Precise Inversion of Logged Slownesses for Elastic Parametersin a Gas Shale Formation

Douglas E. Miller1, Steve A. Horne2, and John Walsh3

ABSTRACT

Dipole sonic log data recorded in a vertical pilot welland the associated production well are analyzed over a200×1100 foot section of a North American gas shaleformation. The combination of these two wells enablesangular sampling in the vertical direction and over arange of inclination angles from 54◦to 90◦. Dipole soniclogs from these wells show that the formation’s aver-age properties are, to a very good approximation, ex-plained by a transversely isotropic medium with a ver-tical symmetry axis and with elastic parameters satis-fying C13 = C12, but inconsistent with the additionalANNIE relation (C13 = C33 − 2C55). More impor-tantly, these data clearly show that, at least for fastanisotropic formations such as this gas shale, sonic logsmeasure group slownesses for propagation with thegroup angle equal to the borehole inclinationangle. Conversely, the data are inconsistent with aninterpretation that they measure phase slownesses forpropagation with the phase angle equal to the boreholeinclination angle.

INTRODUCTION

With increased interest in gas production from shale for-mations, there has been a corresponding increase in theneed to make accurate geophysical measurements of theseformations for use in planning and interpreting formationtreatments. Because these shale formations are largelycomposed of microscopically aligned platelets that are also

1Department of Earth, Atmospheric, and Planetary Sciences,MIT, formerly Schlumberger-Doll Research, Cambridge MA, USA.E-mail: demiller@mit.edu

2Chevron Corporation, Perth, Australia, formerly SchlumbergerK.K., Fuchinobe, Japan. E-mail:HorneS@chevron.com

3Schlumberger DCS, Houston, Texas, USA. Email:Walsh@slb.com

significantly laminated at a macroscale, they are oftenmorphologically anisotropic, with rotational symmetry abouta symmetry axis perpendicular to bedding, typically avertical axis. In such transversely isotropic (VTI) me-dia small perturbations of stress or strain, with respect toa stable reference state, are linearly related via an elastictensor with five free parameters.

Using the Voigt notation (C11 for C1111, C13 for C1133,C55 for C1313, etc.) for elastic moduli, and identifyingthe symmetry axis as the (vertical) 3-axis, the density-normalized moduli Cij/ρ have units of velocity squared.Only five elastic moduli are required to define VTI anisotropy,C11, C33, C55, C66, and C13. The first four of these fivemoduli are related to the squared speeds for wave prop-agation in the vertical and horizontal directions. V11 =√C11/ρ is the wavespeed for horizontally propagating

compresional vibration; V12 =√C66/ρ, the wavespeed

for horizontally propagating shear vibration with horizon-tal polarization; V31 = V13 =

√C55/ρ, the wavespeed

for vertically propagating shear vibration, as well as forhorizontally propagating shear vibration with vertical po-larization; V33 =

√C33/ρ, the wavespeed for vertically

propagating compressional vibration. (Table 1 in a latersection summarizes these values for our field data.)

The remaining parameter, C13, cannot be estimateddirectly and cannot be estimated at all without eithermaking off-axis measurements or invoking a physical orheuristic model with fewer than five parameters. Nev-ertheless, accurate measurements of C13 are essential forinterpreting the results of small-scale hydraulic fractur-ing tests (Thiercelin and Plumb, 1991), for calibratingthe relation between sonic measurements and other reser-voir characterization measurements (Vernik, 2008), for ge-omechanical studies (Amadei, 1996; Suarez-Rivera et al.,2006), and for accurate location of hydrofracture-inducedmicroseismicity (e.g., Warpinski et al., 2009).

Dipole sonic logs recorded in deviated wells have beenused for the determination of elastic parameters in a num-ber of studies (e.g., Hornby et al., 1995; Walsh et al.,

1

2 Miller et al., 2012 To Appear in Geophysics

2007). Somewhat surprisingly, there has been a lack ofconsensus on how the logged sonic wavespeeds are relatedto the elastic parameters in deviated wells. One importantintention of this paper is to resolve this situation based onan argument from fundamental principles and to confirmthat understanding using field and synthetic data.

PHASE AND GROUP VELOCITIES

Wavefronts (surfaces of constant traveltime) generated bya point source in a homogeneous anisotropic elastic mediumare not in general spherical, leading to two natural notionsof “propagation direction” and “propagation speed.” Thedirection connecting the source to a point on the wavefrontis the group (or ray) direction and the apparent speed inthis direction is the group (or ray) velocity. The directionnormal to the wavefront is the phase (or planewave) di-rection and the apparent speed in this direction is phasevelocity.

Mathematically, the relationship between phase and groupvelocities for VTI anisotropy can be written as

vG2(θ) = vP

2(θ) + [∂vP∂θ

]2

(1)

where θ, the phase angle, is the angle of the wavefront nor-mal relative to the symmetry axis; vP is the plane wave(phase) velocity; and vG(θ) is the group velocity associ-ated with phase angle θ. Note that this equation definesonly the magnitude of the group velocity and that thegroup velocity vector is not aligned to the phase velocityvector. The group angle, φ = φG(θ), is the angle of thegroup velocity vector, relative to the symmetry axis. Thetwo angles satisfy

tan(θ − φG(θ)) =[∂vP∂θ ]

vP (θ)(2)

It is of critical importance to distinguish the functionvG, which gives group velocity as a function of phase angle,from the related function vg, which gives group velocityas a function of group angle. The function vg is typicallycomputed indirectly by using equations (1) and (2), ortheir equivalents, to calculate both vG and φG as functionsof phase angle and then to iteratively solve or interpolatethe equation

vg(φG(θ)) = vG(θ) (3)

to determine vg at arbitrary group angles φ.This is illustrated in Figure 1. In the upper plot a

point source is located at the origin in a homogeneousanisotropic medium with elastic parameters that fit ourfield data (Table 2). Successive positions of the quasi-compressional wavefront excited by the point source areindicated by the dotted and solid red curves. Note thatthe noncircular appearance of the wavefront is indicativeof anisotropic wave propagation. The lower figure is acloseup with some added features. The dotted and solid

Figure 1: qP wavefronts and the construction of group andphase velocity surfaces for the medium with parametersfrom Table 2. The dotted and solid curves respectivelyrepresent wavefronts after 0.9 ms and 1 ms of propaga-tion time. The dotted line at 72◦ is aligned to the groupdirection at point a. The solid line at 55◦ is aligned to thephase direction at point a. As described in the text, whenthe curves are normalized by division by the propagationtime, the red curve has the shape of a polar plot of groupvelocity as a function of group angle, while the cyan curvehas the shape of a polar plot of the phase velocity as afunction of phase angle.

Precise Dipole Sonic Log VTI Inversion 3

curves, respectively, represent wavefronts after .9 ms and1 ms of propagation time. Because the propagation timefor the solid red curve is T = 1ms, it can be regarded asa polar plot of group velocity as a function of group anglein units of m/ms.P is the point on the wavefront tangent that has mini-

mum distance from the origin. As the point of tangency,G, varies over the wavefront surface, the set of all suchpoints forms a polar plot of the phase velocity as a func-tion of phase angle, again in units of m/ms. This surfaceis indicated in cyan on the lower part of Figure 1. Thisis the familiar geometric construction of a phase velocitysurface as the τ -p transform of a wavefront surface. Itmay be found, for example, in Postma (1955). Dellinger(1991) cites McGullagh (1837) as possible first reference.

It is a consequence of the definitions that triangle OPGis a right triangle with hypoteneuse OG and sides OPand PG. Equations (1) and (2) are consequences of thefact that the length |GP| of the segment GP is equal to[∂vP∂θ ]. It is evident from this relationship that for all phasedirections θ, and all modes in all anisotropic media

vG(θ) ≥ vP (θ) (4)

with equality occuring only when phase and group direc-tions coincide.

Note that the phase velocity surface lies outside thewavefront surface. That is because the wavefront surfaceis convex, and a tangent to a convex surface intersects thesurface only at the point of tangency. It is a property ofall anisotropic media that both the group and phase sur-faces for the fastest mode are convex (See, e.g., Chapman(2004), p. 168.). For VTI media, this is also true for thehorizontally polarized shear (SH) mode in which case thewavefront surface is an ellipse. Thus, for the fastest modein arbitrary anisotropic media and for the SH mode inVTI media, for any angle ψ,

vP (ψ) ≥ vg(ψ) (5)

with equality occuring only when phase and group direc-tions coincide.

The dotted line at 72◦ in Figure (1) is aligned to thegroup direction at point a. The solid line at 55◦ is alignedto the phase direction at point a. Thus, for phase angleθ = 55◦, φG(θ) = 72◦. Traveltime between the dotted andsolid red curves is dT= .1 ms.

vP (55◦) = |OP|/T = |ba|/dT

vG(55◦) = |OG|/T = |Ga|/dT = vg(φG(55◦)) = vg(72◦)

vg(55◦) = |Og|/T = |gc|/dT

A small array at a aligned with the wavefront normalab would see an apparent propagation speed equal to

the phase velocity vP (55◦) = 4.31 m/ms. An array ata aligned with the direction aG would see an apparentpropagation speed equal to the group velocity vG(55◦) =vg(72◦) = 4.51 m/ms. An array aligned along OP wouldsee an apparent propagation speed equal to the group ve-locity vg(55◦) = 4.08 m/ms. A long array at a aligned toab would see a non-linear apparent velocity that starts atvP (55◦) and asymptotically approaches vg(55◦).

It is also important to distinguish the angular dispersionequation 1 from the temporal dispersion equation

VG(ω) =∂ω

∂k= VP (ω) + k

∂VP∂k

(6)

which arises, for example, in solving for boundary-coupledpropagation in fluid-filled boreholes. Here VP (ω) = ω/kis the temporal phase velocity. For this temporal disper-sion it is the frequency dependence of the wave velocitiesthat gives rise to a difference between the temporal phasevelocity VP and the temporal group velocity, VG. It is ourbelief that this overloaded meaning of phase and groupvelocities has led to some of the confusion in the litera-ture.

Finally, it seems that one cannot discuss sonic loggingwithout speaking about slownesses. As scalars, they arethe reciprocals of the corresponding velocities. As vectors,they are aligned to the corresponding velocities, but withreciprocal magnitude. We use subscripted s to denotethe reciprocal of the corresponding velocity. Thus, forexample, in Figure 1, the phase slowness vector at 55◦ isOP|OP|2 and has magnitude sP (55◦) = .232 ms/m.

In order to recover elastic parameters from sonic data,one needs a correspondence rule relating velocities Vl(ψbh)extracted from sonic waveforms in a borehole with incli-nation angle ψbh to the underlying elastic moduli.

Hornby et al. (2003a) argued that logged compressionalspeeds were group velocities and found good agreementwith field data. Hornby, et al., (2003b) reported synthetictests confirming this correspondence rule, concluding ‘weare measuring the group velocity for all wave modes excitedby the dipole sonic tool.’

Sinha et al., (2004) disclosed a variety of ways to deriveelastic moduli from logged wavespeeds, based on a weakanisotropy assumption that logged speeds are phase veloc-ities for propagation with phase direction aligned to theborehole axis. Sinha et al. (2006) reported synthetic testsapparently confirming this correspondence rule, conclud-ing ‘Processing of synthetic waveforms in deviated well-bores using a conventional STC algorithm or a modifiedmatrix pencil algorithm yields phase slownesses of the com-pressional and shear waves propagating in the nonprinci-pal directions of anisotropic formations.’

Thus, there appear to be two conflicting correspondencerules reported in the literature. However, because theborehole inclination can be matched either to group orphase angle, there are three. For synthetics created witha borehole inclination angle ψbh, Hornby et al. (2003b)compared vP (ψbh) with vg(ψbh) and determined that the

4 Miller et al., 2012 To Appear in Geophysics

latter gave a better match to Vl(ψbh). Under similar cir-cumstances, Sinha et al. (2006) compared vP (ψbh) withvG(ψbh) and determined that the former gave a bettermatch to Vl(ψbh).

In view of the equations 4 and 5, these observations arenot inconsistent with one another. Moreover, for the qPand SH modes, they are consequences of 4 and 5 and thefundamental principal that no energy can propagate inany direction faster than the group velocity in that direc-tion. The introduction of a fluid-filled borehole or otherheterogenetity which only supports propagation at slowervelocity can only lower the propagation speed. That is,

Vl(ψbh) ≤ vg(ψbh) ≤ vP (ψbh) ≤ vG(ψbh) (7)

When vg(ψbh) and vP (ψbh) are distinct, the logged veloc-ity must be a better approximation to the former than thelatter. Both rules considered by Sinha et al. (2006) are in-consistent with propagation in strongly anisotropic media.Their conclusion that the phase velocity agrees better withsynthetic data than the group velocity is due to the use ofvG(ψbh) rather than vg(ψbh). (Figures 2 and 10 of Sinhaet al. (2006) show horizontal axes labeled“Propagationdirection θ (◦)” with no distinction made between groupand phase angles. For qP and qSV the group curves arefaster than the phase curves and are evidently plots ofvG(θ). For SH the group curves are slower and are evi-dently plots of vg(θ). The conclusion seems to be drawnfrom qP results shown in their Figures 6 and 7, wherevalues from processing the synthetic data are comparedto sP (60) and sG(60). The group slowness at group angle60◦, sg(60) = 341.1µs/m, is slower than either of theseand would fit better than either to their synthetic log re-sult.)

In weakly anisotropic media, the distinction betweenvP (ψbh), vG(ψbh) with vg(ψbh) has no practical signifi-cance. However, for shales or other strongly anisotropicmedia, the difference can lead to extreme differences inestimated elastic parameters, particularly for C13. Horneet al. (2011) described a two-well field example from agas shale formation where data were fit to high accuracyassuming the group correspondence rule Vl = vg(ψbh).In the remainder of this paper we review that example,showing that for this case, this group correspondence ruleis uniquely correct. Using the phase rule (Vl = vP (ψbh)),the SH data cannot be fit at all, and the qP and qSV datacannot be consistently interpreted. If only qP data areinterpreted, the phase rule leads to an unrealistic valuefor C13.

SONIC LOG DATA

The vertical pilot well and the horizontal production wellwere drilled from the same pad into a North Americangas shale formation as shown in Figure 2. The pilot wellencounters a 60-m (200-ft) interval in the gas shale. Thehorizontal production well, drilled from the same surfacelocation, encounters the gas shale at the same depths as

the vertical pilot well, implying near horizontal layering,at offsets from the pilot well of about 115 m (380 ft) to350 m (1150 ft), the last 120 m (400 ft) horizontal. Thebuild section of the horizontal production well had a build-radius of 120 m, or equivalently, a build-rate of 8◦/100ft.

Figure 2: (Upper) Vertical section showing the geometryof the two wells. Vertical depth is measured relative tothe top of the gas shale formation, indicated by the yel-low dotted line. The section of the well marked in greencorresponds to the build section of the horizontal produc-tion well. (Lower) Lithology of the build section.

The sonic log data were conventionally acquired usingthe Schlumberger Sonic Scanner4 tool and processed us-ing a standard Slowness Time Coherence algorithm toprovide compressional, fast and slow shear slownesses ateach depth in each well, as shown in Figure 3. The ve-locity data from the build section of the horizontal wellare plotted at Vl(sin(ψbh), cos(ψbh)) where Vl is the loggedvelocity and ψbh is the borehole inclination angle. Com-pressional is red; fast shear (horizontally polarized, SH)and slow (sagitally polarized, qSV) shear are cyan andgreen, respectively. The logged values in the vertical andhorizontal sections are remarkably consistent and are sum-marized by histograms plotted left of and below the axes,respectively.

4Mark of Schlumberger

Precise Dipole Sonic Log VTI Inversion 5

Figure 3: Dipole sonic log data. Logged values from thethe vertical pilot well and horizontal sections from theproduction well are summarized by the histograms plottedto the left and below the axes, respectively.

Only one shear speed is observed in the vertical welland that speed matches remarkably well with the slowshear speed (2.03 km/s) observed in the horizontal section.The lack of shear splitting in the vertical well, togetherwith the consistency of the slow shear speed over thevertical section and the match between vertical and slowhorizontal shear, is strong evidence that the medium is,within measurement accuracy, transversely isotropic witha vertical axis of symmetry (VTI). The five observed ax-ial wavespeeds, together with the observed density (2520kg/m3) yield a precise estimation for the four axial VTIparameters, as summarized in Table 1. Mean variationis about 2.5%. From these vertical and horizontal veloci-ties two of the Thomsen anisotropy parameters (Thomsen,1986) can be readily computed; Thomsen’s ε = C11−C33

C33=

0.48 and Thomsen’s γ = C66−C55

C55= 0.43.

Vertical Well Horizontal Well UnitsVelocity V33 V31 V11 V13 V12

Mean 3.39 2.03 4.76 2.03 2.77 km/secRMS 0.13 0.07 0.11 0.03 0.05 km/sec

variation

Modulus C33 C55 C11 C66

29.0 10.4 57.0 19.3 GPa

Thomsen α0 β0 ε γ3.39 2.03 0.48 0.43

Table 1: Velocities and corresponding elastic constantsmeasured in the vertical pilot well and the horizontal sec-tion of the production well. The first two Thomsen pa-rameters have units km/sec; the others are dimensionless.

Horizontally polarized shear mode: SH

For VTI media, the group and phase velocity surfacesfor the horizontally polarized shear-wave mode (SH) arecompletely determined by the axial shear velocities V55and V66, which are equal to

√C55/ρ and

√C66/ρ, respec-

tively. As noted previously, the group velocity surface isan ellipsoid; the phase velocity surface is not. The phasevelocity, vP (θ), is systematically faster than the group ve-locity, vg(φ), when θ = φ.

Figure 4: Sonic log data overlain with phase and groupsurfaces for SH mode.

Figure 4 shows the fast shear data from Figure 3, over-lain by the SH group and phase surfaces determined bythe measured C55 and C66. It is clearly evident thatthe group velocities are a better fit to the log data thanthe phase velocities. This can be quantified by referringto the root mean square (RMS) misfits defined as χg =√∑

(Vl(ψbh)− vg(ψbh))2/N and χP =√∑

(Vl(ψbh)− vP (ψbh))2/N ,the sums of length N being taken over all data for thegiven mode in the build section of the well. The RMSmisfit for the group surface is χg = 0.029 km/s comparedwith an RMS misfit for the phase surface χP = 0.082 km/s.

It is remarkable that the two shear speeds, measuredin the horizontal well, accurately predict the logged val-ues for the vertical and deviated sections, hundreds offeet away, through significant changes in inclination andlogged wavespeed.

Modes with polarization in the vertical plane:qP and qSV

Four of the five VTI parameters are fixed by the axialdata obtained from the vertical pilot well and the hori-zontal section of the production well. The remaining elas-tic parameter, C13, can be determined using qP and qSVlog data recorded over the build section of the produc-

6 Miller et al., 2012 To Appear in Geophysics

tion well. Thus the determination of C13 becomes a one-parameter inversion problem. Because both qP and qSVdata must be fit at each inclination angle, the problem isvery well conditioned.

RMS misfit as a function of C13 for both correspondencerules and both qSV and qP modes is shown in Figure 5.A C13 value of 16.4 GPa (Thomsen’s δ = 0.35) minimizesRMS misfit for both modes under the group correspon-dence rule. Using the phase correspondence rule, the sameC13 value minimizes RMS qSV misfit to the slow sheardata; however, the compressional data are significantlymisfit by the qP phase velocity surface. The qP misfit un-der the phase rule decreases with decreasing C13 until thevalue becomes significantly negative and the correspond-ing medium becomes significantly unrealistic.

Figure 5: RMS misfit to log data as a function of C13 forqP (top) and qSV (bottom) modes.

Figure 6 shows log data for all the modes overlain withphase and group velocity surfaces using the best-fit valuefor C13. The group surface fits remarkably well. Thephase surface fits only the qSV data. Evidently, for theqSV mode in this medium, the phase and group velocitysurfaces are nearly coincident, the difference being lessthan .5% of the mean for all angles sampled.

Figure 7 shows log data for all the modes overlain withphase and group velocity surfaces using C13 = −5.0GPa.With this value, the qP phase surface is a fair match to thelogged compressional data, but the SV data are in starkdisagreement with the modeled SV phase surface. Note,in particular, that this model predicts that the two shearspeeds should match (with a crossover) at phase anglenear 55 ◦, whereas the measured data at this inclination

Figure 6: Sonic log data overlain with phase and groupsurfaces for qP and qSV modes using C13 = 16.4 GPa(Thomsen’s δ = 0.35). This is the best-fit estimate ofC13. The group velocity surface is a good fit for all modes.The qSV phase and group surfaces are nearly coincident,hence the model is also a good fit to the qSV phase surface,but the qP phase surface is inconsistent with the loggedcompressional data.

Figure 7: Sonic log data overlain with phase surfacesfor qP and qSV modes using C13 = −5.0GPa (Thom-sen δ = −0.29). This model fits the qP phase surface tothe logged compressional data but is inconsistent with thelogged qSV data and is physically implausible.

Precise Dipole Sonic Log VTI Inversion 7

angle differ by more than 0.5% and both are slower thanthe modeled speed at crossover.

Conclusion from Sonic Log Data

It is clear that the group velocity correspondence rule isthe correct rule for this data. Using this rule, it is possibleto fit all the data from all modes, both wells, and all angleswith a single VTI medium description. The phase veloc-ity correspondence rule is demonstrably false. Using thatrule, the SH data cannot be fit at all and there is no valuefor C13 that comes at all close to fitting both qP and qSV.Worst of all, if only P data are used, the phase correspon-dence rule yields a reasonable fit using a best-fit value for

C13 (or equivalently, Thomsen δ = (C13+C55)2−(C33−C55)

2

2C33(C33−C55))

which is far from the correct value and has the wrong sign.The near-perfect fit of the logged data using the group

correspondence rule does not guarantee that the rule isuniversally valid, but it is certainly strong evidence forwide applicability. As a further aid to understanding, wehave performed full-waveform synthetic modeling whichwill be described in the next section.

SYNTHETIC MODELLING

Using a 3D finite-difference code developed at the MITEarth Resources Laboratoy (Cheng, 1994), we created afull-waveform synthetic similar to those used by (Hornbyet al. 2003b) and (Sinha et al. 2006), but based on param-eters from our gas shale model. The elastic parameters forthe modeled formation are the same as those derived fromour inversion (see Table 2, “raw”) and the formation den-sity is 2520 kg/m3. The borehole has a diameter of 0.20m(8 in.), is inclined 55◦ from vertical, and is filled with aliquid having a velocity of 1500 m/s and density of 1000kg/m3. A simulated monopole source was placed at theorigin and driven with an 8 kHz Ricker wavelet.

Figure 8 shows a pressure snapshot at time 1.080 ms(540 timesteps) from the start of the simulation. Over-lain are the geometry of the experiment, together withtwo copies of the analytic wavefront surface for the mod-eled formation, scaled to represent travelimes of .813 msand .693 ms. Away from the borehole, the shape of thefinite-difference wavefront matches the analytic surface,an indication that the source radiates into the solid as anapproximate point-source. Near the borehole there is asmall distortion of the wavefront shape and a loss of en-ergy to the somewhat complicated reverberant signal inthe borehole. In successive snapshots, the pattern movesoutward, but does not change, an indication that the cou-pling is at the axial slowness associated with the wavefrontin the direction aligned to the borehole. That is, it is atthe group slowness associated with a group angle equalto the borehole inclination angle. Careful observers willnote a plane wave connecting a bright spot on the bore-hole wall between the red curves to a point at about 2m along the horizontal axis. That is a quasi-shear wavewhose phase slowness, projected onto the borehole axis,

Figure 8: Snapshot of the wavefield at 1.080 ms, overlainby experimental geometry and wavefronts correpsondingto the phase (blue dotted line) and group (red continuousline) velocities.

matches the group slowness of the qP signal and boreholepressure signal to which it is coupled. There is also someevident direct qSV signal above and below the boreholeat about x = 1.4 m, z = 1 m. A bright Stoneley wave inthe borehole is evident starting at about x = 1 m, z = 0.7m.

Figure 9: Waveforms overlain by parallel lines correspond-ing to the phase (blue dotted lines) and group (red con-tinuous lines) velocities from Figure 8.

Figure 9 shows synthetic waveforms from 13 centeredmonopole pressure receivers at the locations indicated bygray squares in Figure 8. These are spaced to match thetool used to collect our field data. Overlain are two redparallel lines with slope equal to 4.08 m/ms, the groupvelocity for the modeled formation at group angle equalto 55◦. Also shown are two blue dotted lines with slopes

8 Miller et al., 2012 To Appear in Geophysics

equal to 4.31 m/ms, the phase velocity for the modeledformation at the phase angle equal to 55◦. It is evidentthat the signal is aligned to the group velocity and that,although it has an extended signature, it exhibits no sig-nificant temporal dispersion. Sonic modelers will recog-nize this as a ‘Partially Transmitted’ (PT) compressionalsignal.

Figure 10: semblance of the waveforms from Figure 9.Vertical lines indicate slownesses sG(55◦), sP (55◦), andsg(55◦).

The field logs were processed using the conventionalprocessing technique described by Kimball and Marzetta(1984), known as Slowness Time Coherence (STC) to quan-tify the velocity of the compressional arrival. Because oursynthetic is, a priori, windowed in time, it can be ana-lyzed with a simplified semblance calculation which usesa fixed time window.

Given a window function w(t), an array of N waveformsD(t, ri) as in Figure 9, and a slowness, s, we can form ashifted, muted array

Ds(t, rn) = w(t)D(t+ s(rn − r1), rn) (8)

and calculate semblance

semb(s) =

∑t(∑nDs(t, rn))2

N∑t

∑nDs(t, rn)2

(9)

Figure 10 plots semblance of waveforms from Figure 9 asa function of slowness, using a 2.6 ms rectangular win-dow function, centered on 1.3 ms, with a 1 ms raised co-sine taper at each end. The peak semblance occurs ats = Smax = 0.248 ms/m. Solid vertical lines indicateslownesses sP (55◦) = 0.232 ms/m, and sg(55◦) = 0.245ms/m. The dotted black line in Figure 10 shows sG(55◦)= 0.222 ms/m. It is clear that the group rule gives an

excellent match and the phase rule does not.The small difference between the semblance peak and

the formation group slowness is consistent with our equa-tion 7 and similar to the small bias observed in syntheticstudies of isotropic media (e.g. Paillet and Cheng, 1991,pp. 164-167). To confirm this observation we made anotherwise identically created and processed synthetic sub-stituting an isotropic model with Vp and Vs matched tothe gas shale group velocities (4.073 km/sec and 2.108km/sec, respectively). The isotropic synthetic gave a sim-ilar small bias with respect to the .245 ms/m mediumslowness, with a semblance peak at .251 ms/m.

The source of the bias can be analyzed by performinga temporal dispersion analysis. Semblance, as defined byequation 9, can be decomposed as an energy-weighted av-erage of semblance as a function of temporal frequency

semb(s) =∑f

semb(f, s) E(f) (10)

where

semb(f, s) =(∑nDs(f, rn))2

N∑nDs(f, rn)2

(11)

and

E(f) =

∑nDs(f, rn)2∑

f

∑nDs(f, rn)2

(12)

with Ds(f, rn) denoting the temporal Fourier transformof Ds(t, rn). The function Smax(f) defined as the slow-ness which maximizes semb(f, s) provides an estimationof temporal phase slowness as a function of frequency thatis similar to what would be obtained with the variation ofProny’s method used by Sinha et al. (2004) (Lang et al.,1987; Ekstrom, 1995).

Figure 11 plots Smax(f) for the waveforms from Fig-ure 9. The bar graph at the bottom of the figure showsa scaled plot of E(f). Note that the estimated slow-nesses lie at or above sg(55◦). That is, they are at ax-ial wavenumbers that correspond to evanescent qP andoblique outgoing qSV or SH in the solid. This is as ex-pected for the PT signal. The decay and small disper-sion result from the partial conversion of energy into thetransmitted shear modes each time the signal reflects fromthe fluid/solid boundary. The energy-weighted averageSmax =

∑f (Smax(f) E(f)) agrees with Smax to four sig-

nificant digits.Evidently, the inversion for elastic parameters and anal-

ysis of synthetic forward models could be iterated (atsubstantial computational cost) to account for the smallbias that results from using the logged semblance maximaSmax(ψbh) as proxies for vg(ψbh). We have not done this.However, it should be noted that a uniform 1% overes-timation of all slownesses would result in a uniform 2%underestimation of all moduli. That is, a rescaling with-out change of shape of the anisotropy would have the sameeffect as would result from a 2% underestimation of den-

Precise Dipole Sonic Log VTI Inversion 9

Figure 11: Temporal phase slownesses of the wave-forms from Figure 9. Horizontal lines indicate slownessessG(55◦), sP (55◦), and sg(55◦).

sity.Similar results were also obtained with an SH synthetic

using the gas shale model with the borehole and source-receiver geometry as previously detailed. The semblancepeak Smax = 0.414 ms/m was 1 % slower than the grouprule prediction of 0.409 ms/m, and 6 % slower than thephase rule prediction of 0.392 ms/m.

Using the same elastic model, we made monopole andboth horizontal and vertical dipole synthetics at the nineborehole inclination angles indicated in Figure 12. Pro-cessing all these synthetics, we found close agreement be-tween the semblance maxima and sg(ψbh)/1.01, evaluatedat all modes and angles. As noted previously, these areexactly the values of sg(ψbh) associated with a model inwhich all the moduli are 2% larger than our syntheticmodel. This is the “bias-corrected” model shown in Table2 and is our best estimate of the true elastic moduli to fitthe field data. The error estimates are derived from theRMS misfits of the data to the raw group slownesses.

COMPARISON WITH SHALE MODELS

There have been a variety of suggested methods for pre-dicting one or more of the elastic moduli in shales frommeasured values of the remaining parameters (e.g., Schoen-berg et al., 1996; Suarez-Rivera and Bratton, 2009). Inparticular, the ANNIE approximation of Schoenberg et al.(1996) proposes two extra constraints:

C13 = C33 − 2C55 (13)

Figure 12: semblance peaks for processed synthetic data.Dots indicate sP , and sg, evaluated at the borehole incli-nation angles and uniformly increased by 1%.

Modulus C11 C13 C33 C55 C66

raw 57.0 16.4 29.0 10.4 19.3Corrected 58.1 16.6 29.6 10.6 19.7

±2.5 ±1.5 ±2.0 ±0.3 ±0.7

Thomsen α0 β0 ε δ γraw 3.39 2.03 0.48 0.35 0.43Corrected 3.43 2.05 0.48 0.35 0.43

±0.11 ±0.05 ±0.05 ±.025 ±.015

density ρ kg/m3

2520± 50

Table 2: Elastic constants (top) and corresponding Thom-sen parameters (bottom) measured using the vertical pi-lot well and the horizontal production well dipole soniclog data. Elastic Moduli are reported in GPa, α0 and β0are P and S velocities along the vertical direction and arereported in km/s. ε, δ and γ are dimensionless.

10 Miller et al., 2012 To Appear in Geophysics

C13 = C11 − 2C66 (14)

The first constraint is equivalent to Thomsen δ = 0.The second constraint is equivalent to C13 = C12. To-gether, they are inconsistent with the axial measurementsreported herein because the measured C33 − 2C55 is lessthan half of the measured C11−2C66. Our measured valuefor C13 is far from satisfying the first constraint but iswithin statistical error of satisfying the second constraint(our best-fit value satisfies C13 = 0.89 C12).

Another approximation that constrains the five elasticparameters is the fractured isotropic model described bySchoenberg and Douma (1988), which is determined byisotropic moduli λ and µ plus normalized normal and tan-gential fracture excess compliances EN and ET . Sayers(2008) fit an equivalent four-parameter model to measure-ments of muscovite. Sayers’ ratio of excess compliancesBN/BT is equivalent to the ratio EN/ET of Schoenbergand Douma (1988) multiplied by µ/(λ + 2µ).) This typeof medium satisies the extra constraint

(C13 + C33)(C13 + 2C66) = C33(C13 + C11) (15)

which entails

C13 = −C66 +√C2

66 + (C11 − 2C66)C33 (16)

For our gas shale medium, the right-hand side aboveevaluates to 10.8 GPa, which is significantly smaller thatthe measured value of 16.4 GPa. Thus, the gas shalemedium cannot be approximated by a fractured isotropicmedium.

In fact a somewhat stronger statement can be made.Backus (1962) defined quantities S and T by

S =C2

13 + 2C66C33 − C12C33

4C33(17)

and

T =C33 − C13

2C33(18)

and showed that any transversely isotropic medium thatis equivalent to a stack of thin isotropic layers must satisfy

(3

4− T )2 < (

3

4C55− 1

C33)(

3C66

4− S) (19)

For our gas shale, the left-hand side of equation 19 eval-uates to 0.284 while the right-hand side evaluates to 0.267.It follows that the best-fit estimated gas shale cannot beconstructed by an effective medium formed from thin in-terbedded isotropic layers. Note, however, that the in-equality of equation 16 would be satisfied if the value ofC13 were 15.8 GPa or lower, so the possibility of a three-

isotropic-constituents equivalent medium is within exper-imental error. The value of C13 given by equation 14is inconsistent with any layered isotropic approximation.The value given by equation 16 is the upper bound forvalues of C13 consistent with a two-isotropic-constituentsapproximation.

CONCLUSIONS

This gas shale formation, as sampled by this pair of bore-holes and logged with a sonic tool, shows strong anisotropyand remarkable homogenetity. The formation’s averageproperties are, to a very good approximation, explained bya transversely isotropic medium with a vertical symmetryaxis and with elastic parameters approximately satisyingC13 = C12, but inconsistent with any representation bya fractured isotropic medium. More importantly, thesedata clearly show that, at least for fast anisotropic for-mations such as this gas shale, sonic logs measure groupslowness for propagation with the group angle equalto the borehole inclination angle. The dipole sonicdata, taken as a whole, are inconsistent with the assump-tion that they represent phase slownesses for propagationwith phase angle equal to borehole inclination angle.

In this example, the shear speeds are significantly higherthan the fluid speeds, so caution should be used in in-terpreting logged shear data in slow anisotropic forma-tions. The uniform velocity-bias correction should also bechecked using carefully made synthetics based on match-ing models when used in contexts where precise values ofelastic moduli are required.

ACKNOWLEDGMENTS

This material was presented at the 1st International Work-shop on Rock Physics, August 7-12, 2011 at the ColoradoSchool of Mines.

The authors gratefully acknowledge the contribution ofour colleagues in the many stimulating conversations thatled to this paper. We are particularly thankful to YangZhang of MIT’s ERL group for aiding the first author ininstalling a working copy of the 3D finite-difference codeon his Mac and to Chris Chapman, Philip Christie, JakobHaldorsen, David Johnson, and Colin Sayers for commentson earlier drafts of the paper. We are especially gratefulto Chris Chapman for discussions clarifying the argumentfrom first principles presented in the section on phase andgroup velocities.

We are most grateful to the anonymous operating com-pany for permission to publish the data.

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