Simulation of borehole sonic waveforms in dipping ... · Simulation of borehole sonic waveforms in dipping, anisotropic, and invaded formations Robert K. Mallan1, Carlos Torres-Verdı´n2,

Simulation of borehole sonic waveforms in dipping, anisotropic,and invaded formations

Robert K. Mallan1, Carlos Torres-Verdın2, and Jun Ma2

ABSTRACT

A numerical simulation study has been made of borehole

sonic measurements that examined shoulder-bed, anisotropy,

and mud-filtrate invasion effects on frequency-dispersion

curves of flexural and Stoneley waves. Numerical simulations

were considered for a range of models for fast and slow forma-

tions. Computations are performed with a Cartesian 3D finite-

difference time-domain code. Simulations show that presence

of transverse isotropy (TI) alters the dispersion of flexural and

Stoneley waves. In slow formations, the flexural wave becomes

less dispersive when the shear modulus (c44) governing wave

propagation parallel to the TI symmetry axis is lower than the

shear modulus (c66) governing wave propagation normal to the

TI symmetry axis; conversely, the flexural wave becomes more

dispersive when c44> c66. Dispersion decreases by as much as

30% at higher frequencies for the considered case where

c44< c66. Dispersion of Stoneley waves, on the other hand,

increases in TI formations when c44> c66 and decreases when

c44< c66. Dispersion increases by more than a factor of 2.5 at

higher frequencies for the considered case where c44< c66. Sim-

ulations also indicate that the impact of invasion on flexural and

Stoneley dispersions can be altered by the presence of TI. For

the case of a slow formation and TI, where c44 decreases from

the isotropic value, separation between dispersion curves for

cases with and without the presence of a fast invasion zone

increases by as much as 33% for the flexural wave and by as

much as a factor of 1.4 for the Stoneley wave. Lastly, presence

of a shoulder bed intersecting the sonic tool at high dip angles

can alter flexural dispersion significantly at low frequencies.

For the considered case of a shoulder bed dipping at 80�, ambi-

guity in the flexural cutoff frequency might lead to shear-wave

velocity errors of 8%–10%.

INTRODUCTION

Although the analysis of sonic wave propagation in a fluid-filled

borehole has been documented extensively, relatively little attention

has been given to studying the combined effects of dipping bed

boundaries, anisotropy, and invasion. Studies examine the effects of

isotropic, dipping layered formations on sonic waveforms (Yoon and

McMechan, 1992; Cheng et al., 1995; Liu et al., 1996), effects of

invasion on Stoneley dispersion in isotropic formations penetrated by

a vertical well (Baker, 1984), effects of a dipping transversely iso-

tropic (TI) formation on dipole and monopole waveforms (Wang et

al., 2002) and flexural and Stoneley dispersions (Leslie and Randall,

1992; Sinha et al., 2006a), and the effects of azimuthal shear-wave

anisotropy (Tang and Patterson, 2001) and orthorhombic anisotropy

(Cheng et al., 1995) on flexural dispersion. Peyret and Torres-Verdın

(2006) study the combined effects of shoulder beds and invasion on

dipole and monopole waveforms and slowness-time-coherence plots

for an isotropic formation penetrated by a vertical well.

This paper considers a range of 3D models to quantify the

relative influence of layer dip angle, invasion, bed boundary,

and transverse isotropy (TI) on borehole sonic measurements.

Our objective is to characterize the nature of these effects and

quantify their relative impact on borehole sonic measurements.

We focus our analysis on the impact of these effects on slow-

ness-frequency dispersion curves that are processed from the

simulated time-domain waveforms. Flexural and Stoneley dis-

persion data are used commonly to infer formation properties.

Furthermore, these data are sensitive to radial variations of

properties extending several borehole radii away from the well

(Sinha et al., 2006b). Models considered in our study include

Manuscript received by the Editor 2 March 2010; revised manuscript received 17 February 2011; published online 16 June 2011.1Formerly at University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas, U.S.A.; presently at Chevron Energy

Technology Co., Houston, Texas, U.S.A. E-mail: [email protected] of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas, U.S.A. E-mail: [email protected];

[email protected] 2011 Society of Exploration Geophysicists. All rights reserved.

E127

GEOPHYSICS, VOL. 76, NO. 4 (JULY-AUGUST 2011); P. E127–E139, 17 FIGS., 2 TABLES.10.1190/1.3589101

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both soft and hard formations, and the elastic properties are con-

sistent with a gas-bearing layer invaded with water-based mud

filtrate.

The sensitivity analysis is performed with a 3D finite-differ-

ence time-domain (FDTD) code. This second-order explicit

algorithm solves the first-order, coupled velocity-stress elastic-

wave equations with staggered-grid central differencing in both

space and time. In addition, we implement perfectly matched-

layer (PML) absorbing boundary conditions to truncate the sim-

ulation domain, thereby significantly reducing computation time.

Validation of the simulation algorithm is performed against 1D

and 2D computations that include presence of the borehole and

a wide range of soft and hard formations and source frequen-

cies. Presence of a rigid sonic tool is not included in this study;

the simulation code assumes point sources and receivers with

source-receiver array dimensions adapted from a commercial

array sonic tool (Pistre et al., 2005).

In the following sections, we first describe the method used

to perform the numerical simulations. This includes details of

the 3D FDTD algorithm used in the study, along with a descrip-

tion of the source-receiver configuration and the formation

models examined. Next, we compare the 3D FDTD results to an

independent 1D code to benchmark the accuracy of the simula-

tions. Finally, we present results that examine the effects of ver-

tical and dipping TI, invasion, and presence of a shoulder bed

intersecting the tool at a high angle. Effects on simulated sonic

waveforms are illustrated through frequency-slowness dispersion

curves of flexural and Stoneley waves.

SIMULATION METHOD

Borehole sonic measurements are numerically simulated using

a Cartesian, 3D FDTD algorithm that solves the coupled veloc-

ity-stress differential equations (following Leslie and Randall,

1992)

qov

ot¼ r � s; (1)

and

osot¼ C � e; (2)

where q is density, t is time, v is the velocity vector

v ¼ vx; vy; vz

� �T; (3)

and s and e are the stress and strain tensors, respectively,

expressed in vector form as

s ¼ sxx; syy; szz; syz; sxz; sxy

� �T(4)

and

e ¼ ovx

ox ;ovy

oy ;ovz

oz ;ovz

oy þovy

oz ;ovx

oz þovz

ox ;ovy

ox þovx

oy

h iT

: (5)

Here, the superscript T denotes transpose, and C is the fourth-

rank stiffness tensor describing a transversely isotropic medium,

namely,

C ¼

c11 c12 c13 0 0 0

c12 c11 c13 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0 0 0 0 0 c66

0BBBBBB@

1CCCCCCA: (6)

Appendix A describes the rotation of the stiffness tensor and the

expanded differential equations for tilted TI media.

Equations 1 and 2 (or equations A-1 and A-2) are discretized

using staggered-grid, second-order central finite differences in

both space and time. Figure 1 shows the locations of the stress

and velocity components on the staggered grid. The coupled

system of velocity-stress FD equations are solved explicitly,

where time-stepping is performed in a leapfrog fashion in time

intervals 1=2Dt.Stability of the FD scheme is ensured by taking time steps

Dt < Dmin=ðvmax

ffiffiffi3pÞ, where Dmin is the smallest grid spacing,

and vmax is the maximum wave velocity in the model. Grid dis-

persion is mitigated by maintaining a maximum grid spacing

Dmax � vmin=ð2:5fcNÞ, where vmin is the minimum wave velocity

in the model, fc is the central frequency of the source wavelet,

and N ¼ 20. The FD grid used in the simulations consists of

1260� 181� 92 cells in the z-, x-, and y-directions, respec-

tively; grid dimensions in the x- and y-directions are identical

with the exception that, because of model symmetry in the

y-direction (for the simulation examples described in this paper),

the grid in the y-direction is truncated at y ¼ 0, and it is desig-

nated with Neumann boundary conditions. Cell size is Dz ¼ 0:4cm across the source-receiver array, and Dx ¼ Dy ¼ 0:4 cm

near the well, incrementally increasing outward from the well to

extend the grid to approximately 1 m. Simulations define the

circular cross section of the borehole with a staircase approxi-

mation; therefore, small cell sizes across the borehole in the x-yplane are necessary to attain accurate simulation results. A non-

splitting perfectly matched layer (PML) is implemented at the

boundaries, following that described by Wang and Tang (2003),

to mitigate spurious reflections. This implementation decreases

the grid size and shortens the computation time.

Figure 1. Stress and velocity components in the staggered grid.Stress component saa represents the normal stress components sxx,syy, and szz.

E128 Mallan et al.


Borehole sonic measurements are numerically simulated for

variants of the model illustrated in Figure 2, which depicts a

fluid-filled borehole penetrating a sand formation layer that is

shouldered by shale. The borehole radius is 11.1 cm, with the

borehole fluid assumed to have a density of 1000 kg=m3 and a

compressional velocity of 1500 m=s. For cases with presence of

invasion, the radius of the invaded zone is 26.2 cm, where the

invasion zone exhibits a circular, piston-shaped front.

Velocities and densities are assigned to the sand layer, repre-

senting fast and slow rock formations (Table 1). A formation is

described as fast or slow when the shear-wave velocity of the

formation is greater or less, respectively, than the borehole fluid

velocity. Formation properties listed in Table 1 are chosen to be

consistent with 30% (slow-formation) and 10% (fast-formation)

porosity, gas-bearing sands invaded with water-base mud filtrate.

We assume water saturation close to 100% for the invaded

zone, which yields higher compressional velocities in the

invaded zone relative to the uninvaded zone. When an appreci-

able amount of trapped gas remains in the invaded zone, then

compressional velocities in the invaded zone can be slower than

in the uninvaded zone when calculated using Gassmann’s fluid-

substitution model (Gassmann, 1951). However, assuming

“patchy” saturation for the invaded zone, velocities calculated

with the patchy fluid-substitution model (Hill, 1963; Dvorkin

and Nur, 1998) still would yield higher compressional velocities

in the invaded zone than in the uninvaded zone.

Transverse isotropy (TI) is assigned to the formation layer,

with the axis of symmetry normal to layer bedding. A degree of

TI is assumed, where wave velocities perpendicular to layer

bedding, V\, are lower with respect to wave velocities parallel

to layer bedding, Vjj (the isotropic velocities listed in Table 1),

such that V\ = Vjj ¼ 0.8.

Elements of the stiffness tensor are calculated from the

assumed compressional and shear-wave velocities, VP and VS,

respectively, using:

c11 ¼ qv2Pjj;

c33 ¼ qv2P?;

c44 ¼ qv2S?

c66 ¼ qv2Sjj

and

c12 ¼ c11 � 2c66:

Even though a value of vP in the direction of 45� with respect

to the TI symmetry axis is necessary to calculate a true value

for c13, we use the approximation (Schoenberg et al., 1996)

c13 ¼ c33 � 2c44:

Table 2 summarizes the calculated elastic constants.

Presence of a rigid sonic tool is not considered in the simula-

tions; instead, we assume point sources and receivers. Simulations

assume a near source-receiver offset of 2.736 m, and a receiver

array length of 1.824 m, with receiver spacings of 15.2 cm (Figure

3). This source-receiver array configuration is adapted from a com-

mercially available borehole sonic-logging tool (Pistre et al., 2005).

Figure 2. Layered velocity model with borehole and invasion.Radius of the borehole and invasion are 11.1 and 26.2 cm, respec-tively. Borehole dip angle is located in the x-z Cartesian plane andis measured from the normal direction to the layers. Tables 1 and2 summarize the assumed elastic properties of the formationlayer.

Table 1. Formation properties assumed in the numerical sim-ulations considered in this paper. Borehole fluid density andvelocity are 1000 kg=m

3and 1500 m=s, respectively. For cases

where the formation layer is transversely isotropic (TI), veloc-ities parallel to the TI symmetry axis are decreased by a factorof 1.25 relative to tabulated velocities.

ModelUninvaded

q=qf

vP

(m=s)vS

(m=s)Invaded

q=qf

vP

(m=s)vS

(m=s)

Slow 1.858 1970 1283 2.180 2450 1184

Fast 2.389 3160 2067 2.496 3660 2022

Table 2. Summary of the formation elastic constants calcu-lated from the properties listed in Table 1.

Modelc11

(GPa)c12

(GPa)c13

(GPa)c33

(GPa)c44

(GPa)

Slow uninvaded 7.21 1.09 0.70 4.61 1.96

Slow invaded 13.1 6.97 4.46 8.37 1.96

Fast uninvaded 23.9 3.44 2.20 15.3 6.53

Fast invaded 33.4 13.0 8.33 21.4 6.53

E129Simulation of 3D sonic waveforms


The simulated sonic source is a Ricker wavelet with a center fre-

quency of 3 kHz. Simulated time-domain waveforms are converted

to frequency-domain phase velocities using the extended Prony

method described by Donghong et al., (2008).

The 3D FDTD code is validated against an independent, ana-

lytical 1D sonic code (Chi and Torres-Verdın, 2004). Simula-

tions are compared to those of a vertical well in an infinitely

thick, anisotropic layer with presence of invasion. Figure 4a

shows waveforms simulated with a monopole source in the slow

formation model. Displayed waveform data are pressures. Figure

4b displays Stoneley dispersion curves processed from these

waveforms. Figure 5a shows waveforms simulated with a dipole

source in the slow formation model. Displayed waveform data

are the pressure gradient. Flexural dispersion curves processed

from these waveforms are shown in Figure 5b. Similarly, com-

parisons between 3D and 1D code results are examined for the

fast formation model. Figure 6a shows synthetic waveforms

simulated with a monopole source, and Figure 6b shows proc-

essed Stoneley dispersions. Figure 7a shows synthetic wave-

forms simulated with a dipole source, and Figure 7b shows

processed flexural dispersions. For additional reference, we dis-

play the corresponding analytical dispersion curves, which are

generated directly by solving the dispersion equation described

by Tang and Cheng (2004) for isotropic or TI, radially layered

Figure 3. Sonic tool configuration assumed in the numerical sim-ulations. The near source-receiver offset is 2.736 m, and the re-ceiver array length is 1.824 m, with 15.2-cm receiver spacings.

Figure 4. (a) Synthetic waveforms simulated for the case of amonopole source in the slow formation. Results are displayed for3D and 1D codes. (b) Stoneley dispersion processed from wave-forms shown in (a). The analytical dispersion solution is alsoshown for reference. Simulations consider the case of a verticalwell penetrating an infinitely thick, anisotropic, slow formationwith presence of invasion.

Figure 5. (a) Synthetic waveforms simulated for the case of adipole source in the slow formation. Results are displayed for 3Dand 1D codes. (b) Flexural dispersion processed from waveformsshown in (a). The analytical dispersion solution also is shown forreference. Simulations consider the case of a vertical well pene-trating an infinitely thick, anisotropic, slow formation with pres-ence of invasion.

E130 Mallan et al.


media. Results indicate that the 3D FDTD simulations match

within 1% in phase slowness the results obtained with the 1D

sonic code.

Although the 3D simulations compare extremely well to the

1D simulations, discrepancies are observed near the flexural cut-

off frequency between the flexural dispersion curves processed

from simulated waveforms (from the 3D and 1D codes) versus

the analytic flexural dispersion curve (Figure 5b and Figure 7b).

We believe this behavior arises from the fact that the analytical

solution does not consider the finite dimension of a source-re-

ceiver array and considers the ideal case of a single mode. By

contrast, dispersion curves obtained from processing waveforms

collected across a finite receiver array in the proximity of the

source must cope with diminished resolution at low frequencies

(longer wavelengths) and resolving the presence of additional

modes. We ran simulations, using the analytical 1D code, for

the cases with an extended receiver array. Resultant flexural dis-

persion curves processed from 50 receivers, with the nearest

receiver at approximately 10 m from the source, compare (with

significant improvement) to the analytic dispersion curves.

SIMULATION RESULTS

Vertical well

We first investigate the relative impact of the presence of ani-

sotropy and invasion on borehole sonic simulations in a vertical

well. The models consider a circular, piston-shaped invasion

front. Simulations are compared for the cases of slow and fast,

infinitely thick formations, where the formation is either iso-

tropic or transversely isotropic, with and without the presence of

invasion.

Figure 8 displays flexural and Stoneley dispersions, processed

from numerically simulated waveforms, for the slow formation

involving the different cases. These results indicate that pres-

ence of TI alters the degree of dispersion observed in the

Figure 6. (a) Synthetic waveforms simulated for the case of amonopole source in the fast formation. Results are displayed for3D and 1D codes. (b) Stoneley dispersion processed from wave-forms shown in (a). The analytical dispersion solution also isshown for reference. Simulations consider the case of a verticalwell penetrating an infinitely thick, anisotropic, fast formationwith presence of invasion.

Figure 7. (a) Synthetic waveforms simulated for the case of adipole source in the fast formation. Results are displayed for 3Dand 1D codes. (b) Flexural dispersion processed from waveformsshown in (a). The analytical dispersion solution is also shown forreference. Simulations consider the case of a vertical well pene-trating an infinitely thick, anisotropic, fast formation with pres-ence of invasion.



flexural and Stoneley waves, relative to the cases of an isotropic

formation. The amount of dispersion in the flexural wave is

decreased by as much as 30%, whereas Stoneley-wave disper-

sion is increased by more than a factor of 2.5. Flexural and

Stoneley dispersions exhibit sensitivity to the presence of inva-

sion; furthermore, this sensitivity appears enhanced when the

formation is transversely isotropic. Separation observed between

dispersion curves for cases with and without the presence of

invasion increases with respect to cases in isotropic formations,

by as much as 33% in the flexural wave and by as much as a

factor of 1.4 in the Stoneley wave.

Figure 9 displays flexural and Stoneley dispersions, processed

from simulated waveforms, for the fast formation involving the

different cases. Results indicate that the presence of TI, as with

the slow formation, alters the degree of dispersion observed in

the flexural and Stoneley waves, relative to the isotropic forma-

tion. Similar to slow formation models, the amount of dispersion

decreases in the flexural wave. However, in contrast to slow

formation models, Stoneley-wave dispersion decreases signifi-

cantly, to be closely null. Flexural and Stoneley dispersions

show sensitivity to presence of invasion; however, unlike for the

cases of soft formation models, the sensitivity to invasion

appears unchanged when the formation is transversely isotropic.

In addition to the Stoneley mode, a pseudo-Rayleigh mode is

present in the isotropic case, whereas a leaky compressional

mode is present in the anisotropic case.

The results in Figure 8 and Figure 9 consider the case where

the TI is introduced by reducing c44 relative to the isotropic

value, such that c44< c66. To further elucidate the nature of the

dispersion in response to presence of TI and invasion, we simu-

late the case where c44 is increased from the isotropic value

(c44> c66) by a factor of 1.25, and we simulate the anisotropic

cases where c44 is fixed and c66 is increased (c66> c44) and

decreased (c66< c44) by a factor of 1.25. Figure 10 displays ana-

lytic dispersions calculated for these anisotropic cases applied to

Figure 8. (a) Flexural and (b) Stoneley dispersions processedfrom simulated waveforms. The model is a vertical well in aninfinitely thick, slow formation layer. Simulations consider thecases of isotropic and anisotropic layers, with and without pres-ence of invasion. Analytical dispersion solutions (red line) arealso shown for reference.

Figure 9. (a) Flexural and (b) Stoneley dispersions processedfrom simulated waveforms. The model is a vertical well in aninfinitely thick, fast formation layer. Simulations consider thecases of isotropic and anisotropic layers, with and without pres-ence of invasion. Analytic dispersion solutions (red line) are alsoshown for reference.

E132 Mallan et al.


the slow formation, with and without the presence of invasion.

Although the low-frequency slowness of the flexural mode is

controlled primarily by c44, the amount of dispersion with

increasing frequency is governed by c66. Relative to the iso-

tropic case, when c66 is decreased (c66< c44), the amount of dis-

persion increases, whereas when c66 is increased (c66> c44),

dispersion decreases.

In contrast, the low-frequency slowness (approaching the tube

wave at 0 Hz) of the Stoneley mode is controlled by c66, and

c44 governs the degree of dispersion with increasing frequency.

Specifically, dispersion of the Stoneley mode increases when c44

decreases (c44< c66), and dispersion decreases when c44

increases (c44> c66). These results are consistent with flexural

and Stoneley modes sensitivities to V\ = Vjj presented by

Tang and Cheng, 2004. Regarding presence of invasion, the

cases show that presence of TI can alter the impact to the dis-

persion curve relative to the impact observed in the isotropic

case.

Deviated well

Next, we examine the effects of dipping TI on flexural and

Stoneley dispersion curves, and we assess the impact on the

invasion effect. Inline dipole (x-dipole), crossline dipole

(y-dipole), and monopole configurations are simulated for a

borehole dip angle of 60� measured from the normal to the

layer. For cases of TI dipping in the x-z plane, the y-dipole pro-

duces a pure shear wave (SH) with particle motion perpendicu-

lar to the x-z plane, and the x-dipole produces a quasi-shear

wave (qSV) with particle motion in the x-z plane.

Figure 11 displays dispersions processed from simulated

waveforms for the case of the slow formation, with and without

the presence of invasion. Although the SH and SV flexural dis-

persion curves are roughly parallel, the SH flexural wave

appears slightly more dispersive than the SV flexural wave.

Figure 10. (a) Flexural and (b) Stoneley dispersions calculatedanalytically. The model is a vertical well in an infinitely thick,slow formation layer. Simulations consider the cases of isotropicand anisotropic layers, with and without presence of invasion. Inaddition to the anisotropic cases considered to produce the disper-sion curves in Figure 8, where c44 is reduced from the isotropicvalue (c44< c66iso

), dispersions are shown for the case where c44 isincreased from the isotropic value (c44< c66iso

) and the anisotropiccases where c44 is fixed and c66 is increased (c66> c44iso

) anddecreased (c66< c44iso

).

Figure 11. (a) The x- and y-dipole flexural dispersions for caseswith and without presence of invasion, and (b) Stoneley disper-sions for cases with and without presence of invasion, processedfrom simulated waveforms. The model is a well dipping at 60� inan infinitely thick, slow formation layer exhibiting transverseisotropy.



Regarding the impact of the fast invasion zone, separation

between dispersion curves appears less discernable than for the

case of the vertical TI. Regarding Stoneley dispersion,

the amount of dispersion observed is between that observed for

the cases of a vertical well in isotropic and anisotropic forma-

tion layers. The effect of invasion is ambiguous, as dispersion

curves tend to overlay one another.

Figure 12 displays the corresponding results for the case of

the fast formation layer, with and without the presence of inva-

sion. Similar to the slow-formation cases, the SH flexural wave

appears slightly more dispersive than the SV flexural wave.

However, the effect of invasion appears greater than for cases

of a vertical TI, as a larger separation at higher frequencies

arises between the dispersion curves for cases with and without

the presence of invasion. Regarding the Stoneley dispersion, the

amount of dispersion observed is similar to that observed for

the cases of a vertical well in the isotropic formation layer. The

effect of invasion is also similar to that observed for the cases

of a vertical well in an isotropic formation layer.

At low frequencies, simulated flexural dispersion curves

approach the theoretical ray slowness (or group slowness in the

anisotropic sense) described by Thomsen (1986). We believe this

is a result of the strongly TI formation, which produces a strongly

anelliptic wave surface (Thomsen anisotropy parameters e¼ 0.28

and c¼�0.16 for both soft and hard formations considered in the

analysis). This effect also is discussed by Hornby et al. (2003).

Shoulder bed

Finally, we examine the impact of the presence of a layer

boundary (shoulder bed) on borehole sonic measurements. Sim-

ulations are performed for cases of the slow and fast formation

layers overlying the shale shoulder bed, such that the layer

boundary intersects the sonic tool between the source and the

receiver array. The source is located in the shale layer, 121.76

cm below the layer boundary, and the receiver array is located

in the formation layer. Simulations consider cases of the layer

boundary intersecting the borehole at 0�, 60�, 80�, and 90�.Figure 13 displays the half-space models and the locations of

the source and receiver array.

Figure 12. (a) The x- and y-dipole flexural dispersions for caseswith and without presence of invasion, and (b) Stoneley dispersionsfor cases with and without presence of invasion, processed fromsimulated waveforms. The model is a well, dipping at 60� in an infin-itely thick, fast formation layer that exhibits transverse isotropy.

Figure 13. Half-space formation models dippingat (a) 0�, (b) 60�, (c) 80�, and (d) 90�. The upperhalf-space has elastic properties equal to thoseof the formation layer (listed in Table 1 andTable 2), whereas the lower half-space (shoulderbed) has the properties shown in Figure 2. Thehalf-space boundary intersects the borehole atz¼ 0 cm. Source location (indicated with a white*) is z¼ 121.76 cm, and the array of 13receivers (indicated by white x marks), uniformlyspaced 15.2 cm apart, is located 273.6 cm abovethe source.

E134 Mallan et al.


Figure 14. (a) The x-dipole flexural, (b) y-dipole flexural, and(c) monopole Stoneley dispersions processed from simulatedwaveforms. Simulations consider the cases of half-space forma-tion models dipping at 0�, 60�, 80�, and 90� (Figure 13). The for-mation layer is slow and isotropic, and it does not includeinvasion.

Figure 15. (a) The x-dipole flexural, (b) y-dipole flexural, and(c) monopole Stoneley dispersions processed from simulatedwaveforms. Simulations consider the cases of half-space forma-tion models dipping at 0�, 60�, 80�, and 90� (Figure 13). The for-mation layer is fast and isotropic, and it does not include invasion.



Figure 14 displays x- and y-dipole flexural and monopole

Stoneley dispersion curves processed from simulated wave-

forms. Simulations consider the case of the slow formation

layer, isotropic and uninvaded, overlying the shale layer. As a

reference, the analytic dispersion curves corresponding to the

formation layer (solid line) and the shoulder bed (dashed line)

overlay the plot. No appreciable effect is observed in the flex-

ural and Stoneley dispersion curves for cases of 0�, 60�, and

80� dip. However, the compressional wave slowness, observed

in the monopole dispersion plot, is altered for the cases of 60�

and 80� dip, such that the observed slownesses are an average

of the formation layer and shoulder bed compressional wave

slownesses. For the case of 90� dip, dispersion curves are

affected significantly. At the lower frequencies (<5 to 6 kHz),

the flexural and Stoneley dispersion curves tend toward an aver-

age between the dispersion curves representing, respectively, ho-

mogeneous models of the formation layer and the shoulder bed.

At the higher frequencies, dispersion processing discerns two

modes, representative of the respective “homogeneous” cases.

In similar fashion, Figure 15 displays x- and y-dipole flexural

and monopole Stoneley dispersion curves processed from simu-

lated waveforms for the case of the fast formation layer, iso-

tropic and uninvaded, overlying the shale layer. At 0� dip, the

flexural and Stoneley dispersion curves are unaffected by the

presence of the layer boundary between the source and the re-

ceiver array. The Stoneley dispersion curve remains unaltered

for the cases of 60� and 80� dip. Conversely, at 60� and 80�

dip, flexural dispersion curves become irregular at lower fre-

quencies, with the effect becoming more severe with increasing

dip angle. At higher frequencies, dispersion curves are unaf-

fected by the presence of the shoulder bed. For the extreme case

of 90� dip, the Stoneley dispersion curve is altered and exhibits

the character of a slow formation, where the slowness increases

with frequency. The x- and y-dipole dispersion curves appear to

exhibit two flexural modes, characteristic of the two formations

comprising the half-space model.

A closer examination of the case of the 80� dipping half-

space model is shown in Figure 16. Although the simulated

Figure 16. The x-dipole (a) waveforms, (b) amplitude spectra, (c) phase spectra, and (d) dispersion. Simulations consider the case of thehalf-space formation model dipping at 80� (Figure 13c). The formation layer is fast and isotropic, and it does not include invasion.

E136 Mallan et al.


waveforms exhibit no apparent effect from the dipping shoulder

bed, the modulation pattern of the amplitude spectrum

(Figure 16b) clearly indicates mode interference. This behavior

results in the irregular nature of the dispersion (Figure 16d),

where the low frequency part of the dispersion is not well deter-

mined. It is worth noting that locations (in frequency) of the

sharp inflections seen in the dispersion plot at roughly 2, 3, and

4.5 kHz correspond to inflection points in the phase spectrum

(Figure 16c). Dispersion results processed from the simulated

waveforms were confirmed independently by Xiao-Ming Tang

(personal communication, 2009).

We do not believe that dipping shoulder-bed effects toward

the dispersion curves (Figure 15) are the result of refracted

energy along the shoulder bed–formation layer interface (as pre-

scribed by Snell’s law), because the distorting effect is not

observed in monopole dispersion results. Such an effect is

because of the lack of azimuthal symmetry in elastic properties

across the borehole. For the case of the 90� dipping half-space

model, the monopole Stoneley and dipole flexural modes each

appear to separate (or decouple) into two separate modes respec-

tive of either half space, especially at higher frequencies, where

the wavelength becomes small with respect to borehole radius,

thereby allowing the mode to decouple. For the cases of 60�

and 80� dip, the region along the borehole which has nonsym-

metric properties is finite, at least in regard to the radial length

of sensitivity of wave modes. Because the dipole flexural mode

has greater radial sensitivity (and is seemingly more susceptible

to decoupling because of its asymmetric nature compared to the

axisymmetric nature of the Stoneley mode), we believe that the

mode is partially impacted by this “decoupling” effect.

Regarding the slow-formation dispersion curves, the imped-

ance contrast between the slow formation and the shoulder bed

is too small to have an appreciable impact on waveforms, espe-

cially at longer wavelengths. At higher frequencies, the differ-

ence in flexural-mode phase velocity between the two layers

increases (as a result of the dispersive nature of the flexural

mode). This behavior, together with the shorter wavelength, ena-

bles the dispersion processing to resolve the two modes.

CONCLUSION

Numerical simulations performed with the 3D FDTD sonic

code show an excellent agreement with simulations of radially

1D, isotropic, and transversely isotropic formation models.

Benchmarks against frequency-slowness dispersions indicate

that the accuracy of the 3D code is within 1% in phase

slowness.

Simulations for the cases of a vertical well in an infinitely

thick layer show that the presence of transverse isotropy alters

the amount of dispersion observed in flexural and Stoneley

waves. The slowness at the low-frequency endpoint of the flex-

ural dispersion curve is fixed by c44 (the shear modulus govern-

ing wave propagation parallel to the TI symmetry axis), whereas

the change in slowness with increasing frequency is influenced

by c66 (the shear modulus governing wave propagation normal

to the TI symmetry axis). For the case of the soft formation, the

flexural wave is more dispersive when c66< c44 and less disper-

sive when c66> c44. In contrast, the low-frequency endpoint of

the Stoneley dispersion curve is fixed by c66, and the change in

slowness with increasing frequency is influenced by c44, such

that the Stoneley wave becomes more dispersive when c44< c66

and less dispersive when c44> c66.

Furthermore, these simulations suggest that the presence of TI

can alter the impact of invasion effects on dispersion curves.

For the considered case of the soft formation, the effect of the

fast invasion zone on flexural and Stoneley dispersions appears

enhanced in the TI (c44< c66) formation with respect to the iso-

tropic formation. We note that the impact of invasion observed

in this study applies to the modeled circular, piston-like invasion

front. An invasion profile with a considerable transition zone

could impact borehole sonic measurements in a different

manner.

Concerning simulations for cases of a layer boundary crossing

the tool between the source and receiver array, we found that

the Stoneley dispersion was unaffected by the presence of the

layer boundary for dip angles below 90�. On the other hand,

flexural dispersions exhibited significant distortions at lower fre-

quencies for cases of high dip angle. This distortion affects the

low-frequency asymptote of the flexural mode, thereby biasing

the estimation of shear-wave velocity. A modulation pattern in

the amplitude frequency-spectrum of the simulated waveforms

indicates interference of multiple modes in half-space models

dipping at high angles. At low frequencies, dispersion process-

ing is unable to discern individual modes clearly. Moreover, at

low frequencies, the increased wavelength diminishes the resolu-

tion of the receiver array, making it difficult to distinguish mul-

tiple modes that do not differ significantly in phase slowness.

ACKNOWLEDGMENTS

The work reported in this paper was funded by The University of

Texas at Austin’s Research Consortium on Formation Evaluation,

jointly sponsored by Anadarko, Aramco, Baker Hughes, BG, BHP

Billiton, BP, ConocoPhillips, Chevron, ENI, ExxonMobil, Halli-

burton, Hess, Marathon, Mexican Institute for Petroleum, Nexen,

Petrobras, Schlumberger, Statoil, Total, and Weatherford. We are

grateful to Xiao-Ming Tang for his analysis of numerical simula-

tion results. The authors acknowledge the Texas Advanced Com-

puting Center (TACC) at The University of Texas at Austin for

providing high performance computing resources used for the sim-

ulation of sonic waveforms. A note of special gratitude goes to

Gilles Guerin and two anonymous reviewers for their technical and

editorial comments which improved the original version of

the paper.

APPENDIX A

Rotation of the stiffness tensor (Equation 6) through dip angle

h about the y-axis yields (following Auld, 1990)

C0 ¼M � C �MT ¼

c011 c012 c013 0 c015 0

c021 c022 c023 0 c025 0

c031 c032 c033 0 c035 0

0 0 0 c044 0 c046

c051 c052 c053 0 c055 0

0 0 0 c064 0 c066

0BBBBBBB@

1CCCCCCCA;

where the transformation matrix M is given by



M ¼

b211 b2

12 b213 2b12b13 2b13b11 2b11b12

b221 b2

22 b223 2b22b23 2b23b21 2b21b22

b231 b2

32 b233 2b32b33 2b33b31 2b31b32

b21b31 b22b32 b23b33 b22b33 þ b23b32 b21b33 þ b23b31 b22b31 þ b21b32



0BBBBBBBBB@

1CCCCCCCCCA:

Here, bij constitute the elements of the rotation matrix

b ¼cos h cos / cos h sin / sin h� sin / cos / 0

� sin h cos / � sin h sin / cos h

0@

1A;

where h is the dip angle (in the x-z plane) about the y-axis, and

/¼ 0 (for the simulations considered in this paper) is the strike

angle (in the x-y plane) about the z-axis.

Expansion of Equations 1 and 2 by inserting Equations 3

through 5 and C0 yields the coupled velocity-stress differential

equations for a TI media rotated about the y-axis, given by

qovx

ot¼ osxx

oxþ osxy

oyþ osxz

oz;

qovy

ot¼ osxy

oxþ osyy

oyþ osyz

oz;

qovz

ot¼ osxz

oxþ osyz

oyþ oszz

oz;

(A-1)

and

osxx

ot¼ c011

ovx

oxþ c012

ovy

oyþ c013

ovz

ozþ c015

ovz

oxþ ovx

oz

� �;

osyy

ot¼ c012

ovx

oxþ c022

ovy

oyþ c023

ovz

ozþ c025

ovz

oxþ ovx

oz

� �;

oszz

ot¼ c013

ovx

oxþ c023

ovy

oyþ c033

ovz

ozþ c035

ovz

oxþ ovx

oz

� �;

osyz

ot¼ c044

ovz

oyþ ovy

oz

� �þ c046

ovx

oyþ ovy

ox

� �;

osxz

ot¼ c015

ovx

oxþ c025

ovy

oyþ c035

ovz

ozþ c055

ovz

oxþ ovx

oz

� �;

osxy

ot¼ c046

ovz

oyþ ovy

oz

� �þ c066

ovx

oyþ ovy

ox

� �:

(A-2)

Elements of the stiffness tensor associated with calculations of

shear stress are averaged over the finite-difference cells neighbor-

ing the respective shear stress component using a weighted har-

monic average. For example, in calculatingosyz

ot , c044 is averaged

accordingly:

c044 ¼Dyj þ Dyj�1

� �Dzk þ Dzk�1ð Þ

DyjDzk

c044ði;j;kÞþ DyjDzk�1

c044ði;j;k�1Þ

þ Dyj�1Dzk

c044ði;j�1;kÞ

þ Dyj�1Dzk�1

c044ði;j�1;k�1Þ

:

In the finite-differencing of velocities at nodes for which the re-

spective velocity component does not exist, a weighted, linearly

interpolated velocity is used. For example, Figure A-1 shows the

vx velocities interpolated to find ~vx, where o~vx

oz is used in calculating

the stress field sxx (see equation A-2).

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Simulation of borehole sonic waveforms in dipping ... · Simulation of borehole sonic waveforms in dipping, anisotropic, and invaded formations Robert K. Mallan1, Carlos Torres-Verdı´n2,