyreduces central processing unit (CPU) time and memory require-ments (with respect to conventional 3D problems). The weak cou-pling occurring between different 2D problems constitutes the mainadvantage of the proposed method. At the same time, we are stillable to solve the original 3D problem, as opposed to the case oftraditional 2.5D formulations (e.g., Torres-Verdín and Habashy,1994) consisting of uncoupled 2D problems, which are inadequateto simulate borehole-eccentered measurements.The resulting sequence of coupled 2D problems is solved with a

goal-oriented self-adaptive hp finite-element (FE) method (Pardoet al., 2008), where h denotes the element size and p the polynomialorder of approximation within each element. This numerical algo-rithm constructs an optimal grid for the problem via local-meshrefinements. By properly combining h and p refinements, the algo-rithm delivers highly accurate simulation results in limited CPUtimes, along with an estimate of the numerical error. The resultingFourier hp FE method enables detailed simulations of borehole-eccentered triaxial measurements with unprecedented accuracy.

TRIAXIAL INDUCTION TOOL

For the estimation of electrical conductivity and anisotropyin vertical and deviated wells, a triaxial induction tool providesnine magnetic field measurements per logging position, i.e., threeorthogonal magnetic fields (Hx,Hy,Hz) excited by each of the threeorthogonal impressed volumetric magnetic currents (Mx, My, Mz).We denote these measurements as (Hxx,Hxy,Hxz), (Hyx,Hyy,Hyz),and (Hzx, Hzy, Hzz), corresponding to sources Mx, My, and Mz,respectively.The logging instrument under consideration incorporates three

collocated orthogonal transmitter coils and several sets of three col-located orthogonal receiver and bucking coils, whose offsets fromthe source coils are different from each other. These sets of receiverand bucking coils are designed to measure the secondary magneticfield with different radial lengths of investigation. For simplicity,

this study considers only two different offsets denoted as shortand long offsets, respectively. The short offset has bucking and re-ceiver coils located at a distance equal to 15.25 and 22.86 cm awayfrom the transmitter, respectively. For the long offset, the buckingand receiver coils are located at a distance equal to 68.46 and99.06 cm away from the transmitter, respectively (Figure 1b). Theoperating frequency for the short offset is 105.3 kHz, and it is26.325 kHz for the long offset. Likewise, the set of receiver andbucking coils in the long offset is located above the transmittercoils, whereas in the short offset it is located below the source coil,whose vertical position is assumed to be at the center of the mandrel(Figure 1b). We assume that the coils have a finite-sized cross sec-tion with dimensions of 0.8 cm × 1 cm in the ρ-z plane in a cylind-rical system of coordinates (ρ, φ, z), whereas the center of bothfinite-size antennas is located at ρ ¼ 3.5 cm.In the numerical simulations, we consider triaxial induction re-

sistive and conductive mandrels, whose dimensions are 6 cm in dia-meter and 4 m in length (Figure 1b). Resistivities of these mandrelsare assumed to be 106 and 10−6 Ωm, respectively, with correspond-ing relative magnetic permeability and electrical permittivity equalto one for both mandrels. Notice that the resistive mandrel is a math-ematical model used to represent one of the most common types ofinduction tools, in which transmitter and receiver systems are cov-ered with an insulating sleeve to protect the coils (Rabinovich et al.,2006; Davydycheva et al., 2009).

SIMULATION METHOD

Maxwell’s equations

Assuming a time dependence of the form eþjωt, Maxwell’s equa-tions in a Cartesian system of coordinates defined over a domain Ωare given (for a nonzero frequency ω) by

∇ × E ¼ −jωμH −Mimp; ðFaraday’s lawÞ;∇ ×H ¼ ðσ þ jωεÞEþ Jimp; ðAmpere’s lawÞ; (1)

where H and E are the magnetic and electric fields, respectively,Jimp andMimp are the impressed electric and magnetic current den-sities, respectively, and ε, μ, σ are the dielectric permittivity, mag-netic permeability, and electrical conductivity tensors, respectively.We impose n × E ¼ 0 in the boundary Γ ¼ ∂Ω of a large enough

domain Ω, which is a conventional technique for truncation of thecomputational domain in lossy media. By multiplying μ−1 by Fara-day’s law in equation 1, premultiplying the resulting equation by∇ × F (where F ∈ HΓ ðcurl; ΩÞ ¼ fF ∈ L2ðΩÞ;∇ × F ∈ L2ðΩÞ∶ðn × FÞjΓ ¼ 0g is an arbitrary test function), integrating over do-main Ω by parts, and applying Ampere’s law, we obtain the follow-ing variational formulation:

8<:

Find E ∈ HΓðcurl;ΩÞ such that:

h∇ × F;μ−1∇ × EiL2ðΩÞ − hF;k2EiL2ðΩÞ¼ −jωhF; JimpiL2ðΩÞ − h∇ × F;μ−1MimpiL2ðΩÞ ∀ F ∈ HΓðcurl;ΩÞ;

(2)

where k2 ¼ ω2ε − jωσ, and hf1; f2iL2ðΩÞ ¼ ∫ Ωf1f2dV is the L2-

inner product of two complex-valued functions f1 and f2 (and the *symbol represents the complex conjugate).

D42 Nam et al.

New system of coordinates

We now consider a vertical well with a borehole-eccentered triax-ial induction tool embedded in a 2D-layered earth. The 3D geometryof a borehole-eccentered logging instrument in a vertical borehole(Figure 2a) can be represented in the following system of coordinatesζ ¼ ðζ1; ζ2; ζ3Þ (Figure 2b; Pardo et al., 2008; Nam et al., 2010c):( x1 ¼ f1ðζ1Þ þ ζ1 cos ζ2

x2 ¼ ζ1 sin ζ2x3 ¼ ζ3

; (3)

where f1 is defined as

f1ðζ1Þ ¼8<:

ρ0 ζ1 < ρ1ρ0ðζ1 − ρ2Þ∕ðρ1 − ρ2Þ ρ1 ≤ ζ1 ≤ ρ2ζ1 ζ1 > ρ2

; (4)

with ρ1 and ρ2 (ρ1 < ρ2) being the interfaces between subdomains 1and 2, and between subdomains 2 and 3, respectively, and ρ0(ρ0 þ ρ1 < ρ2) being the distance from the center of the tool tothe center of the borehole (Figure 2b). Mathematically speaking,ρ0 can be made arbitrarily close to the value of ρ2 − ρ1 (whichcorresponds to the situation in which the logging instrument istouching the borehole wall). However, as ρ0 approaches thevalue ρ2 − ρ1, the number of required Fourier modes needed tomaintain the discretization error bounded tends to infinity. Thus,from the numerical point of view, it is advisableto leave some distance between the logging in-strument and the borehole wall (e.g., a bore-hole-eccentricity equal to or below 90%–95%,where 100% indicates the tool touching the bore-hole wall).The above system of coordinates is such that

material properties remain invariant with respectto the quasi-azimuthal direction ζ2, making it op-timal for a spectral discretization (e.g., Fourierseries expansion) along ζ2. In addition, thedescribed change of coordinates (given by themapping x ¼ ϕðζÞ) is globally continuous, bijec-tive, and with a positive Jacobian determinant,therefore suitable for FE computations. Forany arbitrary scalar-valued function g, we denote~g ≔ g ∘ ψ . According to the chain rule, we have

∇u ¼X3i;n¼1

∂ ~g∂ζn

∂ζn∂xi

exi ¼ J−1T∂ ~g∂ζ

; (5)

where exi is the unit vector in the xi-direction,∂ ~g∂ζ

is the vector with the nth component being ∂ ~g∂ζn,

and the Jacobian matrix is given by

J ¼∂xi∂ζj

i;j¼1;2;3

¼ f 0

1 þ cos ζ2 −ζ1 cos ζ2 0

sin ζ2 ζ1 cos ζ2 0

0 0 1

!;

(6)

where ζ1, ζ2, and ζ3 are those coordinates used in equation 3, and f1is defined in equation 4.When switching to the new system of coordinates, Maxwell’s

equations remain invariant with the exception of material propertiesand source terms, which now incorporate information on the newcoordinate system. For example, the impressed magnetic current inthe new system of coordinates becomes ~Mimp

NEW ¼ J−1 ~MimpJ−1jJj,where J is the Jacobian matrix of the coordinate transformation, jJjis its determinant, and ~Mimp ¼ Mimp ∘ ϕ. Similar expressions areobtained for ~μNEW, ~kNEW , and ~Jimp

NEW . For additional details, seePardo et al. (2008).

Fourier series expansion inthe new system of coordinates

Any function G in the above system of coordinates is periodic inthe ζ2-direction with the period equal to 2π (Pardo et al., 2008).Thus, it can be represented using its Fourier series expansionwith respect to ζ2 as G ¼Pl¼∞

l¼−∞ flðGÞejlζ2 , where ejlζ2 are theso-called Fourier modes and flðGÞ ¼ 1ffiffiffiffi

2πp ∫ 2π

0 Ge−jlζ2dζ2 are theFourier modal coefficients.To derive our Fourier FE formulation, we apply the change of

coordinates defined in equation 3 to the original variational formu-lation (equation 2), and we expand all its terms as a Fourier serieswith respect to ζ2. By multiplying the resulting variational formula-tion in terms of Fourier modes by a monomodal test function

Figure 1. Configuration of the assumed triaxial logging instrument. The diameter of themandrel is equal to 6 cm. (a) Transmitter-receivers configuration. (b) Triaxial tool.

Borehole-eccentered EM effects D43

F ¼ frðFÞejrζ2 and using the orthogonality of Fourier modes inL2ð½0; 2πÞÞ, we arrive at the final Fourier FE formulation (see Pardoet al., 2008)

8>>>><>>>>:

Find frð ~EÞ ∈ Vrð ~Ω2DÞ such that:

hfrð∇ × ~FÞ; fr−lð ~μ−1NEWÞflð∇ × ~EÞiL2ð ~Ω2DÞ − hfrð ~FÞ; fr−lð ~k2NEWÞflð ~EÞiL2ð ~Ω2DÞ

¼ −jωhfrð ~FÞ; frð ~JimpNEWÞiL2ð ~Ω2DÞ

− hfrð∇ × ~FÞ; fr−lð ~μ−1NEWÞflð ~Mimp

NEWÞiL2ð ~Ω2DÞ ∀ frð ~FÞ ∈ Vrð ~Ω2DÞ;(7)

where the Einstein summation convention with −∞ ≤ l, r ≤ ∞ isused, Γ is assumed to be independent of ζ2, and Vrð ~Ω2DÞ¼ frðHΓðcurl;ΩÞÞ, with

frð∇ × ~FÞ ¼ ½∇ × ðfrð ~FÞejrζ2Þe−jrζ2 and

flð∇ × ~EÞ ¼ ½∇ × ðflð ~EÞejlζ2Þe−jlζ2 : (8)

For a bilinear form b and a linear form l, one can define

brl flðuÞ ¼ bðflðEÞ; frðFÞÞ¼ hfrð∇ × ~FÞ; fr−lð ~μ−1

NEWÞflð∇ × ~EÞiL2ð ~Ω2DÞ

− hfrð ~FÞ; fr−lð ~k2NEWÞflð ~EÞiL2ð ~Ω2DÞ; (9)

and

lr ¼ lðfrðFÞÞ ¼ −jωhfrð ~FÞ; frð ~JimpNEWÞiL2ð ~Ω2DÞ

− hfrð∇ × ~FÞ; fr−lð ~μ−1NEWÞflð ~Mimp

NEWÞiL2ð ~Ω2DÞ: (10)

Based on the above definitions, we express the final Fourier FEformulation (equation 7) in matrix form for the case of seven Four-ier modes (−3 ≤ r ≤ 3) as

26666666666664

b−3−3 b−3−2 b−3−1 b−30 b−31 b−32 b−33b−2−3 b−2−2 b−2−1 b−20 b−21 b−22 b−23b−1−3 b−1−2 b−1−1 b−10 b−11 b−12 b−13b0−3 b0−2 b0−1 b00 b01 b02 b03b1−3 b1−2 b1−1 b10 b11 b12 b13b2−3 b2−2 b2−1 b20 b21 b22 b23b3−3 b3−2 b3−1 b30 b31 b32 b33

37777777777775

26666666666664

f−3ðuÞf−2ðuÞf−1ðuÞf0ðuÞf1ðuÞf2ðuÞf3ðuÞ

37777777777775

¼

26666666666664

l−3

l−2

l−1

l0

l1

l2

l3

37777777777775: (11)

For subdomains 1 and 3 (see Figure 2), fr−lð ~μNEWÞ ¼ fr−lð ~k2NEWÞ ¼ 0 when r − l ≠ 0, and thus interaction among different2D problems in the new system of coordinates occurs only withinsubdomain 2. The resulting stiffness matrix in subdomains 1 and 3becomes diagonal (in terms of the Fourier modes interaction),which constitutes a major advantage of the Fourier FE formulationover traditional 3D formulations (Nam et al., 2010c). More pre-cisely, our formulation delivers a weak coupling among different2D problems. Thus, it can be interpreted as a 2.75D formulationin the sense that it is neither a full 3D formulation (in which all2D problems are fully coupled) nor a 2.5D formulation (in whichall 2D problems are uncoupled). Our formulation is at the same timemore efficient than traditional 3D formulations (due to the weakcoupling among different 2D problems) and capable of simulatingborehole-eccentered measurements employing a finite number ofFourier modes, which is not possible with 2.5D formulations. How-ever, note that as the tool surface approaches the borehole wall[ρ0 → ðρ2 − ρ1Þ in equation 4], a larger number of Fourier modes

is needed in the simulation.

3D sources and receivers

The new system of coordinates reduces exactlyto the cylindrical system of coordinates withinsubdomain 1, that is, within the subdomain con-taining the triaxial logging instrument along withthe sources and receivers. Source Mz can be ap-proximated by a small solenoidal coil (Jφ) (2Dsource), whereas 3D sources (namely, Mx andMy) can be modeled as approximations ofDirac delta functions in the ζ2-direction, thatis, δðζ2 − ζ2;sÞ (where ζ2;s is 0 and π∕2 forMx and My, respectively; i.e., Mx ¼ δðζ2Þ andMy ¼ δðζ2 − π∕2Þ [Nam et al., 2010b]). Specifi-cally, 3D sources (and receivers) are simulatedas Dirac delta functions in the quasi-azimuthaldirection ζ2, whereas they exhibit a finite area

Figure 2. Geometry of the borehole simulation problem, composed of three spatial sub-domains: (1) logging instrument, (2) borehole, and (3) formation. (a) Cross section.(b) Top view.

D44 Nam et al.

in the ζ1 − ζ3 plane (i.e., they are sheet sources). The same conceptis applied for modeling receivers measuring the Hx, Hy, and Hz

components of the solution.

A self-adaptive goal-oriented hp-FEM

The final variational formulation (equation 7) is solved using the3D direct solver MUMPS (MUMPS, 2012), and the quasi-optimalgrid is built employing a 2D self-adaptive, goal-oriented, high-or-der hp-FE algorithm (Pardo et al., 2006), where h indicates theelement size and p is the polynomial order of approximation.We allow for p to vary from one to eight. The self-adaptive hp-refinement strategy automatically performs optimal (and local)mesh refinements in h and p and employs the basis functionsfor electromagnetics described in Demkowicz (2007). It is empha-sized that the hp-FEM algorithm provides numerical solutions withhigh accuracy in a very limited amount of time, because the itera-tive process converges exponentially fast in terms of the error in thequantity of interest (solution at the receivers) versus the problemsize (number of unknowns).In triaxial EM induction logging, field intensities at receiver an-

tennas become several orders of magnitude smaller than at thesource. Thus, a small absolute error in the overall solution doesnot imply a small relative error at the receiver. The self-adaptivegoal-oriented strategy generates optimal hp refinements intendedto reduce the error in the quantity of interest (either the field inten-sity at the receiver or the difference in intensities between buckingand receiver coils). In other words, it minimizes the error of aprescribed quantity of interest mathematically expressed in termsof a linear functional (Paraschivoiu and Patera, 1998; Prudhommeand Oden, 1999; Heuveline and Rannacher, 2003). Thus, the self-adaptive goal-oriented refinement strategy is suitable for simulationof most borehole resistivity logging measure-ments. Several publications have documentedthe outstanding performance of the 2D hp-FEM algorithm for simulation of different resis-tivity logging measurements (e.g., Pardo et al.,2006, 2007; Nam et al., 2009, 2010a).The grid refinement algorithm is finalized

when reaching a user-prescribed tolerance error.Typically, when the relative error between the so-lution at the receiver delivered by a given gridand a globally hp-refined grid is below 1%,where the globally hp-refined grid is built by uni-formly breaking each element of the currentmesh into four (in two dimensions) and increas-ing the polynomial order of approximation ofeach node by one. At the end of this process,the final grid contains elements that vary in sizeby several orders of magnitude and may containa very different number of unknowns dependingupon the nature of the problem (typically below300,000).

VERIFICATION

The accuracy of hp-FEM-based simulationsfor all components of borehole-centered triaxialmeasurements was reported in Nam et al.(2010b), even for models with high conductivity

contrasts. Thus, the verification in this paper is focused to the 3Dformulation for triaxial measurements in the presence of tool eccen-tricity; we refer to Nam et al. (2010b) for further verifications oftriaxial measurement simulations including high conductivitycontrasts.

In the absence of a mandrel

To validate the simulation method, we consider a five-layer for-mation model with an open borehole (Figure 3). The radius andresistivity of the borehole are 0.1016 m and 1 Ωm, respectively.For this validation, we do not include the logging instrument inour model, and we consider a Dirac delta source and a receiver with-out a bucking coil. The operating frequency is 20 kHz. Source andreceiver are located 1.016 m apart and at an eccentered distanceequal to 0.0508 m from the borehole center. Simulated resultsfor Hxx and Hzz using our algorithm are compared to resultsobtained with a 2.5D axial hybrid method developed by Wanget al. (2009). This comparison indicates a very good agreement(Figure 3).

In the presence of a mandrel

For further verification of the Fourier formulation in the presenceof a mandrel, we assume a homogeneous formation and no bore-hole. The resistivity is set to 1 Ωm, and the diameter of the boreholeis 33 cm. Because the resulting model constitutes a homogeneousmedium except for the presence of a mandrel, solutions should beindependent of the position (borehole-eccentered distance) of themandrel. However, the method will provide different values for dif-ferent number of Fourier modes (discretizations), and the exactsolution will only be achieved as one approaches an infinite number

–0.15 0 0.25 0.5 0.75 1–2

–1

0

1

2

3

4

5

Apparent conductivity (S/m)

Rel

ativ

e de

pth

(m)

Wanghp FEM

–0.15 0 0.25 0.5 0.75 1–2

–1

0

1

2

3

4

5

Apparent conductivity (S/m)

Rel

ativ

e de

pth

(m)

Wanghp FEM

Hzz Hxx

a) b)

Figure 3. Comparison between results obtained with the Fourier FE method and a 2.5Dmethod developed by Wang et al. (2009) for a five-layered formation in an open bore-hole. (a) Vertical magnetic fields excited by a vertical magnetic moment (Hzz) and(b) horizontal magnetic fields in the x-direction excited by a horizontal magnetic mo-ment in the x-direction (Hxx). The formation has five layers (whose conductivities are0.5, 0.004, 1, 0.002, 0.0033, and 0.5 S∕m) with mud-filtrate invasion in the second,fourth, and fifth layers (whose conductivities are 0.05, 0.01, and 0.02 S∕m).

Borehole-eccentered EM effects D45

of Fourier modes. The difference between the exact solution(centered-cased) and the finite Fourier modes solution for bore-hole-eccentered measurements not only enables verification ofthe code, but also quantifies the error delivered when using a finitenumber of Fourier modes.

We consider a short offset with a bucking coil and a conductivemandrel with borehole-eccentered distances equal to 6.5 and9.1 cm, respectively. Relative errors for Hzz, Hxx, and Hxy are com-puted by comparing numerical results to those corresponding toborehole-centered measurements (Figure 4). Notice that bore-

hole-eccentered distances of 6.5 and 9.1 cm withHzz measurements achieve relative errors below1% when using 7 and 11 Fourier modes, respec-tively (Figure 4a). Moreover, computations using13 Fourier modes result in relative errors below0.1% for borehole-eccentered distances of 6.5and 9.1 cm. It is also noted that Hxx measure-ments with a borehole-eccentered distance of9.1 cm have a relative error equal to 1.03% whenusing 15 Fourier modes, and this error fallsbelow 1% when decreasing the borehole-eccentered distance to 6.5 cm and using only13 Fourier modes. As expected, the relativeerrors of Hxx measurement smoothly decreasewith an increase in the number of Fourier modes(Figure 4b).Similar results are obtained for the case of Hyy

measurements (Figure 4c). Simulations of Hxy

measurements with borehole-eccentered distanceequal to 9.1 cm require the greatest number ofFourier modes for convergence, namely, 17 Four-ier modes (Figure 4d), which is still a rather lownumber.

NUMERICAL EXPERIMENTS

For the main numerical experiments with resis-tive and conductive mandrels (Figure 1b), we con-sider a formation with six horizontal layers, whoseresistivities are (from top to bottom) equal to 100,1, 300, 1, 0.1, and 1 Ωm, respectively (Figure 5).

The thicknesses of the second, third, fourth, and fifth layers are 1.5, 3,1, and 2 m, respectively, where the third (with a resistivity of 300Ωm)and fourth layers represent hydrocarbon- and water-bearing zones,respectively. Additionally, borehole diameter is assumed equal to0.2 m, whereas mud (borehole) resistivity is either 1 (water-basedmud [WBM]) or 1000 Ωm (OBM), respectively.The wellbore penetrates the formation vertically, and we consider

borehole-eccentered tools with eccentered distances equal to 1.8,3.0, and 4.2 cm. We investigate the sensitivity of triaxial measure-ments to various borehole-eccentered distances for resistive andconductive mandrels, different offsets, and different borehole muds(WBM and OBM).Results display only the imaginary part of the simulated mag-

netic field triaxial measurements because this component retainsmost of the information about the electrical properties of the for-mation. Moreover, we do not perform postprocessing of responsesto apparent conductivities to facilitate the comparison of differentborehole-eccentered effects among different cases. In the numericalresults, we use 11 Fourier modes except for cross components, forwhich we employ 17 Fourier modes. For the display of numericalresults at each offset, we use the depth of the center point betweenthe transmitter coil and the center point of bucking and receivercoils to reference the depth of logging with respect to the probedformation.

3 5 7 9 11 13 15 17

10–1

100

102

Number of Fourier modes

Rel

ativ

e er

ror

(in %

) 6.5 cm9.1 cm

3 5 7 9 11 13 15 17

10–1

100

102

Number of Fourier modes

Rel

ativ

e er

ror

(in %

) 6.5 cm9.1 cm

3 5 7 9 11 13 15 17

10–1

100

102

Number of Fourier modes

Rel

ativ

e er

ror

(in %

) 6.5 cm9.1 cm

3 5 7 9 11 13 15 17

10–1

100

102

Number of Fourier modes

Rel

ativ

e er

ror

(in %

) 6.5 cm9.1 cm

a) b)

c) d)

Figure 4. Convergence behavior as a function of the number of Fourier modes used inthe simulation of (a) Hzz, (b) Hxx, (c) Hyy, and (d) Hxy measurements with borehole-eccentered distances equal to 6.5 and 9.1 cm, respectively. Relative errors are computedwith respect to the corresponding borehole-centered measurements.

Figure 5. Formation composed of six horizontal layers with resis-tivities (from top to bottom) equal to 100, 1, 300, 1, 0.1, and 1 Ωm,respectively. The borehole radius is equal to 0.1 m, whereas theborehole resistivity is either 1 (WBM) or 1000 Ωm (OBM).

D46 Nam et al.

In the numerical simulations of triaxial measurements using thedeveloped Fourier hp FE method, CPU time varies from point topoint, and it depends upon the specific computer configuration.In our personal computer equipped with a 2.2 GHz dual-coreCPU and 4 GB RAM, the time required per one logging positionvaried from 2 to 20 min.

A look at the performance ofthe hp-adaptive algorithm

Figure 6 displays the electric field componentsEx and Ez in logarithmic scale excited by an Mx

source attached to a conductive mandrel in a re-sistive borehole with short offset for the modeldescribed in Figure 5 at a depth of 3 m. Becausethe mandrel is very conductive and the boreholefluid is resistive, the electric field inside the man-drel is almost zero. Indeed, currents rapidly de-crease as they enter the mandrel due to the skineffect. This behavior can be clearly observed inFigure 6. Such a physical effect has to be repro-duced at the numerical level by employing verysmall elements in the proximity of the mandrel.The use of an hp-adaptive algorithm seems anoptimal choice for that purpose. We also observe a rapid decreaseof the solution as one moves away from the transmitter (by fiveorders of magnitude), which requires the use of goal-oriented adap-tive algorithms aimed at minimizing the error in the quantity of in-terest (solution at the receivers) rather than in a global norm.The final hp-optimized grid for Ex (Figure 7) shows how the hp-

adaptive algorithm refines an initial grid to minimize the error inthe quantity of interest in the proximity of the mandrel, wherethe solution decreases rapidly. Specifically, the figure shows finalhp-optimized grids with different amplifications toward the loggingpoint z ¼ 3 m. With increasing amplifications, we observe smallerelements and higher orders of approximations (different colorsindicate different polynomial orders of approximation [p], fromp ¼ 1 [blue] to p ¼ 8 [pink]). Notice that the largest element sizeof the final hp grid is above 50 m (Figure 7a), whereas the smallestone is below 0.001 m (Figure 7c).

Short-offset Hxx

Conductive mandrel

Figure 8a and 8c displays the imaginary part ofHxx simulated forthe short offset (Figure 1b) assuming a conductive mandrel in con-ductive and resistive boreholes, respectively. Even though borehole-eccentered tool effects increase with an increase of the distancebetween the tool center and the borehole center, these effects ontriaxial EM induction logging measurements are relatively smallfor conductive and resistive boreholes. In the most conductive (fifth)layer, in which the effects are most prominent, maximum relativedifferences (with respect to the value of Hxx computed withoutborehole-eccentered distance) are below 10% (Figure 8b and 8d)when the borehole-eccentered distance is maximum (4.2 cm). Thisvalue is slightly larger in resistive boreholes than in conductiveones. In all remaining layers of the formation, measurementsare insensitive to the borehole-eccentered distance of the tool,independently of borehole conductivity.

The number of unknowns for the final optimal hp grid dependsnot only upon the initial grid design, but also upon the solution it-self. For example, for Hxx measurement at a depth of −2.1143 m ina conductive borehole without tool eccentricity, the initial gridcontained 5005 unknowns, and this number increased to 9559

Figure 6. Logarithm of the absolute values of the electric field components (a) Ez and(b) Ex excited by anMx source attached to a conductive mandrel in a resistive borehole.The ratio between the maximum (red color) and minimum (blue color) value is fiveorders of magnitude. The dimensions are 30 × 20 cm.

Figure 7. Final adapted hp grid for Hxx measurement simulatedusing five Fourier modes at a depth of 3.038 m for the model pro-blem described in Figure 5 with a conductive mandrel and a resis-tive borehole (1000 Ωm). Different panels show progressiveamplifications of the final hp grid by factors of (a) 1 (b) 30, and(c) 600 toward the logging instrument. The colors identify differentpolynomial orders of approximation (p), ranging from p ¼ 1 (blue)to p ¼ 8 (pink).

Borehole-eccentered EM effects D47

for an optimal grid. Because the simulation at a specific loggingdepth uses as an initial grid the optimal one from the previous log-ging position, the number of unknowns increases. Specifically, at alogging depth of 2.8 m, the number of unknowns for an optimal gridwas 15,639. In the presence of tool eccentricity, the number of un-knowns further increased according to not only the number of Four-ier modes but also the existing couplings between 2D problems.

Resistive mandrel

When employing a resistive mandrel (Figure 9), borehole-eccenteredtool effects increase considerably. In a conductive borehole, sensitivityof the measurements to the borehole-eccentered distance in the fifthlayer remains similar (16%) to the previous case of a conductivemandrel (see Figure 8b and 8d). However, in the first and third

resistive layers, effects due to the borehole-eccentered locationof the tool are clearly identified when considering a resistive man-drel. Specifically, relative differences in Hxx due to shoulder-bedborehole-eccentered tool effects become as large as 100%. We notethat these effects on Hxx for the 1 Ωm layers are negligible becausethe resistivity of these layers is the same as that of the borehole.Borehole-eccentered tool effects in a resistive mud increase with

the formation conductivity, becoming the largest in the mostconductive (fifth) layer of the formation (around 16% when theborehole-eccentered distance is 4.2 cm), and the smallest in themost resistive (third) layer. Such results are opposite to thoseencountered in a conductive borehole because the main cause forthe corresponding measurement effects is the resistivity contrastbetween the borehole and formation. In a resistive borehole, themost conductive (fifth) and the most resistive (third) layers exhibitthe largest and smallest resistivity contrasts to that of the borehole,respectively, whereas the situation is opposite in a conductive

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Figure 8. Imaginary part of Hxx simulated with a short offset forthe model problem described in Figure 5. Conductive mandrel.Different curves correspond to various borehole-eccentered dis-tances: 0 (centered tool), 1.8, 3.0, and 4.2 cm. (a) Conductive bore-hole. (b) Conductive borehole. (c) Resistive borehole. (d) Resistiveborehole.

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Figure 9. Imaginary part ofHxx simulated with a short offset for themodel problem described in Figure 5. Resistive mandrel. Differentcurves correspond to various borehole-eccentered distances: 0 (cen-tered tool), 1.8, 3.0, and 4.2 cm. (a) Conductive borehole. (b) Con-ductive borehole. (c) Resistive borehole. (d) Resistive borehole.

D48 Nam et al.

borehole. Even though the conductivity contrast is larger in theresistive borehole than in the conductive one, the latter gives riseto larger measurement effects due to the borehole-eccentered tooldistance. Hence, we can conclude that these measurement effectsare more prominent in a conductive borehole than in a resistiveone when employing resistive mandrels. At layer interfaces, thesame effects are also larger in the presence of a conductive borehole.

Long-offset Hxx

Conductive mandrel

Shoulder-bed effects on Hxx are highly controlled by theborehole-eccentered tool distance when considering a long offset.

However, within each layer, borehole-eccentered tool effects arehardly noticeable for the case of a long offset for conductive andresistive boreholes (Figure 10).

Resistive mandrel

For the case of a conductive borehole, we observe small borehole-eccentered tool effects in the most resistive layers (Figure 11). Theopposite situation occurs with a resistive borehole, in which themost affected layers due to the borehole-eccentered tool distanceare those with larger conductivity (Figure 11). Shoulder-bed effectsdue to the borehole-eccentered tool distance decrease as oneincreases the offset between transmitter and receivers (compareFigures 8 and 9 to Figure 10).

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Figure 10. Imaginary part of Hxx simulated with a long offsetfor the model problem described in Figure 5. Conductive mandrel.Different curves correspond to various borehole-eccentered dis-tances: 0 (centered tool), 1.8, 3.0, and 4.2 cm. (a) Conductive bore-hole. (b) Conductive borehole. (c) Resistive borehole. (d) Resistiveborehole.

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epth

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a) b)

c) d)

Figure 11. Imaginary part of Hxx simulated with a long offset forthe model problem described in Figure 5. Resistive mandrel. Dif-ferent curves correspond to various borehole-eccentered distances:0 (centered tool), 1.8, 3.0, and 4.2 cm. (a) Conductive bore-hole. (b) Conductive borehole. (c) Resistive borehole. (d) Resistiveborehole.

Borehole-eccentered EM effects D49

Hyy with short offset

Figure 11 displays borehole-eccentered tool effects on Hyy read-ings in a conductive borehole when employing conductive and re-sistive mandrels. For a conductive mandrel, these effects becomealmost negligible (below 2% differences in all cases; see Figure 12aand 12b). On the other hand, when using a resistive mandrel, theHyy component becomes highly affected by an increase in theborehole-eccentered tool distance (Figure 12c and 12d).

Short-offset Hzz

Even though Hzz measurements are usually less sensitive to bore-hole-eccentered tool effects than are other triaxial measurements, Hzz

measurements are modified by approximately 15% in the mostconductive (fifth) layer when using a conductive mandrel that is

eccentered 4.2 cm from the borehole axis (Figure 13a and 13b). Thesedifferences decrease to a level below 2% for the case of a resistivemandrel (Figure 13c and 13d). Notice that, as reported in Davydyche-va et al. (2009), the corresponding effects can increase significantly asone considers larger boreholes and tool-eccentered distances.

Cross measurements at short offsetwith a conductive mandrel

Simulated Hzx measurements in a conductive borehole exhibitlarge effects (above 80%) due to borehole-eccentered distancesof 4.2 cm in the most conductive (fifth) layer (0.1 Ωm), whichhas a higher conductivity than the borehole. The correspondingeffects on the remaining layers are negligible (Figure 14a). In a re-sistive borehole, eccentricity effects become more prominent than ina conductive borehole. Specifically, we observe relative differences

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Figure 12. Imaginary part of Hyy simulated with a short offset forthe model problem described in Figure 5. Conductive borehole (1 Ωm). Different curves correspond to various borehole-eccentered dis-tances: 0 (centered tool), 1.8, 3.0, and 4.2 cm. (a) Conductive man-drel. (b) Conductive mandrel. (c) Resistive mandrel. (d) Resistivemandrel.

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Figure 13. Imaginary part of Hzz simulated with a short offsetfor the model problem described in Figure 5. Conductive borehole(1 Ωm). Different curves correspond to various borehole-eccentereddistances: 0 (centered tool), 1.8, 3.0, and 4.2 cm. (a) Conductivemandrel. (b) Conductive mandrel. (c) Resistive mandrel. (d) Resis-tive mandrel.

D50 Nam et al.

over 100% in the most conductive layer (Figure 14b). Further, weobserve borehole-eccentered effects also across the second (1 Ωm)layer (relative differences above 8%). Similar to the case of Hzx

measurements, Hxy measurements exhibit the largest borehole-eccentered effects in the most conductive layer. Measurementsacquired in a resistive borehole exhibit larger borehole-eccentricityeffects than in a conductive borehole (Figure 15). Eccentricityeffects onHxy in the conductive borehole are measurable in resistiveand conductive formation layers with conductivity different fromthat of the borehole (Figure 15a).

In Figures 14 and 15, cross components Hzx and Hxy for a cen-tered tool are nonzero. This behavior is attributed to the fact that, inour simulations, sources and receivers are located at the mandrel-mud interface. Thus, coils for Hx (and Mx) and Hy (and My) areslightly (as much as the value of the tool radius) off the tool axis,whereas Hz (and Mz) correspond to axisymmetric rings with theircenter located along the tool axis. Due to the small nonsymmetry ofsources/receivers,Hzx and Hxy have nonzero values, as verified anddiscussed by Nam et al. (2010b).

CONCLUSIONS

We successfully simulated borehole-eccentered triaxial inductionmeasurements (including the logging instrument itself) using ahighly accurate numerical method that combines a Fourier seriesexpansion in a new system of coordinates with a high-order,self-adaptive, 2D hp FE method. Simulations considered triaxialEM induction logging measurements for conductive and resis-tive mandrels. Numerical results confirmed that large borehole-eccentered tool distances increase shoulder-bed effects at layerinterfaces. Furthermore, a conductive mandrel can significantlyreduce borehole-eccentered tool effects on triaxial measurements,mainly in the Hxx and Hyy readings. More precisely, when usinga conductive mandrel, the maximum relative differences at layerinterfaces are approximately 22%. For a resistive mandrel, thesedifferences increase to over 100%. For conventional induction log-ging (Hzz), borehole-eccentered tool effects become negligible forconductive and resistive mandrels. Additionally,Hxy measurementsacquired with a conductive mandrel contain significant borehole-eccentered effects across layers (either conductive or resistive), whichexhibit different resistivities from that of the borehole, although theseeffects are more prominent in the most conductive layers of theformation, and they could be larger than 100%.

ACKNOWLEDGMENTS

The work reported in this paper was funded by the University ofTexas at Austin’s Research Consortium on Formation Evaluation,jointly sponsored by Anadarko, Aramco, Baker-Hughes, BG, BHPBilliton, BP, Chevron, ConocoPhillips, ENI, ExxonMobil, Hallibur-ton, Hess, Marathon, the Mexican Institute for Petroleum, Nexen,ONGC, Petrobras, Schlumberger, Shell, Statoil, TOTAL, andWeatherford. The first author was supported by a National ResearchFoundation of Korea (NRF) grant funded by the Korean govern-ment (MEST) (no. 20110014684), and the second author was par-tially funded under the Spanish Ministry of Sciences and Innovationproject MTM2010-16511. We thank Gong Li Wang for sharinghis numerical solutions with us. A note of special gratitude goes toSonya Davydycheva, Mark Everett, and an anonymous reviewer fortheir constructive technical and editorial comments that improvedthe first version of the manuscript.

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