Static and dynamic pressure sensitivity anisotropy of a...

Geophysical Prospecting, 2016 doi: 10.1111/1365-2478.12403

Static and dynamic pressure sensitivity anisotropy of a calcareous shale

Oliver N. Ong1, Douglas R. Schmitt1∗, Randolph S. Kofman1 and Kristine Haug2

1Institute for Geophysical Research, Department of Physics, 4-138 CCIS, University of Alberta, Edmonton, AB T6G 2E1, Canada,and 2Alberta Energy Regulator, Alberta Geological Survey, Edmonton, AB T6B 2X3, Canada

Received November 2015, revision accepted March 2016

ABSTRACTOptimizing the productivity of nonconventional, low-permeability “shale” reservoirsrequires detailed knowledge of the mechanical properties of such materials. Theserocks’ elastic anisotropy is acknowledged but usually ignored due to difficulties inobtaining such information. Here we study in detail the dynamic and static elasticproperties of a suite of calcareous mudstones from the nonconventional Duvernayreservoir of Alberta, Canada. The complete set of transversely isotropic elastic con-stants is obtained from strategically oriented ultrasonic transducers to confining pres-sures of 90 MPa. Wave speed anisotropies of up to 35% are observed at even thehighest confining pressures. Furthermore, the stress sensitivity of the wave speeds, andhence moduli, is itself highly dependent on direction with speeds taken perpendicularto the bedding plane being highly nonlinearly dependent on pressure, whereas thosealong the bedding plane show, unexpectedly, nearly no pressure dependence. Theseobservations are in qualitative agreement with the preferentially oriented porosityand minerals seen in scanning electron microscope images. These results may be sig-nificant to the interpretation of sonic logs and azimuthal amplitude versus offset forprincipal stress directions, for the concentration of stress within such formations, andfor estimation of static engineering moduli from sonic log wave speeds.

Key words: Acoustics, Anisotropy, Petrophysics, Rock Physics, Seismics.

INTRODUCTIO N

Numerous workers have commented that sedimentary basinsare predominantly filled with rocks that are generically called“shales”, that such rocks are expected to be elasticallyanisotropic, and that ignoring this anisotropy affects the qual-ity of seismic images even in regions with flat-lying stratigra-phy. This last aspect, reinforced by Thomsen (1986), moti-vated many theoretical, experimental, and modelling effortsto better understand shale seismic anisotropy over the last30 years. To some degree, assimilation of these findings intogeneral practice has been slow even though the utility of in-corporating anisotropy has been recognized.

However, at this writing, the growth of hydrocarbon pro-duction from these “shales” formerly at best considered as

∗E-mail: [email protected]

source and seal rock has revolutionized the petroleum industryin North America. Direct recovery of high-quality oil and nat-ural gas that are made accessible by hydraulic stimulations hasopened up new reserves that were not previously accounted.Such production demands that we have improved knowledgeof the anisotropic physical properties of these rocks as thisinformation is, in addition to the more traditional needs of re-flection seismic imaging, necessary to comprehending in situ

states of stress (e.g., Gholami et al. (2015) and Amadei (1996))and the initiation and growth of hydraulic fractures (e.g.,Aghighi and Rahman (2010); Vahid and Ahmad (2011); andWaters, Lewis, and Bentley (2011)); to predicting the stabilityof boreholes drilled through such formations (e.g., Jin et al.

(2012); Jia et al. (2014); Ong and Roegiers (1996); Gaedeet al. (2012); and Holt et al. (2015b); to improving the loca-tion of micro-seismic events (e.g., King and Talebi (2007)and Li et al. (2014)) and the interpretation of their focal

1C© 2016 European Association of Geoscientists & Engineers

2 D.R. Schmitt et al.

mechanisms (e.g., Vavrycuk (2005) and Kawasaki andTanimoto (1981)); and to evaluating seismic hazard(Maxwell, Zhang, and Damjanac 2014). Properly address-ing these issues also often requires that we have knowledgeof the “static” properties that usually differ from the cor-responding dynamic moduli that are determined from wavepropagation.

Aside from notable exceptions (Kaarsberg 1959), therewere few direct laboratory measurements of the anisotropicproperties of “shales” probably because they are difficult towork with and characterize and they were not consideredreservoir rock. It is usually presumed that such rocks havetransversely isotropic (i.e., hexagonal) symmetry and as suchmust be described by five independent elastic parameters.Thomsen (1986) was able to collect from the open literatureand internally reports 17 examples only with a completeset of transversely isotropic elastic constants, a number thathad rapidly increased substantially in the compilation ofCholach and Schmitt (2006) that incorporated comprehensivemeasurement campaigns by Wang (2002b), Vernik and Liu(1997), and Johnston and Christensen (1995). The numberof experimental studies has dramatically grown since thattime, and the bulk of these studies measure P- and S-wavespeeds through core plugs with axes cut at a minimumof three strategic orientations with respect to the shalesymmetry by placement of ultrasonic transducers arounda single cylindrical sample (Dewhurst and Siggins 2006;Wang 2002a; Sarout and Gueguen 2008a; Sarout et al. 2014;Jakobsen and Johansen 2000; Holt et al. 2015a; Holt et al.

2012), at the ends of at least three cored cylinders or onthe flat faces of specially machined polyhedra (Wong et al.

2008; Cheadle, Brown, and Lawton 1991; Homand et al.

1993; Melendez Martinez and Schmitt 2015). With regardto multi-faced polyhedral, the measurements of Nara et al.

(2011) with a 98-faced sample and of Lokajicek and Svitek(2015) on spherical samples allowing 132 different directionsto be sampled are worth noting although these authorshave not studied sedimentary rocks. The same sphericalapparatus was used by Nadri et al. (2012) on a shale samplefrom NW Australia. Recent developments that employ laserinterferometry show promise (Blum, Adam, and van Wijk2013) for increasing the number of directions sampled.These last methods are important as sufficient informationis collected that no a priori assumptions need be made withregard to the material symmetry. Other recent contributionsthat made measurement on plugs taken only parallel andperpendicular to the bedding planes cannot, without assump-tions, completely describe the presumed transverse isotropy

but still provide useful information (e.g., Miller, Plumb, andBoitnott (2013) and Sone and Zoback (2013)).

Under pseudo-static loading conditions, there are numer-ous studies that examine the anisotropic rock strengths (e.g.,Heng et al. (2015) and McLamore and Gray (1967)), butusually, the strains are not analysed to provide elastic mod-uli. Exceptions to this include Chenevert and Gatlin (1965)who clearly demonstrated the anisotropy by measuring thestrains induced by uniaxially loading cylinders and rectangu-lar prisms of two shales and Niandou et al. (1997) who deter-mined from three appropriately oriented samples the completeset of Poisson’s ratios and Young’s moduli of Tournemireshales during uniaxial loading. Gautam and Wong (2006)use a combination of uniaxial compression and torsion mea-surements on vertically and horizontally oriented cores of aColorado shale. Most recently, Islam and Skalle (2013) havedemonstrated the anisotropy of a compressible Pierre shalebut were not able to obtain a full set of elastic constants.

In this study, we focus on stiff calcareous shales of the Du-vernay Formation, Alberta, Canada. The “Duvernay” holdsparticularly large resource estimated to hold upwards of 44.1bbl–82.9 bbl of oil, 7.5 bbl–16.3 bbl of natural gas liquids, and353 Tcf–540 Tcf of natural gas (Rokosh et al. 2012). Even asmall improvement in recovery rates translates to a large bulkeconomic gain. More urgently, however, this shale formationis garnering a great deal of interest because of the possible link-age between hydraulic fracture stimulations within the Duver-nay, and felt-induced seismicity as large as 4.3Mw (AGS 2015)and understanding this material’s mechanical properties arealso of interest. In this contribution, we describe a series ofmeasurements of the static and dynamic elastic properties onDuvernay Formation cores. We find that not only the Du-vernay shale is highly anisotropic but also that the sensitivityof the elastic moduli to stress strongly varies with direction.We organize this paper into sections that briefly review theessential equations of anisotropic elasticity, characterize thematerials, describe the experiments that show these rocks tobe highly anisotropic, and discuss the implications for com-parison between static and dynamic moduli.

B A C K G R O U N D

Forms of Hooke’s law

There are now many discussions of anisotropic elasticity in theliterature, and we direct the reader to any of the many contri-butions available for overviews of general elasticity, includingthe definitions of stress and strain, the statement of Hooke’s

C© 2016 European Association of Geoscientists & Engineers, Geophysical Prospecting, 1–23

Pressure anisotropy of calcareous shale 3

x3

x2

x1

y

θ

Figure 1 Reference Cartesian x1–x2–x3 coordinate system with therotational axis of symmetry parallel to x3 from which angle θ ismeasured. The x1–x2 plane is the bedding or isotropic plane.

law (e.g., Jaeger, Cook, and Zimmerman (2007)), or the solu-tion of Christoffel’s equation in determining the phase speedsand polarizations of the three plane waves that will generallypropagate in any direction through an anisotropic medium(e.g., Auld (1973)). We presume here that the shales studiedare transversely isotropic with the rotational axis of symmetryparallel to the x3-axis with the material’s isotropic “bedding”and x1–x2 coordinate system planes identical (Fig. 1). In thissituation, the material is described by five independent elasticparameters that, depending on the situation, will be either thestiffness values C11, C33, C44, C66, and C13 or the compliancevalues S11, S33, S44, S66, and S13 that in Voigt’s (1928) vectorand matrix notation allow the generalized Hooke’s law relat-ing stress σ = [σ 11 σ 22 σ 33 σ 23 σ 13 σ 12] [σ 1 σ 2 σ 3 σ 4 σ 5 σ 6]to strain ε = [ε11 ε22 ε33 ε23 ε13 ε12] [ε1 ε2 ε3 ε4 ε5 ε6]

σ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

σ 1

σ 2

σ 3

σ 4

σ 5

σ 6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

C11 C11 − 2C66 C13 0 0 0C11 − 2C66 C11 C13 0 0 0

C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

ε1

ε2

ε3

2ε4

2ε5

2ε6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

= Cε, (1)

or

∈ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∈1

∈2

∈3

2∈4

2∈5

2∈6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

S11 S11 − S66/2 S13 0 0 0S11 − S66/2 S11 S13 0 0 0

S13 S13 S33 0 0 00 0 0 S44 0 00 0 0 0 S44 00 0 0 0 0 S66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

σ 1

σ 2

σ 3

σ 4

σ 5

σ 6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

= Sσ , (2)

with

S = C−1. (3)

Sayers (2013), among others, provides more direct rela-tions between Cij and Sij.

The stiffnesses of equation (1) are preferred when work-ing with dynamic wave propagation and are most often seenby geophysicists. In contrast, the compliances of equation (2)are more conveniently obtained by measuring strains inducedby applied stresses, and as such, the compliance matrix S interms of engineering moduli

S =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1/E1 −ν21/E1 −ν31/E3 0 0 0−ν12/E1 1/E1 −ν31/E3 0 0 0−ν13/E1 −ν13/E1 1/E3 0 0 0

0 0 0 1/μ31 0 00 0 0 0 1/μ13 00 0 0 0 0 1/μ12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4)

with the various Young’s Ei and shear μij moduli and the Pois-son’s ratios ν ij described in Fig. 2. Examination of equation (4)in light of the symmetry of S together with the requirementthat there be at most five independent moduli demands ν21 =ν12, ν13/ν31 = E1/E3, and μ13 = μ31. Melendez Martinez andSchmitt (2015) review a number of the relationships betweenequations (1), (2), and (4) useful for converting between stiff-nesses, compliances, and engineering moduli; we include thesein Appendix A for the sake of completeness.

In this study, we obtain a full set of dynamic elastic stiff-nesses by measuring the wave speeds in a number of strategicdirections with respect to the material’s axis of symmetry, asshown in Fig. 3; the linkages between the six observed speedsand the five stiffness values required to define the transversely



2

1 3

1

3

3

1

1

1

1

2

(a)

(c) (d)

(e)

(b)

x2

x1

x1

x3

x1

x3

x1

x1

x3

x1

x2

Figure 2 Illustration of the deformations associated withthe definitions of the engineering moduli of the trans-versely isotropic block in Fig. 1. Applied stresses repre-sented as blue arrowheads applied to the surface of thematerial. Orange edges represent surfaces that cannot bemoved during the experiment. Original and final shapesrepresented by light and dark grey zones, respectively,and the changes in the lengths of appropriately orientedfibres in the material are the differences between the reddashed lines and the black arrows that would define thestrains. Moduli in the isotropic x1–x2 plane with (a) E1

= –σ 1/ɛ1 and ν12 = –ɛ2/ɛ1 from application of a uniaxialcompression σ 1 and (b) μ12 = σ 6/ɛ6 from application ofshear stress σ 6. Moduli associated with the vertical planeinclude (c) E3 = –σ 3/ɛ3 and ν31 = –ɛ1/ɛ3 from applicationof uniaxial compression σ 3, (d) μ13 = σ 5/ɛ5 from the ap-plication of shear stress σ 5, and (e) ν13 = –ɛ3/ɛ1 obtainedduring the same application of σ 1 as in panel (a).

isotropic material have been described many times in the lit-erature (e.g., Auld (1973) and Sarout et al. (2007)), and weprovide only the final relationships that are used later to cal-culate the stiffness values:

C11 = ρVP2 (90) . (5)

C33 = ρVP2 (0) . (6)

C44 = ρVSH2 (0) . (7)

C66 = ρVSH2 (90) . (8)

C13 = −C44 + 12

[(4ρVP

2 (45) − C11 − C33 − 2C44

)2

−(C11 − C33)2] 1

2. (9)

In the above equations, subscripts P, SV, and SH indi-cate the polarization of the wave to be longitudinal, trans-verse within any plane containing the X3-axis, and trans-verse within the X1–X2 plane. The propagation direction isdenoted by the angle q, as indicated in Fig. 1. Examination ofequations (5)–(9) shows that these elastic constants can bedetermined with relative ease and accuracy through measure-ments of wave speeds along the principal axes of the medium.Estimation of C13, however, is not as straightforward dueto the complexity of the equation yielding large uncertaintiesthrough the propagation of errors.

Alternative expressions for C13 can be derived, whichinclude the 45° quasi-shear velocity in addition to the quasi-compressional (Hemsing 2007)

C13 = −C44

+ 12

√4ρ2

((VP

2 (45) − VSV2 (45)

)2 − (C11 − C33)2, (10)



X1

X3

VP(0°)

VSH(0°) VSv (45°)

VSH (0°)

VP(45°)

VP (90°)

θ

Figure 3 View of hypothetical wave speed surfaces for the longitudi-nal polarization P in red, the horizontal polarization SH (i.e., parallelto the x1–x2 plane) in purple, and the vertical polarization SV (i.e.,parallel to the x1–x3 plane) in blue. The coloured dots represent thesix wave speeds measured here in directions along the vertical (θ =0°), the horizontal (θ = 90°), and obliquely (θ = 45°). The light-filledellipse represents the elliptical SH wave surface, and the dark-filled el-lipse represents the special case of elliptical anisotropy for the P-wavesurface when Thomsen’s (1986) δ = ɛ.

or use VSV2(45) exclusively in the cases where (VP

2(45) maybe unavailable

C13 = −C44

+12

√(C11 + C33 + 2C44 − 4ρVSV

2 (45))2 − (

C11 − C33

)2.

(11)

as will be necessary later in one of our analyses due to trans-ducer failure.

Elastic moduli can be either estimated from directly mea-suring the deformation (static) or calculated using the phasevelocities and bulk density of the material (dynamic). Histor-ically, studies have shown discrepancies between static anddynamic moduli in materials of TI symmetry (e.g., Hofman(2006); King (1969); Sone and Zoback (2013); Cheng andJohnston (1981); and Mashinsky (2003)) with the dynamicmoduli almost always exceeding the static significantly al-though this may not always be the case (Ciccotti andMulargia 2004). This difference is problematic in that it is notclear how one should employ seismic or sonic log observationsto supply quantitative moduli for engineering deformationcalculations. Here we compare simultaneous strain (static) toultrasonic (dynamic) moduli under a hydrostatic confiningpressure.

The linear compressibility λi (Brace 1965) or, conversely,the linear stiffness Kli (Ruiz Pena 1998; Melendez Martinez2014) describes the relationship between the linear strains

εi induced parallel to direction i subject to a change in thehydrostatic stress dp

λi = 1Kli

= −dεi

dp. (12)

It is much easier to apply a hydrostatic pressure to a rocksample than a full triaxial state of stress, and because of this,many of the earliest rock elasticity and, indeed, anisotropymeasurements are used λι (e.g., Zisman (1933)). Of course,no information on the static shear moduli can be obtained bymeasuring strains under hydrostatic compression.

The measurements in this study are made under hydro-static stresses, and we take advantage of this by measuring,using strain gauges, the strains ε1 and ε3 perpendicular, tothe isotropic bedding plane of the sample. These strain re-sults are quasi-statically measured and can be compared withthe stiffnesses obtained from the ultrasonic wave speeds via(Melendez Martinez and Schmitt 2015)

Kl1 = Kl2 = 2C33 (C11 − C66) − 2C132

C33 − C13. (13)

and

Kl3 = C33 (C11 − C66) − C132

C11 − C66 − C13. (14)

Equations (13) and (14) provide a means to directly comparethe dynamic ultrasonic to the static strain measurements.

SAMPLE CHARACTERIZATION

The Duvernay Formation samples are from depths between2871.60 m and 2882.96 m in borehole 02-17-043-04 W5 inthe Alberta Township System coordinate system (52.670°N,114.470°W). The Duvernay Formation is a depositional unitwithin the upper Devonian Woodbend Group and is consid-ered to be the source rock for many conventional hydrocarbonaccumulations such as the prolific Leduc, Nisku, and SwanHills reservoirs. The deposition of the Duvernay Formation ischaracterized by extensive basin deposits resulting from trans-gression of the Devonian sea and is laterally equivalent to theLeduc reefs. The formation is divided between the East andWest Shale Basins where it is overlain by the Ireton Formationand overlies the upper Majeau Lake Formation except in thesouthern end where it overlies the Cooking Lake Formation.

The East Shale Basin lithology is dominated by organic-rich lime mudstone and grades laterally into shallow-watercarbonate equivalents of the Grosmont shelf complex. Inthe West Shale Basin, the Duvernay Formation thins awayfrom the reef complexes and is dominated by characteristic



radioactive shales seen in the natural gamma ray logs (Fig. 4).As a whole, the formation tends to thicken from east to westwith the same trend in thermal maturity covering a rangefrom immature in the east to overmature in the west. Totalorganic carbon (TOC) ranges of 0.1 wt.%–11.1 wt.% havebeen reported based on samples obtained from 50 wells andfound that samples from the East Shale Basin contained thegreatest amount of TOC. Overall, the formation is estimatedto host approximately 443 Tcf of natural gas and 61.7 bbl ofoil (Rokosh et al. 2012).

The four calcareous mudstone samples used in this studywere taken from the same well at similar but different depths.X-ray powder diffraction analysis was conducted using aRigaku Ultima IV diffractometer to qualitatively identify themajor constituent minerals. TOC is determined through adry combustion method using a Shimadzu TOC-V CHS/CSNanalyser where any inorganic carbon is first removed by acidi-fying the sample with a 6% HCl solution. These characteristicsare displayed in Table 1. As all samples are from the same welland at similar depths, TOC values are expectedly very simi-lar as well, with samples 7–13 having a slightly lower value.Mineral modes were not determined, but a typical Duver-nay sample (Anderson et al. 2010) would principally consistof 30% quartz and feldspar, 50% carbonates, and 20%phyllites (primarily muscovite and illite). There were no de-tectable montmorillonite series clays in these samples.

Petrophysical properties (Table 2) of the samples weredetermined through mercury intrusion Porosimetry using aMicromeritics AutoPore IV capable of resolving pores of di-ameters between 360 μm and 0.003 μm. The porosities sodetermined are relatively low falling in the range from 2.5%to 3.8%.

The microscopic structure and porosity of such rockshas received a great deal of attention recently with a num-ber of workers using a wide variety of techniques (Andersonet al. 2010; Wust, Nassichuk, and Bustin 2013; Clarkson et al.

2013; Sondergeld et al. 2010), including scanning electron mi-croscopy (SEM). SEM images were also acquired here fromsample 5-13 but more with a view to studying the rocks fabric,and as such, images were made at different orientations. Bed-ding planes are readily apparent at low magnification (Fig. 5a),whereas the preferential orientations of illite subparallel to thebedding are seen at greater magnification (Fig. 5b). This ori-entation persists when seen from a direction perpendicularthe bedding plane where the material appears as sheet-likesurfaces (Fig. 5c). These images appear to suggest that bothbedding layers and mineralogic orientations influence the ma-terial’s anisotropy.

010

020

030

0

2600

2650

2700

2750

2800

Logging Depth (m

)

2850

2900

2950

2600

2650

2700

2750

Logg

ing

Dep

th (m

)

2800

2850

2900

2950

0 4.5

5 10 N NE S W0

9.0 13.5

Win

terb

urn

Ireto

nD

uver

nay

MB

VR

H

Natural Gamma (API)

Photelectric Z (barns) Fast Shear Azimuth

Slowness Anisotropy %

Figure 4 Geology and logs from 02-17-043-04 W5. Generalizedstratigraphy delineated by colour includes the Winterburn Group(dolomites and anhydrite), the Ireton Formation (calcareous and grey-green shales), the target Duvernay Formation (bituminous and cal-careous shales), the Majeau Lake Formation (M) (shale), and theBeaverhill Lake Group (limestone and shale). The left panel containsthe standard natural gamma radioactivity log and the photoelectric ef-fect log. The right panel shows two attributes derived from the dipolesonic log of the direction of the fast shear wave and the “slowness-based anisotropy”.



Table 1 Sample composition

X-ray Diffraction Analysis Dry Combustion

TOC Kerogen SampleSample Quartz Calcite Muscovite Pyrite Kaolinite Ankerite Clinochlore Orthoclase Epidote (wt.%) (wt.%)1 Depth (m)

2-13 3.86 4.63 2871.655-13 3.74 4.49 2875.667-13 2.64 3.17 2878.0510-13 3.83 4.6 2882.88

1. Kerogen weight fraction estimated from TOC values using conversion factor of 1.2 as provided by Tissot and Welte (1978)

Table 2 Petrophysical properties

Sample Bulk density ρb (kg/m3) Skeletal density ρs (kg/m3) Porosity (%) Avg Pore Diameter (μm)

2-13 2464 2556 3.61 0.0425-13 2427 2523 3.80 0.0227-13 2482 2579 3.78 0.02910-13 2476 2541 2.56 0.167

EXPERIMENTAL SE T UP A N DINSTRUMENT A T I ON

In this contribution, the degree of anisotropy and dynamicmoduli are measured using the ultrasonic pulse transmissionmethod with piezoelectric ceramic elements. This method hasbeen extensively used to estimate the velocities in geologicmaterial at ultrasonic frequencies.

Core samples (Fig. 6a) were trimmed into a prism-likegeometry where the opposing faces allow for wave propa-gation along the x1- and x3-axes well as at an off-axis an-gle 45° from the bedding plane. This configuration, adaptedfrom Wong et al. (2008), allows us to reduce the heterogene-ity issues of taking measurements of separate samples takenfrom the same core; the propagation paths of the differentwaves all intersect. Each opposing face on the core configu-ration is grounded parallel to better than 0.1 mm and pol-ished with fine sandpaper to ensure proper adhesion of thepiezoelectric elements and strain gauges. For the same rea-son, samples were vacuum dried at 65 °C as not to compro-mise the integrity of the bonding agents used to attach thecomponents.

Lead zirconate titanate (PZT) elements with a resonancefrequency of 1 MHz are glued directly on to the surface of eachface and foil strain gauges oriented parallel and perpendicularto the bedding (Fig. 6b). The sample is then coated with a non-conductive urethane compound in order to test the sampleunder jacketed conditions.

Experimental setup for the dynamic measurements con-sists of a pulser/receiver system, a digital oscilloscope, a dig-itizer, and a pressure vessel (Fig. 7). The pulser repeatedlyapplies a fast-rise time voltage step to drive the transmittingceramic. The subsequent pulse propagates through the mate-rial as an elastic wave and is converted to a voltage by thereceiver transducer. The signal then passes through an 8-bitdigitizer before being recorded by the digital oscilloscope ata rate of 10 ns/sample. A LabVIEW R© program displays andstacks each signal to produce an averaged waveform in orderto reduce electrical and random noise. Detailed example ofthree waveforms (Fig. 8) highlights the initial RF noise arisingfrom the activation pulse but with a clear and strong (2-Vpeak-to-peak) waveform transmitted in all cases. There is noambiguity in picking the first amplitude extremum in theseimages. These waveforms also point to an additional advan-tage of the geometry that consists of relatively large aperturetransmitters and receivers directly facing one another. First,the contamination of the signal by “parasitic” arrivals is di-minished because directionality effects are minimized. Thesecan be quite problematic when transmitters and receivers donot face one another such that both elements’ directionalitiesare not optimally aligned. Indeed, Mah and Schmitt (2001a, b)exploited the directionality effects to obtain profiles of shearwave modes by using the stronger shear modes radiated atoblique angles by small-aperture P-mode transducers as partof their inversion for the phase velocities in an orthorhom-bic solid. These parasitic modes cannot be eliminated but are



(a)

(b)

(c)

x1

x3

x1

x3

x1

x2

Figure 5 Scanning electronic microscope images of Sample 5-13 with(a) 23 times magnification along x1 highlighting the bedding planes,(b) 2500 times magnification from the same view showing layering ofillite grains subparallel to bedding planes, and (c) 1000 times magni-fication looking down on to bedding planes along x3.

greatly reduced by the geometry used here. Techniques thatemploy smaller aperture transducers that are not aligned withone another will have waveforms that will be influenced by di-rectionality effects that can cause problems for interpretationof the different shear wave modes, particularly at oblique an-gles where one expects significant generation of longitudinalwaves.

Calibration of the PZT transducers is necessary as there isa slight delay between the excitation and the waveform’s first

(a)

(b)

Figure 6 (a) Section of black calcareous shale core from 02-17-043-04 W5 prior to machining. (b) Sample 5-13 before application ofurethane compound complete with piezoelectric elements, electrodes,and strain gauges (not shown).

amplitude maximum. For the calibration, PZT crystals (com-pressional and shear-mode) are mounted on to the flat endof a set of Aluminium 6061-T6 cylinders of varying lengths.Similar to the sample measurements, waveforms are averagedover 300 instances and the transit time is picked from the firstextremum of the averaged waveform. These values are thenplotted as function of the cylinder length and fitted with alinear regression where the slope is equal to the slowness ofthe propagating ultrasonic wave that gives compressional andshear speeds of 6211 m/s and 3122 m/s, respectively, whichare in good agreement with reported values (Christman et al.

1971). The non-zero intercept of the equation is equal to theexcitation delay of each transducer plus the required time shift



Pressure System

Pulser / Receiver

Oscilloscope

V

X3

X1

ε(0)

ε(90)

P(0) S(0) P(45) SV

(45)

P(90)

SH(90)

a)

b)

Figure 7 Simplified diagram of (a) Arrangement of ultrasonic piezo-electric ceramic elements on polyhedral sample colour coded for com-parison with Fig. 10 and of the two strain gauges colour coded to cor-relate with Fig. 10. (b) The final experimental configuration includingpulser/oscilloscope and Wheatstone bridge arrangements.

to the first arrival, and these time shift values are 1.019 μsand 0.585 μs for the P- and S-wave corrections, respectively.

Samples are subjected to hydrostatic confining pressuresup to 90 MPa where, prior to taking measurements, sam-ples are held at a constant pressure for 20 minutes to allowfor time-dependent changes due to the new stress conditions.These measurements are initially taken at small pressure incre-ments to target the expected non-linear elasticity of the rocksdue to compliant deflects such as cracks, fractures, and graincontacts. As confining pressure is increased, these cracks andfractures close, resulting in a more linearly elastic response.At this stage, measurements are taken at larger pressureintervals.

Static strains are measured through the use of 350-Ω foilstrain gauges attached directly to the sample. The changes inresistance due to deformation of the material are generallyorders of magnitudes smaller than the changes in nominal

resistance of the strain gauge and therefore must be measuredwith a Wheatstone bridge circuit constructed of precision350-Ω resistors. DC input voltages of 2.5 V activate the bridge,and the output voltages are measured and recorded using adigital multi-meter facilitated by the same LabVIEW R© pro-gram. Details of the conversion of the observed unbalancedbridge voltages to strains may be found in Melendez Martinez(2014).

R E S U L T S A N D D I S C U S S I O N

Wave speeds and dynamic moduli

Four different suites of measurements (hereafter, designatedas 2-13, 5-13, 7-13 #1, and 7-13 #2) were made on three ofthe core samples. The fourth core, sample 10-13, was unsuit-able for testing as there was insufficient material to cut intothe multi-faced polyhedron, but it still provided petrophysicalinformation that was useful in correcting depths between thelogs and the core.

For the sake of completeness, we provide the full suiteof waveforms for sample 2-13 (Fig. 9) as an example to ob-serve the shift in the pulse transit times as a function of theconfining pressure and quality check for each waveform. Thefirst extremum of each trace appears to be relatively clean andthus puts confidence in our transit time determination. A hys-teresis effect is also observed in this figure, but it will be moreapparent in the plots of the velocities, elastic constants, andmoduli.

In general, the wave speeds observed between the foursamples were relatively uniform. It is notable that the speedsmeasured on 7-13 #1 and 7-13 #2 sampled adjacently to eachother in the same piece of core differed at all pressures byless than 100 m/s, indicating the relative homogeneity of thematerial. Here, only the speeds from samples 5-13 and 7-13#2 are plotted (as shown in Fig. 10) as, within the set of four,these two differ the most from one another. All of the datafor these samples are provided in the electronic supplement.The 45° P-wave transducer on sample 5-13 failed and pro-vided no data. Subsequently, elastic constants that originallyrequired VP (45) will instead use the off-axis SV-wave velocityequation (11).

From Fig. 10, we see that, except for VSH(90°), velocitiesincrease with confining pressure and exhibit a hysteresis effectbetween the pressurization and depressurization cycles. Pres-sure sensitivity is most pronounced for those wave speeds mea-sured at 0°. In contrast, VP(90°) modestly increases, whereasVSH(90°) remains nearly constant.



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Figure 8 Examples of stacked P-wave signal measured in the x3 direction, SH polarization signal measured in the x1 direction, and SVpolarization signal measured at the θ = 45° direction at 50 MPa of confining pressure and room temperature. Arrow shows time pick at firstextremum. The higher frequency oscillations from time t = 0 are not signal but are an induction effect in the receiver’s circuit arising from therapid change in voltage that activates the transmitter.

It is useful to consider these observations in terms ofthe wave polarization. The observation that VSH(90°) remainsnearly constant during the entire pressurization is particularlyinteresting. It suggests that the bedding plane polarized wavesare not influenced by small-aspect-ratio, crack-like, pressure-sensitive porosity. At the same time, VP(90°) does show weakpressure dependence, but this should not be surprising as itspolarization and the resulting wave-induced strains will be in-fluenced by both the elastic properties both in and perpendicu-lar to the bedding plane, and its weak pressure sensitivity car-ries over from the out-of-bedding-plane pressure sensitivities.

Initial observations show that the velocities are consistentwith what one would expect from a material with verticaltransverse isotropy (VTI) with VP (0) < VP (45) < VP (90)and VSH(0) < VSV(45) < VSH(90). It is interesting to ob-serve, however, that VSH(90°) exceeds VP(0°) at pressures be-low 30 MPa in both samples 7-13 #1 and 7-13 #2.

The elastic stiffnesses are calculated directly from the bulkdensity and the phase velocities of the samples through equa-tions (5)–(11) and plotted as a function of confining pressure(Fig. 11). As the measurements are made under dry condi-tions, no additional corrections for fluid effects were neces-sary, and the moduli reported are for the dry frame. From theinitial observations, the elastic constants are consistent withthe thermodynamic constraints (Auld 1973; Fung 1965) im-posed on media of VTI symmetry stemming from the positivedefinite nature of the strain energy function (Postma 1955).Also included are Young’s moduli E1 and E3 subsequentlydetermined from the stiffnesses according to equations (A1)and (A2). E1 substantially exceeds E3 in all cases. The cor-responding dynamic Poisson’s ratios calculated from equa-tions (A5)–(A7) are given in Fig. 12. Generally and except atthe lowest pressures ν12 ν13, both are significantly smallerthan ν31



Figure 9 Complete set of observed waveforms for sample 2-13 over range of pressures from 0 MPa to 90 MPa and back.

C11, C33, C44, C66 > 0. (15)

C11, C33 > C44. (16)

C11, C33 > C66. (17)

C33 (C11 − C66) > C132. (18)

Thomsen (1986) ε, δ, and γ parameters have become im-portant quantitative measures of the anisotropy of a material.These are calculated using equations (A8)–(A10) and shown,including uncertainties, in Fig. 13. Again, 5-13 and 7-13 #2were chosen as they bound the range of properties seen in thefour different samples measured. These results make it clear

that these rocks are highly anisotropic with 0.35 < ε < 0.5and 0.3 < γ < 0.46; these correspond to simple anisotropiesat the highest confining pressures of 90 MPa. Prior to employ-ing these values for seismic analysis, it would be prudent toassess whether the simplifications remain valid in the analysisof seismic observations within this formation.

Experimental uncertainties were estimated for all vari-ables but only shown for the Thomsen parameters as all othersare approximately equal to or smaller than the markers used inthe figures. A general propagation of error was used to calcu-late the variance in each parameter with consideration of theuncertainties in the pulse transmission offset, parallelism ofopposing faces, and measurements of physical properties suchas bulk density. Limitations in the data acquisition equipment



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VP(0°) VP(45°) VP(90°) VSH(0°) VSV(45°) VSH(90°)Figure 10 Phase velocity for samples 2-13and 7-13 #2 plotted as a function of hy-drostatic confining pressure. Uncertaintiesare approximately the size of the markerswith arrows showing the evolution ofthe pressurization and depressurizationmeasurements.

were also considered in the estimates of error although picksin the first arrival were taken to be absolute as the waveformsprovided clear distinct points of the first extremum.

Stress–strain relations and static linear stiffnesses

As noted, linear strains were measured both parallel (ε1) andperpendicular (ε3) to the bedding plane. In all of these sam-ples, the magnitudes of ε3 > ε1 by factors of 3–6 again indicatethat the bedding planes are stiff. Moreover, ε1 increases al-most linearly with pressure and displays little hysteresis upondepressurization (Fig. 14). The grey-shaded area at the topcorner of the panels delimits the corresponding linear strainthat would be expected for a nonporous isotropic polycrys-tal of quartz with an adiabatic bulk modulus Ks = 37.8 GPa(Schmitt 2015), and although quartz is among the most com-pressible minerals, the fact that the observed values of ε1 areclose to this do indicate that the bedding planes of these rocksare exceptionally stiff. In contrast, the bedding perpendicularstrain ε3 behaves nonlinearly with pressure. Sample 7-13 #2has a high strain gradient at low pressures, and its trend be-comes increasingly linear at higher pressures, indicating theprogressive closure of the crack-like aligned porosity. Saroutand Gueguen (2008b) developed models that are able to in-corporate such aligned porosity into an otherwise intrinsicallyanisotropic medium.

The static linear stiffness of the material is determined byfinding the local slope to the strain–pressure curve in Fig. 14.The “tangent” linear stiffnesses were found by fitting an alge-braic curve that appropriately fits the observed strain–pressurerelationship and then taking its derivative with respect topressure; the inverse of this slope gives the linear stiffnessesKli as described by equation (11) above (see Melendez Mar-tinez and Schmitt (2015)). These are then compared directlywith their dynamic counterparts calculated using equations(13) and (14). It is interesting to note that, in all of the foursamples measured, the dynamic and static linear stiffnessesin the bedding plane are nearly equal despite the number ofvariables necessary to calculate the former. In contrast, thosemeasured perpendicular to the bedding plane diverge. Melen-dez Martinez and Schmitt (2015) recently found similar be-haviour in a series of Cretaceous calcareous mudstones fromthe Western Canadian Sedimentary Basin but those rocks areless anisotropic.

The directionally dependent differences between the dy-namic and static linear stiffnesses may be important. First,perpendicular to the bedding, the two values differ by morethan 50% even at the greatest confining pressure. In contrast,along the bedding plane, the two match one another well. Thissuggests that the degree of difference between the static andthe dynamic moduli also varies with direction.




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C11 C12 C13 C33 C44 / μ23 C66 E1 E3Figure 11 Dynamic elastic stiffnesses andYoung’s moduli calculated from the ob-served wave speeds as a function of hydro-static confining pressure. Arrows show theevolution of pressurization and depressur-ization measurements. Uncertainties are ap-proximately the size of the markers.

As noted earlier, proper static elastic moduli are neces-sary for many engineering calculations, but in the absence ofcore materials they must be estimated from geophysical logs(Asef and Najibi 2013). It has long been known that, at “low”confining pressures typically less than 200 MPa (see review inSimmons and Brace (1965)), the dynamic and static valuesdisagree with one another even in dry rocks (Fjær 2009), as isthe case here, where complications related to undrained versusdrained and to wave-induced fluid motions cannot act. Com-parisons on shales are rare. In an earlier study, Jizba (Jizbaand Nur 1990; Jizba 1991) compared static to ultrasonic dy-namic measurements on a range of siliclatic rocks, findingthat the ratios of the dynamic KD to the static KS bulk mod-uli were great as 5. In the clay-rich rock, this ratio tended tobe independent of the confining pressure. Holt et al. (2012)reinterpreted earlier measurements of static Young’s and dy-namic P-wave moduli (i.e., ρ(VP)2) that upon cross-plottingsuggest that, with more data, useful empirical relationshipscould be developed. Studies that have considered static to dy-namic anisotropies explicitly, however, are even rarer. Wonget al. (2008) compared static, ultrasonic, and log values fora much more friable and still-wet Cretaceous shale but toconfining pressures of only 11 MPa, and in this rock, thedynamic/static Young’s moduli ratios differed by factors of5–6. Although strictly they did not make measurements of

the static moduli, Suarez-Rivera et al. (2001) briefly reportedon a series of measurements on two shales from quasi-staticto ultrasonic frequencies and reported that wave speeds overthis range would differ by 45%. Hofman (2006) was able toobtain full sets of the elastic stiffnesses at “low” frequenciesfrom 3 Hz to over 1 kHz and compared these to ultrasonicmeasurements; one of his shales showed significant dispersion,whereas the second did not. Ruiz Pena (1998) also made di-rect comparisons of linear stiffnesses on a suite of dolomitizedshales from Venezuela with the dynamic easily exceeding thestatic in all cases. More recently, Mbia et al. (2014) made ul-trasonic and strain measurements on oriented cylinders takenfrom Jurassic shales cored from the Norwegian–Danish Basinover depths from 1000 m to 4600 m. Their samples were sub-ject to uniaxial loading to 25 MPa while maintaining a con-stant ratio of 0.38 between the uniaxial stress and the biaxialside pressures. Their static compressibilities always exceededthose determined dynamically except early in the depressur-ization stage.

Comparison between these many studies is complicatedby the fact that they have all been made on a wide variety of“shales”, i.e., that some are under dry and others are in fullsaturation, and at a wide range of triaxial states of stress. Theone common element in all is that, generally, dynamic mod-uli exceed static moduli regardless of direction. In our four




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ν12 ν31 ν13 νlog

Figure 12 Dynamic Poisson’s ratios as a func-tion of hydrostatic confining pressure. Arrowsshow the evolution of pressurization and de-pressurization measurements. Uncertainty isapproximately the size of the markers.

Duvernay samples, as well as the less anisotropic ones mea-sured by Melendez Martinez and Schmitt (2015), the ratioR = Kli(dynamic)/Kli(static) critically depends on the direc-tion. This R is always > 1.5 in the direction perpendicularto the bedding planes. This contrasts with the other observa-tions in sandstones and shales in which the value R 1 onlyupon the initial depressurization of the material (e.g., Mbiaet al. (2014)). The recent study of Holt et al. (2015a) too is ofinterest in this context. They measured the full set of elasticconstants on Mancos (8% porosity, 20%–25% clay), Pierre(25% porosity, 40%–60% clay), and Field (10% porosity,50%–55% clay) shale samples using ultrasonic pulse trans-mission, strain, and low-frequency forced oscillation meth-ods. The finding for these rocks is that the ultrasonic dynamicmoduli exceeded the static moduli in all cases but that thelow-frequency moduli reasonably matched the static moduli.In contrast, within this current study, R is always 1 withinthe bedding plane regardless of whether the sample is in apressurization or a depressurization cycle. One needs to takecare with a direct comparison between these two studies. Theirshales contain substantially more clays that the Duvernay For-mation material. The Duvernay clays too are non-swelling

illite and kaolinite. Holt and coworkers do not report on thekinds of clay, but Pierre shale, for example, can contain con-siderable proportions of montmorillonite (Schultz et al. 1980)and it is possible that the dispersive effects are greater throughsuch materials. Certainly, Wong et al. (2008) report large di-rectional variations in attenuation (which is related to disper-sion) in earlier studies on wet samples of Colorado shale fromAlberta. This anisotropy to R will further complicate attemptsto predict the static moduli from geophysical seismic or logwave speeds.

One final issue rests here with the fact that this studyis carried out under conditions of hydrostatic pressure whilemost studies are carried out subject to triaxial states of stress.While the mean stress within the sample must certainly re-main as a uniform pressure, the thin-section and SEM imagesdo show that the material’s structure is heterogeneous at mi-croscopic scales. As such, we expect that stress concentrationswill arise in the vicinity of such structural irregularities. Thismay partly explain the hysteresis in strains and wave speedsupon depressurization that has long been known. What isanomalous in this study, however, is the lack of significanthysteresis for those deformations of bedding parallel strain



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Figure 13 Thomsen (1986) anisotropy parameters ε, δ, andγ calculated from the dynamic elastic stiffnesses.


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Figure 14 Strains measured perpendicularand parallel to bedding as a function ofhydrostatic confining pressure. Uncertaintiesare smaller than the marker. Arrows show theevolution of the pressurization and depressur-ization measurements. The grey-shaded areadelimits the linear strain that would be ex-pected on a nonporous isotropic quartz poly-crystal for purposes of comparison.

and SH(90°) wave speeds that indicate a lack of crack-likeporosity oriented obliquely to the bedding. This suggests thatthe easily compressible, small-aspect-ratio porosity in this ma-terial is dominantly aligned with the horizontal beds. Such

preferential structures have been anticipated in the modellingof Sarout and Gueguen (2008b) for shales and were necessaryto explain the seismic field observations over a thick biotiteschist by Schijns et al. (2012).



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Kl1(dyn) Kl1(stat) Kl3(dyn) Kl3(stat)Figure 15 Direct comparison of the strain-derived static (stat) and wave speed-derived dy-namic (dyn) linear stiffnesses plotted as a func-tion of hydrostatic confining pressures. Arrowsshow the evolution of the pressurization and de-pressurization measurements.

Core to log comparison

An extensive set of geophysical logs was acquired by the op-erator immediately after drilling of this well, and we are ableto obtain these logs through publicly accessible databases; as

provided directly from the field observations, we do not haveaccess to any subsequently processed log information. Despitethis, some borehole data are available that can be comparedwith the laboratory measurements (Fig. 16). Note that thecore depths in Table 1 were shifted downwards 5.55 m toaccount for incomplete core recovery and differences betweenthe driller’s and geophysical log depths; once this shift waseffected, the core to log densities agreed well except for sam-ple 5-13, which is noticeably less dense. We do not know thereason for this discrepancy, but it may simply be due to het-erogeneity within the formation. Again, we do not know if thedensity log was in any way corrected for mineralogic contentor potential mudcake nor do we have access to the densityerror log.

The only compressional log available is DT4P, i.e., thecompressional wave slowness obtained from the monopolesource, and this is the line in the right panel in Fig. 16 whereit is directly compared with the values of the wave speed inthe different directions. These ultrasonic measurements ap-pear as horizontal lines that represent the full range of valuesmeasured over the confining pressure range from 0 MPa to

90 MPa. The results from sample 7-13 #1 are not shown asthey are largely similar to those for 7-13 #2. Except for sample5-13, laboratory VP(90°) intersects the log-derived curve. Incontrast, both VP(45°) and VP(90°) substantially exceed logVP. This is to be expected because, assuming that the boreholeremains vertical, the sonic logs are enforced by geometry toprovide VP perpendicular to the bedding plane.

There are more shear wave speed measurements availablefrom the logs and three are given in Fig. 16 and include VS de-rived from the transit times of the: (i) monopole source (blackline) and (ii) DT1 and DT2, i.e., the two measurements fromthe upper to the lower dipole transducers and vice versa; theseare delimited by the thickness of the green line. Again, exceptfor sample 5-13, the range of laboratory observed VS(0°) in-tersects these log-derived VS curves. As with VPs, this is alsoexpected as the polarizations provided by the vertically prop-agating directly arriving shear waves, regardless of whether amonopole or a dipole source is used, will be parallel to thebedding plane as is the case for VS(0°). The range of observedVS(45°) also intersects the log VS but only at the lowest confin-ing pressures; at higher confining pressures, the values signif-icantly differ. VS(90°) is always much larger than the log VS.

This matching is encouraging, but some care should betaken with the interpretation particularly with regard to in situ

conditions (fluid saturation state, in situ stress, pore pressure,



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Figure 16 Comparison of log and laboratory wave speeds. Left panel: Red-outlined grey area shows standard γ − γ density log with corre-sponding (shifted) laboratory determinations of the dry bulk (open circle) and grain (filled black circle) densities. The green zone bounds thesimultaneous S-wave measurements provided by the lower and upper dipoles. The black line is the S-wave speed given from the t measurementof the shear wave in from the monopole source. The blue, gold, and purple lines illustrate the range of values of VS(45), VSV(45), and VSH(90)obtained during pressurization. Right panel: The black line is the P-wave speed given from the t measurement of the compressional wavefrom the monopole source. The green, magenta, and red lines illustrate the range of values of VP (45), VP (0), and VP (90) obtained duringpressurization. Line thicknesses are varied only to allow for comparison and have no meaning in themselves.

and temperature), sample heterogeneity and preparation, andthe differences between the monopole and the dipole sonic logspeeds. As such, it is a useful exercise to determine the con-fining pressure at which the laboratory and log wave speedswould be equal, the idea being that, barring dispersive ef-fects, the wave laboratory wave speeds could indicate thein situ conditions. Taken at face value, the results of this com-parison (Table 3) would predict in situ stresses of no morethan 40 MPa with some of the predicted values approachingzero. The estimated overburden effective stress ranges from21 MPa to 27 MPa. While this is close to the depressurizationVP values for samples 7-13 #1 and 7-13 #2, the comparison isrelatively good. However, it is not known to what degree the

differences may arise from dispersion between the ultrasonicand the sonic log frequencies, and this will require additionalstudy (Suarez-Rivera et al. 2001). The wave speeds measuredat the greatest pressure all exceed those observed from thesonic logs and this at least suggests that some care may needto be taken to account for effective pressure variations whenone attempts to determine velocity dispersion.

Implications of stress insensitivity in the bedding plane

It has been long known that the physical properties of mostrocks depend nonlinearly on confining stress with this be-haviour attributed to the progressive closure of compressible



Table 3 Laboratory confining pressure at which log and laboratorywave speeds match

Sample Pressure

Log VP Log VS Pressurization DepressurizationSample (m/s) (m/s) (MPa) (MPa)

2-13 3689 23 32309 5 NA

5-13 NA NA NANA NA NA

7-13 #1 3524 40 192192 8 0

7-13 #2 3524 40 202192 6 0

crack-like pores. A corollary manifestation of this nonlin-earity is that application of deviatoric stress states inducesanisotropy, a topic that has generated a great deal of interestand many literature works in the seismological communitybased on studies in the laboratory (e.g., Nur and Simmons(1969) and Gorbatsevich and Kovalevskiy (2015)), theoreti-cal analyses (e.g., Sayers and Kachanov (1995); Johnson andRasolofosaon (1996); and Collet et al. (2014)), and in thefield (e.g., Crampin (1981); Oda, Yamabe, and Kamemura(1986); Schijns et al. (2012); and Shelley et al. (2014)). Thiseffect is particularly interesting around a vertical boreholedrilled through a formation subject to differing maximum andminimum horizontal compressions SH and Sh, respectively, astheir concentration around the borehole results produces acyclical pattern of opening and closing of microcracks aroundthe borehole, as illustrated in Fig. 17. These stress concen-trations have been shown in the laboratory (Winkler andD’Angelo 2006; Winkler 1996; Fang et al. 2013; Hsu et al.

2011) and field (Balland and Renaud 2009) to induce con-siderable azimuthal variations in the compressional and shearwave speeds. These azimuthal variations are also referred toas an “anisotropy”, but while it is related to the formationanisotropy due to mineral, crack, and fracture alignment, lay-ering, or in situ stress, one must be aware that stress concen-trations contribute to these additional changes in the materialproperties around the borehole (Plona et al. 2002). Special-ized sonic logging tools that incorporate orthogonal dipolesources (e.g., Sinha, Kane, and Frignet (2000) and Tang andCheng (2004)) may be used to find the azimuths at which thefastest and the slowest shear waves propagate, yielding the SH

and Sh directions, respectively (Schmitt, Currie, and Zhang2012).

Normalized Mean Stress

SH SH

Sh

Sh

y

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Figure 17 Illustration of the effect of the concentration of the hori-zontal principal compressions SH and Sh on the opening and closingof vertical cracks around the borehole. After Fig. 9c of Schmitt et al.(2012) with permission granted from Elsevier.

Such changes, however, are predicated on the existenceof compliant crack-like porosity that can be closed by theapplied compressions. Here, the relevant laboratory obser-vations are that VSH(90°) is nearly insensitive to confiningpressure, that VP(90°) is only weakly so, and that ε1 linearlyincreases with pressure and has little hysteresis upon depres-surization. Melendez Martinez and Schmitt (2015) have sug-gested that this insensitivity to stress would reduce any stress-induced anisotropy. Returning to the logs in Fig. 4, the fastshear wave direction (blue line in the right track) is indetermi-nate and the slowness-based anisotropy (green line in the righttrack) is typically < 1% through the shale formations below2642 m. At face value, one could interpret this as indicating anearly uniform state of horizontal stress. However, abundantevidence in the region from the existence of drilling-inducedtensile fractures in borehole image logs and focal plane solu-tions indicating strike-slip indicates that SH cannot be equalto Sh in this area. The lack of stress sensitivity in the hori-zontal plane provides a possible explanation for the inabilityof the fast and slow shear waves to indicate the stress direc-tions. More generally, this may also provide a rationale forTang and Cheng (2004) observation that the fast shear waverdirections cannot be reliably found in shale formations.

CONCLUSIONS

All four samples of calcareous shale from the Duvernay For-mation are strongly anisotropic with both simple dynamicanisotropies AP and AS ranging between 25% and 35% at



the highest confining pressure of 90 MPa. The sensitivity ofthe samples to pressure is also highly dependent on direc-tion. Strains and wave speeds measured perpendicular to thebedding direction are nonlinearly dependent upon confiningpressure indicative of small-aperture crack-like pores. In con-trast, strain measured parallel to bedding increases linearlygoverned by stiffnesses similar to that expected for a non-porous isotropic quartz composite. Wave speeds in this direc-tion also show little pressure sensitivity. This also translatesinto an anisotropy of the ratio between the dynamic and thestatic moduli expressed here as linear stiffnesses. At the highestpressures, the dynamic linear stiffnesses perpendicular to bed-ding exceed the corresponding static stiffnesses significantly.However, somewhat unexpectedly, within the bedding plane,these nearly match.

Ideally, one would like to be able to predict the in situ

static moduli necessary for engineering purposes from proxymeasurements from sonic logs. We found that the speedsof the waves measured perpendicularly to the bedding planedo match the sonic log values particularly once the effectivestress is considered. Consequently, one could, in principle, usesome empirical relation between the observed dynamic and thestatic moduli to make a prediction. However, this predictionbecomes suspect because of the directional differences in thesensitivity of the moduli to stress. The lack of horizontal stressdependence has implications for the analysis of dipole soniclogs for stress directions; this may explain the degeneracy ofthe fast shear directions through these calcareous mudstoneformations. This suggests that care may need to be taken inthe interpretation of seismic azimuthal amplitude versus offsetdata for stress directions over such formations.

We have not attempted to model the pressure-dependentelastic behaviour. However, there are likely two primary gov-erning factors suggested from the scanning electronic micro-scope images. These show small aspect ratio “crack-like”pores primarily oriented parallel to the bedding planes. Thisis consistent with the nonlinear increases in strain and wavespeeds perpendicular to this direction. Second, the images alsoshows strong orientations of the well-bonded phyllitic miner-als into tabular sheets, and these features likely are able tosupport considerable horizontal load. As such, modelling ofthis behaviour is likely premature until a better understand-ing of both the character of the porosity and the preferentialalignment of the constituent minerals is available. That said,application of models such as those developed by Sarout andGueguen (2008bb) could reveal additional information on thestructure of the pore space in such rocks. Furthermore, ashas been done in most previous studies, we assume that our

measured values are phase (plane wave) speeds, and if this isnot completely, the case then the measurements made at 45°and hence the determined values of C13 may be suspect. Webelieve that this can only be answered properly by a full nu-merical simulation that will incorporate material propertiesand transducer apertures. However, the unexpected agree-ment in all of the cases between the dynamic and static linearstiffnesses suggests that any errors introduced by ignoring thebeam propagation are small.

These rocks are highly anisotropic, and they have com-plex stress-dependent behaviour. Its significance for reflectionand microseismic seismology, for stress concentration, and forthe risk of induced seismicity is currently being explored.

ACKNOWLEDGEMENTS

Funds for this work were obtained through the GeothermalStudies Section of the Helmholtz-Alberta Initiative.

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APPENDIX A

For a transversely isotropic medium, we can define expres-sions for the dynamic Young’s modulus E, bulk modulus K,shear modulus μ, and Poisson’s ratio ν as a function of theexperimentally determined effective elastic constants

E1 = 4C66

[C33 (C11 − C66) − C13

2]

C11C33 − C132 . (A1)

E3 = C33 (C11 − C66) − C132

C11 − C66. (A2)

K = C33 (C11 − C66) − C132

C33 + C11 − C66 − 2C13. (A3)

μ23 = C44. (A4)

ν12 = C12C33 − C132

C11C33 − C132 . (A5)

ν31 = C13

C11 + C12. (A6)

ν13 = C13 (C11 − C12)

C11C33 − C132 , (A7)

where νi j refers to the negative ratio of transverse strain ε j toaxial strain εi with a uniaxial stress applied as σ i .

Define three dimensionless parameters to describe theanisotropy of a transverse isotropy (TI) medium provided thatthe anisotropy is weak. These parameters are described by thefollowing equations:

ε = C11 − C33

2C33. (A8)

γ = C66 − C44

2C44. (A9)

δ = (C13 + C44)2 − (C33 − C44)2

2C33 (C33 − C44), (A10)

where ε and γ are expressions used to quantify the degree ofP- and S-wave anisotropies, respectively. Parameter δ can beseen as the anellipticity of the P-wavefront. The deviation ofthese parameters from zero characterizes the relative strengthof anisotropy in the material (Yi-Mou et al. 2014). We also de-fine the simpler and more traditional fractional anisotropies,



which are simply the ratio of the difference between the max-imum and minimum wave speeds with their average value, inwhich (Birch 1961) for the P-wave here will be:

AP = VP (90) − VP (0)VP (90) + VP (0)/2

. (A11)

and, for the SH-wave, is

AS = VSH (90) − VS (0)VSH (90) + VS (0)/2

. (A12)

Note that, for a TI medium, VSV(0°) = VSV(90°), andcalculating its anisotropy in a similar manner would be mis-leading as VSV(θ ) may show a great deal of variation.

APPENDIX B: QUA N T I T A T I V EESTIMATION OF ST R ESS

Here we use publicly available data to estimate in situ stressconditions. Borehole 02-17-043-04 W5 was first drilled ver-tically to 2977 m to provide the log information, and thencemented and re-drilled with a long horizontal leg at depthsof 2895 m. The vertical stress was estimated from integration of a den-

sity log formed as a composite of the data from the studywell with those nearby to range between 71.0 MPa and76.7 MPa.

Sh = 74.9 ± 5.9 MPa is constrained from the average ofthe reported instantaneous shut-in pressures from the tensuccessful hydraulic fracture stimulations. This includes the

correction for head pressures as described for the formationpore pressure.

The formation pore pressure is estimated from an 84-hourmonitor pressure build-up test conducted after completionof the hydraulic fracture stimulations with the final shut-

in casing pressure = 18.089 MPa measured at the surface.Using the last reported fluid density of 1090 kg/m3, weestimate the head pressure at the depth of the stimulationsto be 30.96 MPa, suggesting that the formation pressure is49.04 MPa. This formation is highly overpressured.

Taken together, these values suggest that the effective ver-tical stress for wave speeds is 22 MPa–27 MPa.

It is useful to compare this to a sonic log-derived value of theminimum horizontal stress Sh under the lateral constraintassumption that is often carried out in order to predict the“fracture gradient”. For an isotropic rock mass

Sh = ν

1 − νSV . (A13)

and if the isotropic Poisson’s ratio, as estimated from the soniclog VP and VS values, ranges from 0.2 to 0.25, Sh is estimatedto be only 25%–33% of SV. Applying the same conditions fora TI material where (Melendez Martinez and Schmitt 2015)

Sh = ν13

1 − ν12

SV, (A14)

will further reduce the estimated Sh magnitude. Such low val-ues contradict those more directly obtained above.


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