solution of coupled acoustic–elastic wave propagation problems

Comput. Methods Appl. Mech. Engrg. 213–216 (2012) 299–313

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Comput. Methods Appl. Mech. Engrg.

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Solution of coupled acoustic–elastic wave propagation problemswith anelastic attenuation using automatic hp-adaptivity

P.J. Matuszyk a,b,⇑, L.F. Demkowicz c, C. Torres-Verdin a

a Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, TX 78712, USAb Department of Applied Computer Science and Modeling, Faculty of Metals Engineering and Industrial Computer Science, AGH – University of Science and Technology, Kraków, Polandc Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, TX 78712, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 July 2011Received in revised form 28 October 2011Accepted 8 December 2011Available online 27 December 2011

Keywords:Borehole acoustic loggingWave propagationLinear elasticityCoupled problemsHp-adaptive finite elementsPML

0045-7825/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cma.2011.12.004

⇑ Corresponding author at: Department of Petroleuing, The University of Texas at Austin, TX 78712, USA

E-mail addresses: [email protected], pjm@agh

The paper presents a frequency-domain hp-adaptive finite element (FE) method for a class of coupled(visco-)acoustics/anelasticity problems with application to modeling of sonic tools in the borehole envi-ronment. The paper extends methodology, software, and results presented in [1]. A new FE code enablessolution a wider class of problems, including modeling of logging-while-drilling (LWD) tools, viscousdamping in fluids and solids, well- and poor-bonded cased boreholes, and more complicated axially-sym-metrical geometries of the formation. Solutions of non-trivial examples involving modeling of fast andslow formations, a formation with a soft layer, and an LWD tool, are presented. Results of the hp-FE sim-ulations are post-processed to obtain time-domain solutions.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Motivation. Accurate numerical simulations of borehole sonictools enhance the understanding of acoustic logs in complicatedscenarios (presence of the wireline or LWD tool, multilayer forma-tions, casings, mud invasions, etc.) and help to improve the tech-nology and design of new generations of sonic tools. Thesimulations fall into the class of coupled problems: borehole fluidis modeled with inviscid (or viscous) acoustics, tool and casingwith elasticity, and the models for the formation range from rela-tively simple isotropic elasticity through anisotropic elasticity andviscoelasticity to various poroelasticity theories.

Simulations of sonic tools. Numerical modeling of the wave prop-agation problem in a complex borehole environment has nearly a40-years history. Initially, semi-analytical methods were used,with the solution of nonlinear dispersion equations and integrationin the complex domain obtained numerically. Almost all of thesemodels assumed radial symmetry. The approach has been docu-mented in various technical papers, but the most comprehensiveexposition of this topic can be found in [2–4].

The Finite Difference Time Domain (FDTD) technique has beenthe most popular approach for numerical simulations of sonic

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m and Geosystems Engineer-..edu.pl (P.J. Matuszyk).

logging measurements. The developments started with a simplemodel of an open borehole surrounded by a homogeneous or hor-izontally layered formation presented by Bhasavanija et al. [5]. In[6], Stephen et al. successfully applied the FDTD method to themonopole acoustic logging in boreholes with washouts and dam-aged zones. A 2D FDTD method for simulation of axisymmetricproblems with a multipole acoustic excitation was presented byRandall et al. [7]. In [8], Leslie and Randall applied a 2.5-D veloc-ity–stress FDTD algorithm to boreholes penetrating a generalanisotropic formation. A parallel 3D version of FDTD method forsimulating borehole wave fields in general anisotropic formationswas developed by Cheng et al. [9]. In [10], Wang and Tang com-bined for the first time the FDTD method with an unsplit PML tosolve the 2D axisymmetric problem of interest. Recently, Sinhaet al. applied the FDTD method to analyze elastic-wave propaga-tion in a deviated fluid-filled borehole in an arbitrary anisotropicformation [11].

The finite element method (FEM) was used in this field muchmore rarely. Finite element analysis of fluid–solid interaction prob-lems was initiated by Zienkiewicz and Bettess [12]. Komatitsch andTromp applied spectral element methods to calculate seismogramsin 3D Earth models [13]. Chang was probably the first to use finiteelements to model the acoustic wave propagation in a borehole[14]. Bermúdez et al. used displacement/displacement formulationwith H1 and H(div)-conforming elements to solve a 3D elasto-acoustic vibration problem [15]. Käser and Dumbser applied

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1 One should emphasize perhaps at this point that adaptivity produces completelydissimilar optimal meshes for different frequencies. It is a common misconceptionthat the mesh resolving higher frequencies can be used for all other frequencies aswell. In the presence of strong material discontinuities, for low frequencies, theenergy of the solution ‘‘sits’’ mostly in point singularities whose resolution requiresdifferent meshes than those used for the high-frequency range.

300 P.J. Matuszyk et al. / Comput. Methods Appl. Mech. Engrg. 213–216 (2012) 299–313

discontinuous Galerkin method to simulate seismic wave propaga-tion in heterogeneous media containing fluid–solid interfaces, bothfor 2D and 3D geometries [16]. And finally, the presented work ex-pands on the results presented in [1], where for the first time, weapplied automatic hp-FEM and PML methods to model acousticwaves in boreholes. More recently, a time domain FEM was pre-sented by Frehner and Schmalholtz to simulate Stoneley guided-wave reflections in the formation with fluid-filled fractures [17].

Method. The presented work is a continuation of [1]. The meth-od of choice is the hp finite element (FE) method with a fully auto-matic hp-adaptivity based on the two-grid paradigm described in[18,19]. Speaking mathematically, the hp method delivers expo-nential convergence (error vs. number of degrees-of-freedom,CPU time or memory use), for both regular and irregular solutions.The problem under study encounters many of such irregularities.At junctions of three or more different materials, stresses are sin-gular, material anisotropies and the use of Perfectly Matched Layer(PML) leads to internal and boundary layers. Foremost, however,the solution ‘‘lives’’ mostly along the borehole/formation in thepresence of strong interface waves. If left unresolved, these irregu-larities pollute the solution at points of interest (receivers) andresult in completely erroneous numerical solutions.

A common misconception is the claim that infinite stresses arenon-physical and, therefore, need not be resolved. Although thesimplified models based on various elasticity models do result innon-physical infinite stresses (in reality, the material undergoes aplastic deformation), they capture correctly the energy distributionand provide meaningful models away from the singularities. Leav-ing the singularities unresolved distorts the energy distributionand produces wrong results away from them. Of course, only acomparison with experiments allows for an ultimate validationof the models, and this is exactly the direction in which we areheading. Assessing the validity of the model requires high-fidelitydiscretizations with negligible discretization error, and this iswhere the use of hp method is critical.

The presented contribution focuses on extending the hp tech-nology to multiphysics coupled problems. A starting point for ourhp code developed in the course of this project, has been a newhp framework [20] designed for coupled problems (different phys-ics in different subdomains) and supporting the hp-adaptivity forelements forming the exact sequence, i.e. H1-, H(curl)-, H(div)-and L2-conforming elements. In this project, only the classicalH1-conforming elements are used. Extending the hp adaptive algo-rithm to coupled problems requires additional modifications of thealgorithm, which are discussed in this contribution.

The main goal of this paper is, however, to demonstrate amaturity of the hp-technology in a context of non-academic, indus-trial-strength examples. This includes solutions of a number ofnon-trivial examples hardly- or non-accessible with other meth-ods. The high-accuracy claims call for a careful verification of thepresented methodology and codes, which has been reported in[21]. For the sake of brevity, the paper skips the verification issueand focuses on new challenging examples.

Mathematics of coupled problems. The subjects of acoustics andelasticity are classical. Discussion on linear problems of elasticstructures coupled to fluids in bounded and unbounded domainscan be found e.g. in [22]. The subject of interest relates to the cat-egory of problems involving a compact perturbation of a continu-ous linear operator, see e.g. [23]. In [24,25], Demkowicz hasstudied the asymptotic stability of those problems and has shownhow it relates to the convergence of the eigenvalues of the discreteproblem to those of the continuous problem. A rigorous mathemat-ical analysis for a related class of coupled problems can be found in[26]. The analysis does not include the PML truncation. Numericalanalysis of the coupled acoustics/elasticity problem with the PMLtruncation is an open problem.

Frequency-domain approach. In the context of acoustic well-log-ging, all time-domain methods deliver the waveforms (time-do-main signals calculated at the receivers’ locations), which in turncan be compared to the real-life waveforms acquired by sonictools. However, the most important obtained information, givinga comprehensive insight into the formation (e.g. formation aniso-tropies, radial properties distribution, permeability and formationalteration), comes in the form of the so-called dispersion curves(plots of the slowness/velocity versus frequency, for detectedwave-modes). The dispersion curves are obtained by postprocess-ing the waveforms using a variety of methods. The most importantand frequently used techniques include the Prony-like methodsand variants of the matrix-pencil method [27]. Numerical evalua-tion of the dispersion curves is prone to additional errors ‘‘pollut-ing’’ the final results.

In this context, the use of the frequency-domain approach isvery beneficial. If one is interested in obtaining high-quality dis-persion information but not necessarily the waveforms, computa-tions in the frequency-domain seem to be much more natural:solving a relatively small number of independent problems corre-sponding to different frequencies is more effective and accurate.Here, we may use for each frequency a dissimilar mesh (coarsefor low frequencies and fine for higher frequencies), which acceler-ates the calculations (lower number of degrees-of-freedom, calcu-lations for all the frequencies can be performed in parallel) keepinga high-quality of the solution.1 In comparison, any time-domainmethod needs to use one mesh for all the frequencies contained inthe signal. Additionally, the time-integration cannot be easilyparallelized.

Moreover, if waveforms are of interest, results in the frequencydomain can be used to produce waveforms corresponding to differ-ent acoustic sources (e.g. Ricker wavelet, chirp signal, etc.) at a lowcost of performing inverse Fourier transform, provided that thespectrum of the excitation is sufficiently resolved with the fre-quencies being used.

Plan of the presentation. The content of the paper is as follows.We begin in Section 2 with the formulation of the problem. Section3 describes briefly the hp technology and discusses necessary mod-ifications and updates for coupled problems. In Section 4, we pres-ent shortly the algorithms implemented to accelerate the method.Examples of non-trivial simulations are presented in Section 5, andwe conclude with some remarks and future work plans in Section6.

2. A class of coupled acoustic–elastic problems with anelasticattenuation

2.1. Acoustic waves in fluid. Time and frequency formulation

Propagation of acoustic waves in a fluid is described by twocoupled equations (continuity and linear momentum laws):

@p@t þ c2

f qfr � v ¼ 0;

qf@v@t þrp ¼ 0;

(ð1Þ

where p(x, t) is the (perturbation of hydrostatic) pressure, v(x, t)denotes the (perturbation of) velocity in the fluid, qf is the fluid den-sity and cf stands for the sound speed in the fluid. The system ofequations must be accompanied with appropriate boundary and

P.J. Matuszyk et al. / Comput. Methods Appl. Mech. Engrg. 213–216 (2012) 299–313 301

(in the case of an unbounded domain) radiation conditions. Uponeliminating the velocity, the scalar wave equation in terms of pres-sure p is obtained:

c2f Dpþ @

2p@t2 ¼ 0: ð2Þ

Applying Fourier transform with respect to time, we obtain the cor-responding coupled problem in the frequency domain:

ixpþ c2f qfr � v ¼ 0;

ixqf v þrp ¼ 0;

(ð3Þ

where pðx;xÞ and vðx;xÞ are Fourier transforms of the pressureand fluid velocity respectively, and x denotes the angular fre-quency. Combining these equations or calculating directly Fouriertransform of Eq. (2), one obtains the Helmholtz equation:

Dpþ k2f p ¼ 0; kf ¼

xcf: ð4Þ

A weak form of the Eq. (4) is given by:

p 2 pD þ Q :RXAðrp � rq� k2

f pqÞdXA ¼R

CANqnf � rpdCAN 8q 2 Q :

(ð5Þ

Here XA is a domain occupied by the fluid, @XA ¼ CA denotes itsboundary, and nf denotes the outward unit normal to the boundaryCA. Function pD stands for an admissible lift of the Dirichlet data,and Q denotes the space of test functions,

Q ¼ fq 2 H1ðXAÞ : qjCAD¼ 0g:

The Neumann boundary CAN � CA is partitioned into two disjointsubsets: boundary Cex where an acoustic source n � rp ¼ gex is de-fined, and interface part CAE where coupling conditions betweenfluid and solid are prescribed. Interaction with a truncated part ofthe domain extending to infinity is modeled with a PML layer (seeSection 2.5), terminated with a homogeneous Dirichlet boundarycondition on the remaining part of the boundary: CAD ¼ CA � CAN .

2.2. Waves propagation in elastic solids. Time and frequencyformulation

Elastic wave propagation in the formation, the tool, and the cas-ing, in the absence of body forces, is described with the equationsof linear elasticity to be satisfied in an elastic domain XE:

�r � rþ qs@2u@t2 ¼ 0;

r ¼ C : e;eðuÞ ¼ 1

2 ðruþrT uÞ;

8><>: ð6Þ

Here r stands for the stress tensor, e is the strain tensor, u denotesthe displacement vector, qs is the solid density, and C denotes theelastic ðCijkl 2 RÞ 4th order compliance tensor. The equations mustbe accompanied with appropriate boundary and (in the case of anunbounded domain) radiation conditions. For the linear isotropicsolid, the tensor C simplifies to:

Cijkl ¼ lðdikdjl þ dildjkÞ þ kdijdkl;

where k and l denote (real) Lamé coefficients. The Lamé coefficientscan be defined through characteristic wave speeds in the solid,namely: P-wave speed vp, and S-wave speed vs,

l ¼ qsv2s ; k ¼ qsðv2

p � 2v2s Þ: ð7Þ

Applying Fourier transform in time, we obtain the correspond-ing equations in the frequency domain:

�r � r� qsx2u ¼ 0;r ¼ C : e;eðuÞ ¼ 1

2 ðruþrT uÞ;

8><>: ð8Þ

where r; e, and u denote the Fourier transforms of the appropriatequantities.

The equations are equipped with two kinds of boundary condi-tions (BCs): prescribed displacement (Dirichlet) BC imposed on thepart of boundary CED � CE; CE ¼ @XE, and (Neumann) tractions BCns � r ¼ t prescribed on the remaining part of boundaryCEN ¼ CE � CED. Here ns stands for the outward unit normal tothe boundary CE.

A weak form of Eq. (8) is given by:

u 2 uD þW :RXEðew : C : eu � qsx2u � wÞdXE ¼

RCEN

t � wdCEN; 8w 2W

(

ð9Þwhere uD is an admissible lift of the Dirichlet data, and W is thespace of test functions defined by:

W ¼ fw 2 H1ðXEÞ : wjCED¼ 0g:

The Neumann boundary CEN is then divided into two disjoint parts:CEA where coupling conditions between solid and fluid are pre-scribed, and the remaining part on which zero traction is pre-scribed. Similarly to the acoustic case, the interaction with anunbounded part extending to infinity is truncated with a PML layer(see Section 2.5) terminated with homogeneous Dirichlet boundaryconditions.

2.3. Coupling conditions between fluid and solid

Coupling conditions at the interface between fluid and solid as-sure the compatibility of displacements, i.e. the equality of the nor-mal component of velocities:

nf � v ¼ nf � ðixuÞ;

and tractions:

t ¼ ns � r ¼ �nsp:

Using Eq. (32), the first condition can be expressed as:

nf � rp ¼ qf x2nf � u:

Thus, finally, the weak form for the coupled acoustic–elasticproblem is given by:

ðu; pÞ 2 ðuD; pDÞ þW � Q :

bAAðp; qÞ þ bAEðu; qÞ ¼ lAðqÞ 8q 2 Q ;

bEAðp; wÞ þ bEEðu; wÞ ¼ 0 8w 2W;

8><>:where

bAAðp; qÞ ¼Z

XA

ðrp � rq� k2f pqÞdXA;

bAEðu; qÞ ¼ �Z

CAE

qf x2qnf � udCAE;

bEAðp; wÞ ¼Z

CEA

pns � wdCEA;

bEEðu; wÞ ¼Z

XE

ðew : C : eu � qsx2u � wÞdXE;

lAðqÞ ¼Z

Cexqgex dCex:

2.4. Modeling of anelastic attenuation

It has been observed that energy of the elastic waves in the for-mation exhibits some loss due to internal friction, presence of fluids,and a nonhomogeneous structure of the formation (grains, crystalimperfections, fractures, etc.) [2]. The attenuation is frequency-dependent and has stronger effects for higher frequencies causing


a decrease in the wave amplitude and change of wave velocity. Onthe other hand, reliable simulations of logging tools require anappropriate model taking into account a sophisticated structure ofthe tool, which is designed to attenuate significantly all collarmodes. Effectively, this can be achieved by introducing anelasticattenuation, keeping the tool geometry simple, which decreasescomplexity of the model and saves computational resources.

Several methods have been developed to deal with the attenu-ation [28]. One of the classical approaches to describe this phe-nomenon is based on the theory of viscoelasticity, whichcomplements the classical elastic theory strain with velocity-dependent terms. Consequently, the stress does not depend solelyon the strain but also takes into account the strain history.

Our work, however, is based on an alternate approach, the so-called constant-Q model, following from a study of causality inwave propagation, and developed by Aki and Richards [29]. Theclassical elasticity equations are modified by introducing complexwavenumbers, complex wave velocities and complex compliancetensor. The Lamé coefficients remain to be defined by the charac-teristic wave velocities in the solid, see (7). Additional parameters,the so-called quality factors Qp and Qs, expressing damping rates inone wave cycle are introduced to model the attenuation. In theconstant-Q model, it is assumed that these factors are frequencyindependent. The complex phase velocity is defined as,

cðxÞ ¼ c0 1þ 1pQ

lnxx0

� �1þ i

2Q

� �; ð10Þ

where c0 is a reference velocity corresponding to an angular fre-quency x0. Aki and Richards suggest to use for the reference fre-quency 1 Hz ðx0 ¼ 2pÞ [29]. The same approach can be used tomodel attenuation in the fluids, water or oil base mud in the bore-hole or hydrocarbon, or brine in cracks present in the formation.The increase of seismic velocities with frequency agrees with exper-iments over the range of frequencies from 100 Hz to 100 kHz [3].

Setting x0 ¼ 2p, and introducing separate quality factorsQp; Q s for compressional and shear waves, we have,

vp ¼ v0p 1þ 1

pQ pln f

� �1þ i

2Q p

� �;

v s ¼ v0s 1þ 1

pQ sln f

� �1þ i

2Qs

� �;

where v0p and v0

s are low frequency limit reference velocities, and fdenotes the frequency. In summary, to account for the attenuation,one simply replaces characteristic wave velocities with the complexvelocities, which makes the solid compliance tensor C complex-val-ued and frequency dependent. The same strategy, resulting in com-plex acoustic wavenumber kf, is used for the fluids.

2.5. Perfectly matched layer (PML)

A general problem of wave propagation in a borehole–forma-tion system is posed in an unbounded spatial domain. An effectivenumerical simulation of this problem using FEM needs a truncatedspatial domain, and special techniques are needed for truncatingboundary conditions to avoid reflections of outward propagatingwaves. For this purpose, we use the PML method [30,1].

In cylindrical coordinates ðr; h; zÞ, for the problem defined in afrequency domain, the PML absorbing layer is constructed througha complex stretching of the axial (z) and radial (r) coordinates,using the following transformations:

xj :¼ Xjðxj; kÞ;@

@xj:¼ 1

X 0j

@

@xj; where X0j ¼

@Xj

@xj; ð11Þ

and xj and Xj represent real unstretched and complex stretchedcoordinates, respectively. Notice that the stretching depends uponthe wavenumber k.

The transformation of coordinates xj results in an analytic con-tinuation of the solution into a complex plane characterized withan exponential decay of the outgoing waves, and an exponentialblow-up of the incoming waves in the stretched direction (r orz) within the PML absorbing region. Consequently, imposing ahomogeneous Dirichlet boundary condition at the end of thePML region does not affect the (‘‘stretched’’) outgoing wave, butit eliminates the incoming one. It is of an utmost practical impor-tance to construct the stretching in such a way that the attenu-ated outgoing wave reaches a machine zero on the PML outerboundary.

The general form of the stretching transformation in direction Xj

can be written as follows:

Xjðxj; kÞ ¼ gjðxj; kÞð1� iÞ þ xj; ð12ÞX0jðxj; kÞ ¼ g0jðxj; kÞð1� iÞ þ 1; ð13ÞX00j ðxj; kÞ ¼ g00j ðxj; kÞð1� iÞ; ð14Þ

where i is an imaginary unit and a function gjðxj; kÞ is defined by:

gjðxj; kÞ ¼2p

kn0

jn0j nm where ½0;1� 3 nðxjÞ ¼

xLj�xj

djxj < xL

j ;

xj�xRj

djxj > xR

j ;

0 otherwise:

8>>><>>>:

ð15Þ

The simultaneous stretching of both real and imaginary parts pro-duces an effect that not only the plane waves but also evanescentwaves are damped within the PML region. Computational domainin xj-direction is contained in ½xL

j � dj; xRj þ dj�; xL

j < xRj , where dj is

the PML width in xj-direction. Parameter p controls the strengthof the wave attenuation. For instance, it can be estimated as

p Plnðd ln 10Þ

ln 2

if we want to decrease an amplitude of the incident wave by a factor10d. For p = 5, the wave amplitude attenuation on the outer bound-ary of the PML is of order 10�14, and for p = 6 the amplitude de-crease is of order 10�28, which (assuming wave amplitudes oforder 1 into the simulated domain) gives values below the machinezero on a standard computer. Finally, the term nm controls the wayin which the attenuation of the wave is increased as we go deeperinto the absorbing layer.

In the presented examples, we use p = 6 and m = 3. The thicknessh of the PML layer, both in r and z direction, is computed for eachfrequency separately, using h ¼minf2kmin;0:15diamðXÞg, wherekmin is the smallest compressional or shear mode wavelength takenover all contributing fluids and solids constituting the boreholeenvironment. Using the above-mentioned values for the PMLparameters, we have found, that in all the performed simulations,the number of degrees of freedom within the PML region (for the fi-nal fine meshes) was roughly about 20% of the total number degreesof freedom in the whole computational domain.

2.6. PML formulation in cylindrical coordinates

The cylindrical coordinates, due to the axial symmetry of thegeometry of the problem, as well as due to convenient approachfor the representation of multipole nonsymmetric sources (pre-sented in the previous sections), are a natural choice for the pre-sented problem.

Formulation for acoustics. First, one has to define gradients of thefunctions p and q under PML stretching. Using definitions (19), weobtain:


rp ¼ @p@r;1r@p@h;@p@z

� �!PML; ð19Þ 1

R0@pn

@r;þinpn

R;

1Z0@pn

@z

� �einh;

rq ¼ @q@r;1r@q@h;@q@z

� �!PML; ð19Þ 1

R0@qn

@r;�inqn

R;

1Z0@qn

@z

� �einh:

Now, taking into account that the jacobian dXA ¼ r dr dhdz is trans-formed, due to the PML stretching, into dXA ¼ RR0Z0 dr dhdz, one ar-rives at the final weak form for the acoustic equation:Z

XA

RZ0

R0

� �@pn

@r@qn

@rþ RR0

Z0

� �@pn

@z@qn

@zþ n2 R0Z0

R� k2

f RR0Z0� �

pnqndXA

� qf x2Z

CAE

qnnf � uRR0Z0 dCAE ¼Z

Cex

qnp0

2RR0Z0dCex 8qn 2 Q :

ð16Þ

The corresponding energy space is defined as a weighted H1 space:

Q ¼ q :RZ0

R0

��

12 @q@r;

RR0

Z0

��

12 @q@z;n

R0Z0

R

��

12

q; RR0Z0�� 12q 2 L2ðXAÞ; qjCAD

¼ 0

( ):

Formulation for elasticity. In here, one has to define first straintensor components expressed in cylindrical coordinates:

err ¼@ur

@r; erz ¼

12

@ur

@zþ @uz

@r

� �;

ezz ¼@uz

@z; erh ¼

12@uh

@r� 1

ruh �

@ur

@h

� �� ;

ehh ¼1r

@uh

@hþ ur

� �; ehz ¼

12

@uh

@zþ 1

r@uz

@h

� �:

Performing an analogous PML stretching for the strains, we obtain:

errðuÞ ¼ a1R0@ur

@r; errðwÞ ¼ b

1R0@wr

@r;

ezzðuÞ ¼ a1Z0@uz

@z; ezzðwÞ ¼ b

1Z0@wz

@z;

ehhðuÞ ¼ a1Rður þ inuhÞ; ehhðwÞ ¼ b

1Rðwr � inwhÞ;

erzðuÞ ¼ a12

1Z0@ur

@zþ 1

R0@uz

@r

� �; erzðwÞ ¼ b

12

1Z0@wr

@zþ 1

R0@wz

@r

� �;

erhðuÞ ¼ a12

1R0@uh

@r� uh � inur

R

� �;

erhðwÞ ¼ b12

1R0@wh

@r� wh þ inwr

R

� �;

ehzðuÞ ¼ a12

1Z0@uh

@zþ inuz

R

� �; ehzðwÞ ¼ b

12

1Z0@wh

@z� inwz

R

� �;

where we have dropped subscript n for displacement components,a ¼ einh and b ¼ e�inh ¼ 1=a. Thus, for the most general case, wehave:

ew : C : eu

¼ ð2lþ kÞ½errðuÞerrðwÞ þ ehhðuÞehhðwÞ þ ezzðuÞezzðwÞ�þ k½errðuÞehhðwÞ þ ehhðuÞerrðwÞ þ errðuÞezzðwÞ�þ k½ezzðuÞerrðwÞ þ ehhðuÞezzðwÞ þ ezzðuÞehhðwÞ�þ 4l½erhðuÞerhðwÞ þ erzðuÞerzðwÞ þ ehzðuÞehzðwÞ�: ð17Þ

The final equation for elasticity is obtained from (9) replacingew : C : eu term according to definition (17), and incorporating thestretch jacobian RR0Z0 in each integral.

In the case of monopole excitation (n = 0, fully axisymmetricproblem), the bilinear form decouples into two independent bilin-ear forms, where the first one depends only on r and z componentsand the second solely on h component. Similar decomposition fol-lows for the load vector. Assuming that uh ¼ 0, the solution reducesto the determination of the ur and uz components only.

The energy space for the stretched elasticity equations isdefined analogously to the acoustic case as a weighted H1 space.

2.7. Treatment of the singularities in the energy functional at r = 0

Inspecting the expressions contributing to the energy norm (16)and (17), we arrive at the problem of securing additional condi-tions to assure finiteness of the energy along the axis of symmetry(r = 0).

Consider first the monopole case. The acoustic bilinear formcontains no singular terms, but the elastic bilinear form includesa singular term ehh ¼ ur=r. Two possible scenarios can occur. Ifthe elastic material is placed off of the axis of symmetry (openborehole or logging–while–drilling (LWD) tools), there is no singu-larity in the energy functional. However, simulation of a wireline(solid) tool leads to the presence of the singular term in the energyfunctional.

Two solutions were implemented to overcome the problem andcompared numerically in [21]. In the first (we call this approach‘‘formulation A’’ hereafter), no conditions are prescribed on the axisof symmetry in the elastic domain. Consequently, for all elementsadjacent to the axis of symmetry, no additional boundary integralsare computed. As the volume integrals are computed using stan-dard (non-adaptive) Gaussian quadrature, integration of the singu-lar term results effectively in an implicit penalty term, whichforces the radial displacement component ur to vanish on the axisof symmetry.

A similar situation occurs for the multipole sources ðn > 0Þ. Inhere, however, there are more singular terms in the energy func-tionals: in acoustics, we have the term p2/r2, and in elasticity, wehave term ehz � uz=r, and terms ehh and erh, which are proportionalto linear combinations of ður þ iuhÞ=r. Therefore, for each possibleconfiguration, the discussed phenomenon occurs.

Another approach (called the ‘‘formulation B’’ hereafter) dealswith the problem through a proper choice of alternate dependentvariables. For monopole excitation, we consider new variablesu0r ; u0z; p0 defined as:

ur ¼ ru0r ; uz ¼ u0z; p ¼ p0:

Therefore, the term ehh is now proportional to u0r , and automaticallyequal zero for r = 0, which in turn makes the elastic energy finite.The stretched strain tensor for monopole excitation has the follow-ing components:

errðu0Þ ¼ a1R0

u0r þ r@u0r@r

� �; errðw0Þ ¼ b

1R0

w0r þ r@w0r@r

� �;

ezzðu0Þ ¼ a1Z0@u0Z@z

; ezzðw0Þ ¼ b1Z0@w0Z@z

;

ehhðu0Þ ¼ aru0rR; ehhðw0Þ ¼ b

rw0rR;

erzðu0Þ ¼ a12

rZ0@u0r@zþ 1

R0@u0Z@r

� �; erzðw0Þ ¼ b

12

rZ0@w0r@zþ 1

R0@w0Z@r

� �:

‘‘Formulation B’’ for a multipole excitation is obtained throughthe following choice of the dependent variables:

ur ¼ u0r; uz ¼ ru0Z ; uh ¼ u0h; p ¼ p0:

Again, as a consequence of the choice of the new variables,acoustic energy and the ehz are finite at the axis of symmetry.We are still left with singular terms ehh and erh. The ansatz toenforce an automatic vanishing of the quantity ður þ iuhÞ=r atr = 0 requires considering vector-valued shape functions forelasticity, and it is much more difficult to implement. Thestretched strain tensor for monopole source has the followingcomponents:


errðu0Þ ¼ a1R0@u0r@r

; errðw0Þ ¼ b1R0@w0r@r

;

ezzðu0Þ ¼ arZ0@uz

@z; ezzðw0Þ ¼ b

rZ0@w0Z@z

;

ehhðu0Þ ¼ a1R

u0r þ inu0h� �

; ehhðw0Þ ¼ b1R

w0r � inw0h� �

;

erzðu0Þ ¼a2

1Z0@u0r@zþ r

R0@u0Z@rþ u0Z

R0

� �;

erzðw0Þ ¼b2

1Z0@w0r@zþ r

R0@w0Z@rþ w0Z

R0

� �;

erhðu0Þ ¼ a12

1R0@u0h@r� u0h � inu0r

R

� �;

erhðw0Þ ¼ b12

1R0@w0h@r� w0h þ inw0r

R

� �;

ehzðu0Þ ¼ a12

1Z0@u0h@zþ inru0Z

R

� �; ehzðw0Þ ¼ b

12

1Z0@w0h@z� inrw0Z

R

� �:

Comparison of the alternative formulations defined above, aswell as a detailed verification of the code, performed both in time(waveforms) and frequency (dispersion curves) domains, can befound in [21]. Our investigation has shown that both versions de-liver practically the same final solutions, thus both approachesare correct. In the sequel, we use only ‘‘formulation A’’.

2.8. Modeling of multipole acoustic sources

Almost all logging tools are based on a combination of mono-pole, dipole or quadrupole acoustic sources. A monopole sourcecan be modeled with an acoustic point source, which exhibits aspherical symmetry pattern of the radiation. A multipole sourceof order n can be constructed with 2n monopole point sourcesalternating in sign and placed periodically along a circle of radiusr0 [31]. Furthermore, such a model can be approximated with aFourier expansion in the azimuthal direction h. The leading termin the radiation pattern for the n-order multipole source exhibitscosðnhÞ dependence in angle h (see Fig. 1)). Therefore, one canapproximate a multipole source of order n by:

gðnÞex ¼ p0 cosðnhÞ ¼ p0

2einh|fflffl{zfflffl}

gþn

þp0

2einh|fflffl{zfflffl}

g�n

; n ¼ 0;1;2; . . . ð18Þ

Use of complex exponentials instead of the cosine function enablesa further simplification of the variational formulation of the prob-lem in cylindrical coordinates. Due to the linearity of the problem,solution ðpn; unÞ of nth azimuthal order can be computed as a super-position of solutions ðpþn ; uþn Þ and ðp�n ; u�n Þ calculated independentlyfor excitations gþn and g�n respectively.

Assuming excitation gþn , we arrive at the following definitions oftrial and test functions:

pþn ¼ pnðr; zÞeinh; un ¼ unðr; zÞeinh;

qþn ¼ qnðr; zÞe�inh; wn ¼ wnðr; zÞe�inh;ð19Þ

Fig. 1. Models for multipole acoustic sources.

where pn; qn; un and wn are solely functions of r and z coordinates.Monopole source (n = 0). The source excites only one mode. Conse-

quently, one can solve the problem with excitation source equal gþn ,and then multiply the solution by 2. The source pattern is axially-symmetric. The solution does not depend upon the azimuthal direc-tion h and, therefore, can be directly solved for in 2D (r,z) domain.Furthermore, the test and trial functions are of the same form, whichimplies the symmetry of the (complex-valued) stiffness matrix.

Multipole source ðn > 0Þ. The source excites two modes. It turnsout, however, that it is still enough to calculate only one solutionðpn; unÞ corresponding to excitation gþn . Denoting by bþn and b�nthe bilinear forms corresponding to einh and e�inh ansatz, we have,bþn ðpþ; uþÞ; ðqþ; wþÞð Þ, and b�n ðp�; u�Þ; ðq�; w�Þð Þ, one can observethat:

b�n ðp�; u�r ; u�h ; u�z Þ; ðq�; w�r ; w�h ; w�z Þ� �¼ bþn ðpþ; uþr ;�uþh ; u

þz Þ; ðqþ; wþr ;�wþh ; w

þz Þ

� �:

The corresponding terms defining load vectors are identical,

gþn qþ ¼ g�n q� ¼ p0

2qnðr; zÞ:

Consequently, the solution for the problem with excitation g�n can bedirectly calculated from the solution of the problem with excitationgþn , which halves the computational time. Components p; ur; uh anduz for constant values of n are solely functions of r and z coordinatesenabling thus the solution only in 2D trace (r,z) domain.

The final solution ðpn; unÞ is calculated in terms of the solutionfor one excitation mode only, as follows:

pn ¼12

pþeinh þ p�e�inh� �

¼ pþ cosðnhÞ; ð20Þ

un ¼12

uþr einh

uþh einh

uþz einh

264

375þ u�r e�inh

u�h e�inh

u�z e�inh

264

375

0B@

1CA ¼ uþr cosðnhÞ

uþh i sinðnhÞuþz cosðnhÞ

264

375: ð21Þ

2.9. Temporal representation of an acoustic sonic source

The most frequently used model for a sonic source in boreholesimulations is the Ricker wavelet, see Fig. 2, due to its fast decay inboth time and frequency domains [29]:

pðtÞ ¼ 2ffiffiffiffipp 1� 2

t � t0

T

� �2" #

e�t�t0

T

� �2

; ð22Þ

pðxÞ ¼ 4Te�ixt0 X2e�X2; X ¼ xT

2¼ pfT: ð23Þ

In the above, t0 is a time at which maximum of the pulse occurs andT ¼ 1=ðpfcÞ is a characteristic period of the pulse defined by theso-called central frequency fc. Given a central frequency fc, the fre-quency spectrum of the Ricker wavelet is contained essentially be-tween 0 and 3fc. The rule is used to estimate the required frequencyrange in typical well-logging applications [4].

The compact and smooth spectrum of the Ricker waveletenables calculation of the full waveforms with a relatively smallnumber of sampling frequencies. One has to emphasize, however,that the ultimate number of frequencies needed depends not onlyupon the spectrum of the source but the response of the wholestructure as well. Excitations with smooth spectrum may producea rugged response, which will require a smaller frequency sam-pling step.

Transforming the solution into time domain. Having calculatedsolutions for sufficiently many frequencies, the inverse Fouriertransform2 is performed to deliver the solution in time domain:

2 We use the inverse FFT in actual computations.

−1 0 1 2 3 4 5 6−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2Ricker wavelet

Time

Ampl

itude

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25Ricker wavelet

Frequency

Ampl

itude

Fig. 2. Ricker wavelet: fc = 2, t0 = 2.


pðx; tÞ ¼ 12p

Z 1

�1pðx;xÞeixtdx � Dx

2pXc

n¼�c

pðx;xnÞeixnt ; ð24Þ

uðx; tÞ ¼ 12p

Z 1

�1uðx;xÞeixtdx � Dx

2pXc

n¼�c

uðx;xnÞeixnt : ð25Þ

Here Dx is the chosen frequency step, c is the number of discretefrequencies used, and xn ¼ nDx is the discrete angular frequency.The frequency spacing Dx should be chosen in such a way thatDx < p=T , where 2T is the anticipated simulation time.

The number of needed frequencies c depends on the spectrumof excitation and the response.

3. hp Technology

3.1. A new hp-FE code for multiphysics problems

The presented work has been implemented within a new ver-sion of our 2D hp code for coupled, multiphysics problems [20].The principal characteristics on the new code include:

support of discretizations using a simultaneous use of all ele-ments forming the exact sequence: H1-, H(curl)-, H(div)-, andL2-conforming elements,3

support of (weakly) coupled problems, energy driven automatic hp-adaptivity [18,19].

In particular, the discussed project led to a non-trivial modifica-tion of the automatic hp-adaptivity algorithm for a coupled prob-lem (different systems of equations in different parts of thedomain) discussed below.

3.2. Modification of the hp algorithm for coupled multiphysicsproblems through an automatic norm scaling

The hp-algorithm, based on a coarse–fine grids paradigm (see[18,19] for a detailed discussion), involves frequent calculationsof relative error along element edges or over element interiorsusing appropriate energy norms.

For the coupled problem, we deal with two different media (so-lid and fluid), and two different sets of unknowns (displacementand pressure) measured in different energy spaces. Even with aproper non-dimensionalization, the energy errors and norms cor-responding to elastic or acoustic subdomains may have dramati-cally different values. In the examples to follow, contributionsfrom the acoustical domain to the total (non-dimensional) energy

3 Only H1-conforming elements are used in this project.

were typically orders of magnitude larger from those correspond-ing to the elastic domain. This implies that the resolution of theelastic part of the domain becomes of a secondary importancefor the energy-based hp algorithm. Large differences in the energylead also to possible conditioning problems, even if a direct solveris used. This can be observed e.g. by monitoring pivots reported bythe solver.

A key-point in a successful application of the automatic hpadaptivity to the coupled problem turned out to be an appropriaterescaling of the governing equations to balance the acoustic andelastic energy norms of the solution. With the norms in balance,comparison of relative (interpolation) errors for both fields be-comes meaningful and results in correct mesh refinements.

Let u ¼ su ~u and p ¼ sp~p, where su and sp are appropriate scalingfactors to be determined. The coupled problem scales as follows:

sp

subAAð~p; qÞ þ bAEð~u; qÞ ¼ 1

sulAðqÞ 8q 2 Q ;

bEAð~p; wÞ þ susp

bEEð~u; wÞ ¼ 0 8w 2W ;

(

with the corresponding scaling for the (energy) norms:

jjjpjjjA ¼ spjjj~pjjjA; jjjujjjE ¼ sujjj~ujjjE:

With sp ¼ jjjpjjjA and su ¼ jjjujjjE, the corresponding energy norms ofthe scaled solutions jjj~pjjjA and jjj~ujjjE are equal 1. Obviously, the ex-act solutions are unknown, so the scaling can be done using approx-imate solutions only. As those change with mesh refinements, thescaling must be executed dynamically.

The modified automatic hp algorithm has now the form:

1. set sp ¼ su ¼ 1,2. solve the problem for ðu; pÞ3. save new values s0p ¼ jjj~pjjjA and s0u ¼ jjj~ujjjE4. FOR each hp step DO

(a) set sp s0p; su s0u,(b) perform classical hp-step, calculate new solution ð~u; ~pÞ,(c) save new values s0p ¼ spjjjpjjjA and s0u ¼ sujjjujjjE.

In Fig. 3, we present convergence curves obtained for the anLWD quadrupole logging in a soft formation, described in detailin Section 5.2. In here we define the final relative error as a squareroot of the sum of squared relative errors calculated in acoustic andelastic domains, respectively.

Fig. 4 demonstrates a sequence of optimal meshes automati-cally generated by the hp-adaptive algorithm for the problem ofthe monopole sonic logging in a fast formation with a soft layerat the frequency equal 12 kHz. We can observe subsequent isotro-pic and anisotropic element subdivisions in the regions with largegradient of the solution (e.g. PML in the borehole, vicinity of theacoustic source, formation inter-layer interfaces) and increase in

104 10510−1

100

101

102

log10(# Dofs)

Rel

ativ

e er

ror (

%)

Fig. 3. Convergence of the hp algorithm for an LWD quadrupole logging in a softformation. We plot a set of 25 convergence curves for the sequence of frequenciesbetween 1 and 25 kHz, with step equal 1 kHz.


order of approximation in the regions where the solution issmooth. In the outermost PML layer, one can observe that thereare nearly no refinements, which can be explained by an efficientdamping of the solution resulting in a very low energy contributionto the total energy over the whole mesh.

4. Performance issues

As mentioned in Section 2.9, recovering the waveforms excitedby the Ricker wavelet with central frequency fc, necessitates com-putations for a number of frequencies within the range [0,3fc].Usually, fc is contained between 2 kHz and 10 kHz, which resultsin the 0–30 kHz simulation range. With a typical frequency stepDf ¼ 50 Hz (found to be sufficient for most of the consideredexamples), the estimated number of discrete frequencies to be ac-counted for is in the range of hundreds. Each single frequencyproblem can be solved independently, and thus all the problemscan be run in parallel. In the context of hp-adaptivity however,the approach is not very effective. A crucial observation is thatthe final optimal coarse mesh obtained for a frequency f, may serveas an excellent starting mesh for neighboring, lower frequencies

Fig. 4. Optimal coarse meshes obtained for the frequency 12 kHz in 2nd, 4th, 6th, 8th, andetail in Subsection 5.1. Color scale indicates degree of the polynomial.

f 0 ¼ f � nDf , typically with n = 1,2, . . . ,10. Going down with the fre-quency keeps us on a ‘‘safe side’’ in terms of resolution of waves.The need for additional hp-refinements, automatically identifiedby the algorithm, may come e.g. from the resolution of singulari-ties. The approach guaranties an excellent efficiency of the overallhp-algorithm and results in significant savings.

The global parallel algorithm reads as follows:

1. Define the range of needed frequencies based on the inspectionof the acoustic source spectrum and chosen central frequency fc.

2. Define frequency step Df . Compute number Nf of all frequencies,and select number Np of frequencies to be processed by oneprocessor.

3. Run independently Nf =Np parallel processes:(a) begin with the highest assigned frequency,(b) run full hp-refinement algorithm starting with a uniform

coarse mesh resolving the frequency (typically 15–35refinement steps),

(c) save the obtained optimal coarse mesh M,(d) perform global hp-refinement, solve the problem on the fine

mesh, and save the fine grid solution,(e) for each of the decreasing frequencies f 0 ¼ f � nDf , where

n ¼ 1; . . . ;Np � 1:i. starting with mesh M, run the hp-refinement algorithm

(typically 1–3 refinement steps),ii. perform global hp-refinement, solve the problem on the

fine mesh and save the solution.

According to our experience, value of Np � 10 leads to areasonable partition of the frequencies. With a preset error toler-ance level, in most cases, solution for consecutive decreasing fre-quencies involved 1–3 hp-adaptive steps only. The total numberof adaptive steps was significantly smaller than starting the fullhp-algorithm for each frequency ‘‘from scratch’’. With themultiple frequency load for each processor, solution of thewhole problem requires typically a couple of tens of processorsonly.

Solution of the problem for single set of Np consecutive frequen-cies was furthermore accelerated by application of the followingtechniques:

d 10th hp-adaptation steps for the problem depicted in Fig. 5a, case A, described in

Fig. 5. Geometries of the presented problems. All the dimensions are given in meters. Acoustic transmitter is denoted by TX.

Table 1Material parameters describing the problem with layers: density q, speeds of waves Vand corresponding slownesses S.

Id P-wave S-wave Density

Vp ½m=s� Sp ½ls=ft� Vs ½m=s� Ss ½ls=ft� q ½kg=m3�

Fast form. 1 3048 100.0 1793 170.0 2200Slow form. 2 2300 132.5 1000 304.8 2000Fluid 3 1524 200.0 – – 1100


implementation of fast integration technique for calculation ofelemental stiffness matrices; this also allows for a significantreduction of memory requirements, and thus enables solvinglarger problems on shared-memory multi-core machines; parallelization of the most time-consuming parts of the hp-FE

algorithm (assembling of the global linear system, projectionsover edges and faces, calculation of elemental norms and errors)via OpenMP directives and additional multicoloring meshpartition; use of the sequential direct sparse linear solvers MUMPS [32,33]

and Pardiso [34] with multi-thread BLAS and LAPACK libraries.

The calculations for all the presented examples were performedon LONESTAR and RANGER clusters from the Texas Advanced Com-puting Center (TACC) [35].

5. Numerical examples

In all presented examples, we proceeded with the hp algorithmuntil the 0.5% relative error4 on the final coarse mesh was achieved.Final results were obtained by postprocessing the corresponding finemesh solutions, which typically exhibit an error of one order of mag-nitude less. In all the presented examples, we use Ricker wavelet atthe central frequency equal 8 kHz as an acoustic source.

5.1. Fast formation with a soft layer

The geometry of the first example is shown in Fig. 5a. A fast for-mation with a soft horizontal layer of thickness equal to 0.5 m ispenetrated by an open borehole. In the borehole, there is a cen-tered ‘‘ring’’ source (TX) of radius 4.6 cm, vertical thickness equal5 cm, and the array of 13 equidistant receivers (RX) located atthe same horizontal offset as the source. The radial thickness ofthe computational domain, excluding the PML layer, is equal to25 cm. We consider two cases (see Fig. 5a), where the receiversarray is facing the soft layer (case A, figure to the left), and whereall the receivers are located below the layer (case B, figure to the

4 Difference between the coarse and fine grid solutions.

right). Material properties of the borehole fluid and the formationare given in Table 1. The simulation was performed for both mono-pole and dipole excitations.

To investigate the influence of the anelastic attenuation in theformation, we also calculated corresponding cases, where bothcomponents of the formation are attenuative media. Formationcomponent quality factors Qp (for compressional) and Qs (for shearwave) were set to 30.

Fig. 6 shows waveforms acquired for the monopole and thedipole excitations, with and without anelastic attenuation, for bothconsidered layer localizations. Fig. 7 shows the comparison of thewaveforms obtained at the first receiver, to investigate an influ-ence of anelastic attenuation.

The waveforms obtained for cases A and B differ significantly.Shape-similar waveforms (for all the receivers) can be observedfor the case B (Figs. 6c and 6d), where all the receivers ‘‘see’’ thesame homogenous formation. The only manifestation of the pres-ence of the soft layer above the array are two families of reflectedwaves situated rearmost in the waveforms. These wave packetstravel in the opposite direction (in the direction of the source)and include the same sequence modes, which are contained inthe incident wave (compressional, shear, etc.), but of significantlysmaller amplitudes.

Waveforms obtained for the geometry A are more complex(Fig. 6a and b)). Lower receivers (located below the soft layer)see the same incident waves as the corresponding receivers in caseB. The difference is only in earlier arrivals of the reflected wavesdue to the closer localization of the inter-layer interfaces. Thewaveforms obtained for the group of central receivers (facing the


soft layer) have smaller amplitude, and the most energetic wavecomponent (Stoneley wave for a monopole and flexural wave forthe dipole) is hardly observed due to the destructive interferenceof multiply-reflected (trapped in the layer) waves between softlayer interfaces. The last group of the waveforms (obtained forthe receivers above the layer) differs from the previously de-scribed. They are retarded due to the presence slower soft layer,and decreased in amplitude, because part of the incident wave en-ergy is reflected back in the direction of the source, and do not con-tain reflected waves.

The presence of nonhomogeneous formation facing the receiv-ers array disturbs significantly in proper identification of materialproperties of the formation. The dispersion processing cannot deli-ver reliable results (see dispersion plot in Fig. 8a and b, and com-pare to the corresponding curves in Fig. 8c and d)). The former

Fig. 6. Problem with layers: waveforms obtai

set of dispersion curves clearly indicate the shear wave velocityin the formation (low frequency asymptotes of the pseudo-Ray-leigh (pR) mode for monopole and flexural mode (Fl) for dipole).In the case A however, the dispersion curves are completely de-stroyed in low and mid frequencies and thus prohibiting correctvelocity identification.

The incoming waves are composed of several modes, which canbe identified by investigating the shape of the waveforms. The firstarrival can be observed for non-dispersive compressional wave(denoted by P), which is the fastest wave in the formation.The amplitude of this mode is small in comparison to consecutivemodes in the wave train, and thus it is hardly seen on the generalplots, however, zooming in the region close to slopes P1 and P2

reveals the presence of P wave (see Fig. 7c and 7d)). The next arri-val in the fast formation corresponds to the shear wave in the

ned for the monopole and dipole source.

Fig. 7. Problem with layers: comparison of the waveforms obtained for case B without and with formation attenuation and for the first receiver.

Fig. 8. Problem with layers: dispersion curves.


0.01ms

z (m

)

0

1

2

3

4

5

0.52ms 1.03ms 1.54ms 2.05ms

2.56ms

z (m

)

r (m)0 0.1 0.2

0

1

2

3

4

5

3.07ms

r (m)0 0.1 0.2

3.58ms

r (m)0 0.1 0.2

4.09ms

r (m)0 0.1 0.2

4.6ms

r (m)0 0.1 0.2

P

S

St

Fig. 9. Problem with layers: monopole sonic logging in elastic slow formation. Consecutive snapshots of the pressure field in time (PML region excluded).

Table 2Material parameters describing the LWD problems: density q, speeds of waves V andcorresponding slowness S.

Id P-wave S-wave Density

Vp ½m=s� Sp ½ls=ft� Vs ½m=s� Ss ½ls=ft� q ½kg=m3�

Fast form. 1 3048 100.0 1793 170.0 2200Slow form. 2 2300 132.5 1000 304.8 2000Fluid 3 1500 203.2 – – 1000Tool 4 5862 52.0 2519 121.0 7800


formation (denoted by S). The highly dispersive wave packets fol-lowing immediately the shear wave are the pseudo-Rayleigh (pR)and Stoneley (St) modes for the monopole excitation and the flex-ural (Fl) modes for the dipole excitation. These are the most ener-getic modes in the wave trains. They overlap each other’s in thetime domain for the actual source-receiver offset, and thus it is dif-ficult to separate them, but in the frequency domain their presenceis unambiguously identified by the separate dispersion curves (seeFig. 8).

Anelastic attenuation in the formation changes the shape of thewaveforms (compare corresponding plots in Fig. 6) and slightly thedispersion curves (Fig. 8). The most prominent effect is a decreasein amplitude. At the same receiver locations (see Fig. 7) shear,Stoneley and flexural wave packet amplitudes are significantlysmaller, and low energy packets, like compressional and reflectedwaves are hardly perceived. The effect of geometric spreading ofthe borehole waves (energy is propagated mainly in 1D, alongthe axial direction) has a small observable effect for the elastic for-mation (one can observe a very little amplitude decrease for fur-ther receivers). However, for the anelastic media, the additionaldissipation of the energy results in larger signal amplitude

decrease, along with the distance from the source. Use of the par-ticular model for anelastic attenuation (Eq. (10)) results in theslightly faster wave arrivals (according to Aki–Richards model, realpart of the velocity increases with the frequency) and lowering ofthe dispersion curves in mid and high frequencies.

Finally, to illustrate the borehole wave propagation in space–time domain, we plot in Fig. 9 a sequence of snapshots of the pres-sure field (excited by the monopole source in elastic formation) atselected time instants. One can observe here a separation of differ-ent direct and reflected wave packets and effect of geometricspreading resulting in amplitude decrease.

5.2. Logging while drilling in a homogenous formation

The next example deals with the simulation of the sonic log-ging-while-drilling (LWD) scenarios for homogeneous fast andslow formations. In comparison to the wireline tool, which is mod-eled as a solid cylinder occupying a part of the borehole, the LWDtool possesses an internal fluid-filled channel and occupies a muchlarger portion of the borehole. In consequence, this results insignificant change of the waves excited in such a modifiedenvironment.

The geometry for the problem is shown in Fig. 5b. Materialproperties of two kinds of formations, the tool, and the fluid aregiven in Table 2. In this case, we model an LWD as an anelasticmedium, characterized by the quality factors Q p ¼ Qs ¼ 25. Themassive tool posses an internal fluid channel of the radius equalto 2.54 cm. It is centered in the borehole, has an array of 6 equidis-tant receivers, with spacing 15.24 cm, and the 5 cm width acoustic‘‘ring’’ source. The source and receivers are localized at the tool’ssurface. The radial thickness of the computational domain, exclud-ing the PML layer, is equal to 25 cm. The simulation was performedfor monopole, dipole, and quadrupole excitations.


Fig. 10 shows the waveforms obtained for the fast and slow for-mation, and for all considered acoustic excitations. For monopoleexcitation, the first wave to arrive is small in amplitude a collarwave (more visible to the slow formation). The next, formation Pwave packet is followed in a fast formation by stronger shear wavetrain. The last and the strongest is Stoneley mode, which in thecase of fast formation dominates completely the waveforms. In aslow formation, it carries less energy, and in consequence enablesto perceive faster weaker modes.

For dipole excitation in the fast formation, one sees the arrivalof the weak compressional mode followed by the dispersive forma-tion flexural mode. The waveforms obtained for the slow formationare clearly separated into two groups: a highly dispersive collarmode, which interferes with the formation compressional mode,followed by the formation flexural mode.

In the case of quadrupole excitation, for both fast and slow for-mations, one observes the strongest formation quadrupole (screw)mode. The arrival time of this mode corresponds to the anticipatedformation S-wave arrival. Waveforms obtained for the slow forma-tion exhibit the presence of additional small in amplitude com-pressional mode.

In all the cases, despite using acoustic source with central fre-quency equal 8 kHz, one can observe that most of the energy islocalized in lower frequencies (compare to the previous examplewith logging in open borehole).

Fig. 11 shows the corresponding frequency dispersion curves.The monopole source excited in both formations the Stoneley

Fig. 10. Waveforms obtained for LWD problem with homogeneous fast and slow formappropriate arrivals of P- and S-waves.

mode. In both cases, the mode exhibits anomalous dispersion(the phase velocity increase with frequency), which is usually thecase only for fast formations. This and additional shift of the curvesto the higher slownesses is the effect of the presence LWD tool(compare with the dispersion curve of the Stoneley mode inFig. 6c)). One can also observe pseudo-Rayleigh (pR) mode excitedin the fast formation. In both cases, we also observe very weak Pmodes.

Dipole LWD sonic logging creates problems in identification ofthe shear of the formation. The reason for that is an inevitableinterference of the first flexural (Fl1) mode with strong tool mode(T). The crossover of these two curves, almost always localized inlow frequencies, prohibits correct identification of the low-fre-quency asymptote of the first flexural mode. Usually, the sourcefrequency spectrum is not wide enough (contains enough energyin high frequencies) to excite higher order flexural modes, whichcould have been used to identify the shear of the formation. Thedifficulties with using LWD dipole led to quadrupole sonic logging.

In contrast to the dipole, the quadrupole collar mode is wellseparated from the formation screw mode (Sc), because the collarmode begins at a cut-off frequency, which is beyond the excitationfrequency range of the source [4]. Thus, the first screw mode is notaffected by the tool. It (theoretically) begins at formation S-waveslowness at low frequencies, which makes it practical in estimatingthe shear velocity of the formation. The shape of the screw modedispersion curve resembles the corresponding curve of the dipoleflexural mode.

ations and different acoustic sources. P and S with indices (see Table 2) indicate

Fig. 11. Dispersion curves obtained for LWD problem with homogeneous fast and slow formations and different acoustic sources. P and S with indices (see Table 2) indicateappropriate arrivals of P- and S-waves.


6. Conclusions

A summary. The paper is a continuation of [1] and focuseson extending the fully automatic hp FE methods to coupled(visco-)acoustic/anelasticity problems posed in the frequencydomain, with a broad range of formation geometry scenarios. Themethodology is applied to a challenging problem of modeling sonictools in the borehole environment. The extension of adaptivehp-algorithm to coupled problems required an automatic energyrescaling procedure. The method delivers a superior accuracy andpractically eliminates numerical dispersion error in their presentedfrequency range. We have parallelized the most time-consumingparts of the hp-FE algorithm to enable efficient use of shared-mem-ory multi-core machines. Finally, we have presented a number ofnon-trivial examples illustrating the potential of the hp-technology.

Challenges and future work. It would be a lie to claim that wehave not encountered any problems. The main challenge comesfrom the construction of Perfectly Matched Layer and its mathe-matical understanding. The PML method is practically theonly technique that we can think of in the case of geometriesinvolving multiple layers and the borehole. The construction of

the implemented PML extrapolates heavily from a simple 1D caseand seems to be insufficient in the case of waveguide geometries,especially in the context of coupled problems and anisotropies[36]. We have not experienced these problems in the examplespresented in this paper.

Our current work is marching in two directions. Non-axisym-metric geometries call for the use of 3D elements, and the 3Dimplementation is indeed underway. On the modeling side, wefocus on more sophisticated models for the formation includingvarious poroelasticity theories. We hope to report new results soonin a forthcoming paper.

Acknowledgments

The work reported in this paper was funded by The Universityof Texas at Austin’s Research Consortium on Formation Evaluation,jointly by: Anadarko, Apache, Aramco, Baker Hughes, BG, BHP Billi-ton, BP, Chevron, ConocoPhillips, ENI, ExxonMobil, Halliburton,Hess, Maersk, Marathon, Mexican Institute for Petroleum, Nexen,Pathfinder, Petrobras, Repsol, RWE, Schlumberger, Statoil, TOTAL,and Weatherford.


The authors acknowledge the Texas Advanced Computing Cen-ter (TACC) at The University of Texas at Austin for providing HPCresources that have contributed to the research results reportedwithin this paper.

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