1. IntroductionGranitic rocks attract attention because of their elastic and mechanical properties. A large number of dedi-cated experimental studies has been initiated due to relevance of granites to development of nuclear waste storage, geothermal projects, rock mechanics, and earthquake mechanics. Quantification of elastic proper-ties of granites is important to determine crustal seismic velocities and stress orientation. Generally, it is as-sumed that granitic rocks are elastically isotropic. To determine whether rock material is isotropic or aniso-tropic ultrasonic methods can be applied (Babuška, 1968; Bajuk et al., 1967; Barruol and Kern. 1996; Kern & Wenk, 1990). Multidirectional (up to 132 independent directions) experiments are necessary in case of low-er symmetry, unknown orientation of anisotropy or simply to diminish the effect of rock heterogeneity (e.g., Martínková et al., 2000; Pros et al., 1998) and to study fabric symmetry (e.g., Přikryl et al., 2007). Recently, such experiments were applied to investigate in detail anisotropy of shear waves (Keppler et al., 2015; Loka-jíček & Svitek, 2015; Lokajíček et al., 2014; Svitek et al., 2014), the influence of thermal treatment to elastic anisotropy (e.g., Lokajíček et al., 2012) and attenuation anisotropy (Svitek et al., 2017). In addition, diffrac-tion methods, such as neutron diffraction (ND), could be used to determine crystal preferred orientations (e.g., Ivankina & Matthies, 2015; Wenk, 2006) in bulk rocks and calculate intrinsic elastic properties using advanced micromechanical models.

One of the classical granitic rocks is Westerly granite (WG), a fine-grained granite from Rhode Island, USA. There are quarries in two main areas—Westerly and Charlestown, and commonly three types of granite are recognized: “Pink Westerly,” “Blue Westerly,” and “Red Westerly” (Dale, 1923). WG was determined to be of Permian age, ∼276 My (Skehan & Rast, 1990). Mineral composition and microstructure were analyzed first

Abstract Westerly granite (WG) has been generally accepted as an isotropic homogeneous rock. Here, we return to WG and observe significant elastic anisotropy using multidirectional ultrasonic sounding on spherical samples at pressures up to 400 MPa. Thermal treatment of WG leads to formation of microcracks that reduce elastic wave velocities and increase its elastic anisotropy. The 3D distribution of P-wave velocities at low pressure is close to orthorhombic symmetry. Application of hydrostatic pressure closes most of thermally induced microcracks and decreases elastic anisotropy of WG, but at high pressure the anisotropy is practically reversed compared to low pressure: maximum P-wave velocity direction at low pressures is near minimum velocity direction at high pressure and vice versa. To understand this effect, microstructures of the rock were investigated by optical and scanning electron microscopy. Preferred orientations of four major rock-forming minerals—quartz, orthoclase, plagioclase, and biotite—were measured by time-of-flight neutron diffraction, which confirms significant crystal alignment. All these data were used to numerically model anisotropic elastic properties of WG. It is shown that WG possesses weak intrinsic elastic anisotropy related mainly to the preferred orientation of feldspars formed during igneous crystallization. Observed microcracks are mostly related to the cleavage planes of feldspars and biotite, and thus also demonstrate preferred orientation. Higher preheating temperatures produce larger quantity of longer microcracks. A numerical model shows that these microcracks act against the weak intrinsic elastic anisotropy of WG, and define the elastic anisotropy at low pressures.

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© 2020. American Geophysical Union. All Rights Reserved.

Intrinsic Elastic Anisotropy of Westerly Granite Observed by Ultrasound Measurements, Microstructural Investigations, and Neutron DiffractionT. Lokajíček1 , R. Vasin2 , T. Svitek1, M. Petružálek1, M. Kotrlý3, I. Turková3, R. Onysko3,4, and H. R. Wenk5

1Department of Physical Properties of Rocks, Institute of Geology of the Czech Academy of Sciences, Prague, Czech Republic, 2Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Russia, 3Institute of Criminalistics Praha, Prague, Czech Republic, 4SG Geotechnika, a.s., Praha, Prague, Czech Republic, 5Department of Earth and Planetary Science, University of California, Berkeley, CA, USA

Key Points:• Intrinsic anisotropy in Westerly

granite is documented by ultrasonics on spheres, crystal preferred orientations by neutron diffraction and shape preferred orientation (SPO) explored with scanning electron microscopy

• Ultrasound experiments at different pressures and heating cycles document microfractures that appear to be dominantly along grain boundaries

• Adding crystal preferred orientations and SPO, we can predict elastic properties that agree with velocity measurements

Correspondence to:T. Lokajicektl@gli.cas.cz

Citation:Lokajíček, T., Vasin, R., Svitek, T., Petružálek, M., Kotrlý, M., Turková, I., et al. (2021). Intrinsic elastic anisotropy of Westerly granite observed by ultrasound measurements, microstructural investigations and neutron diffraction. Journal of Geophysical Research: Solid Earth, 126, e2020JB020878. https://doi.org/10.1029/2020JB020878

Received 22 SEP 2020Accepted 20 NOV 2020

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by Dale (1923), and subsequently by Chayes (1950), Flanagan (1967), Ames (1978), and Moore et al. (1987). It was found that there may be some compositional variations of WG, but in general over 90 Vol.% of WG is composed of quartz and feldspars with average grains sizes of ≈ 1 mm, and the rest are mica and accessory minerals. Goldich and Oslund (1956) made detailed chemical analyses of WG. Intrinsic total porosity of WG was found to be close to 0.01 (Brace, 1965; Nur & Simmons, 1969).

Heard and Page (1982) studied elastic properties and thermal expansion of WG up to 350°C and 55 MPa pressure and found it to be isotropic within experimental error margins. Thermal and mechanical proper-ties of WG were studied by Dwivedi et al. (2008). Heard and Page (1982), Morrow et al. (1986), and Stanchits et al. (2003) investigated its permeability. Spencer and Nur (1976) and Hadley (1976) studied elastic wave velocities in WG at different confining pressures, pore pressures, and temperatures. They considered WG an isotropic rock, though Hadley (1976) noticed a significant decrease of P-wave and S-wave velocities, and compatible changes in velocity ratios with increasing dilatancy in WG under nonhydrostatic loading, indi-cating development of elastic anisotropy.

Subsequently, much attention focused on processes of formation, growth and evolution of cracks in WG, where crack systems were analyzed by optical microscopy, scanning electron microscopy (SEM), or their presence was inferred from their impact on propagation of ultrasonic waves. The fatigue process in WG granite subjected to a cyclic loading under uniaxial compression became a new topic (e.g., Batzle et al. 1980; Chen et al. 2011; Heap & Faulkner 2008). Path dependences of acoustic velocity and attenuation in WG deformed up to the brittle failure were studied by Granryd et al. (1983). They observed an increase in VP anisotropy related to increasing volume of oriented microcracks forming a macrocrack. This effect was found to be location-independent, indicating uniformity of the process through the material volume. Tap-ponnier and Brace (1976) and Stanchits et al. (2003) observed formation of stress induced cracks in WG, mostly oriented parallel to the compression. Tapponnier and Brace (1976) mentioned that subjecting WG to pressure cycles over 100 MPa produces local, but marked damage. Petružálek et al. (2018, 2020) analyzed the acoustic emission source mechanisms to demonstrate that tension microcracks parallel to maximum compressive stress controlled the fracturing of WG under uniaxial compression. The size of induced cracks correlated well with the grain size distribution of WG. Moore and Lockner (1995) applied image-analysis computer techniques to study crack formation in WG. They found that open microcracks in undeformed WG are concentrated within and along the edges of quartz crystals. Longest cracks are located along elon-gated crystals, follow cleavage planes in feldspars and biotite, and simple twin planes in plagioclase. Under increasing axial load, intragranular cracks are formed and crack densities increase in all minerals, but at a greater rate in feldspars than in quartz. They also observed that cracks in feldspar may consist of a stairstep arrangement of smaller cracks that follow the cleavages in the mineral.

Arrays of cleavage cracks in feldspars and biotite at high angles to compression were observed during devia-toric loading at increased pressure and temperature (250 MPa at 150°C and 400 MPa at 350°C) (Wong, 1982); preferred orientation of these minerals would mean preferred orientation of cracks, and, consequently, de-fine bulk elastic anisotropy of cracked WG. In contrast, high angle cracks were absent in quartz, confirming that minerals in WG behave differently with respect to crack formation, with the formation mechanisms be-ing dependent on both mineralogy and grain orientation. Cyclic axial loading experiments at ambient con-ditions (Chen et al, 2011) confirmed presence and growth of fine cleavage cracks in some feldspar grains, but only at initial loading stage. At increased stresses, intragranular cracks preferentially oriented parallel to the loading direction were formed in feldspar grains, subsequently joining into intergranular cracks closer to the faulting. Cracks in quartz had no marked preferred orientation.

Stanchits et al. (2003) gave a detailed description of anisotropic changes in P-wave velocities and attenua-tion due to formation of dilatant cracks. They also noted that WG has a small inherent velocity anisotropy at 50 MPa confining pressure, though it was negligible for the purposes of their study. Thompson et al. (2006) found that even at the onset of deviatoric mechanical loading, WG shows significant elastic anisotropy (7.7%).

Several studies documented formation of thermally induced cracks in WG. Fredrich and Wong (1986) found that the majority of cracks in WG are located at quartz/feldspar grain boundaries and these grain bounda-ries are cracking more readily during heating. Significant thermal cracking occurs already at 100°C–150°C.

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At temperatures over 250°C there is a pronounced increase in the intragranular cracking rate. At tem-peratures over 500°C all grain boundaries are cracked. Cracks originating at grain boundaries propagate through grains, and cleavage planes of feldspars favor the propagation. Nasseri et al. (2007) studied thermal cracking of WG up to 850°C and measured evolution of grain boundary and intragranular crack densities and shapes with temperature. According to their data, grain boundary crack density in WG steadily increas-es during heating, with a rapid jump observed at 850°C; intragranular crack density starts increasing only at 650°C and higher. Widths of both crack types remain practically constant at lower temperatures and increase significantly at temperatures of 650°C and higher. Thus there is an increase of the grain bounda-ry-to-intragranular crack density ratio with preheating temperature increase. Wang et al. (1989) found that the onset of significant thermal cracking of WG increases linearly with pressure. It starts above 100°C at lower pressures (<7 MPa), and above 250°C at 55 MPa. Thermal cracking is caused by thermal expansion mismatch between different minerals, or grains with different orientations. Griffiths et  al.  (2018) noted that observed peculiarities of elastic wave velocities ratio VP/VS during cooling of WG are unusual for the isotropic material.

To summarize, WG is a well-studied rock, which is often considered isotropic, though there are some in-dications of intrinsic weak elastic anisotropy. Deviatoric stresses, increased pressure and temperature may lead to development of different crack systems in WG. Some of these cracks are related to elongated grain boundaries, or cleavage planes of feldspars and mica, and thus acquire preferred orientation, which also reflects intrinsic WG properties, such as microstructure and crystallographic texture.

In this study, we are going to explore elastic anisotropy of WG and its changes invoked by thermal treatment and hydrostatic pressure. We will use multidirectional ultrasonic sounding (US) to study P-wave propaga-tion in preheated spherical samples of WG at different pressures to uncover the influence of formed thermal cracks and their closure with pressure on elastic properties of WG and their anisotropy. Microstructural features, mineral composition and grain preferred orientations in WG will be quantified with scanning elec-tron microscopy, optical microscopy and ND. Identical bulk samples were used for elastic wave propagation study and in ND experiments, demonstrating US and ND capabilities as complementary methods suitable for in-depth analysis of elastic anisotropy of rocks.

2. Samples and Thermal Treatment ProceduresAll samples were cut from a single WG cube with the edge of 50 cm, and for all samples the same Cartesian right-handed coordinate system XYZ was assigned, related to the initial large cube edges because of absence of evident structural features, such as foliation or lineation.

For the SEM analysis of pristine WG, we prepared two cylindrical samples 20 mm high and 25 mm in di-ameter (marked WGA and WGB, see Table 1). Cylindrical samples were taken from the vicinity of the cored samples to be examined by the energy-dispersive spectroscopy (EDS). The axis, along which samples were drilled, was labeled Y. There was preserved the same Cartesian system of sphere orientation. One side of each specimen, used for EDS study, was polished and cleaned by ethanol and cellulose. The polished side was carbon-coated.

A 30 μm petrographic thin section was prepared for preliminary analysis with a petrographic microscope.

Four spherical samples (labeled WG100, WG200, WG400, and WG600, where the number corresponds to the maximum preheating temperature), each with a diameter of 50 mm ± 0.05 mm, were prepared for preheating in different regimes and subsequent US and ND measurements. For ultrasonic measurements at increased pressures, these spherical samples were dried in an oven at 100°C and covered by epoxy resin for pore space protection against infiltration of oil serving as a pressure medium during the high pressure measurements. The thickness of epoxy resin was 0.1 mm. Spherical sample diameter was measured before and after epoxy layer coverage in 18 regularly distributed points. Wave travel time corrections are applied according to epoxy layer thickness. The measured dry density of samples before heating was 2.635 (1) g/cm3. This is consistent with reported values for WG: 2.635 and 2.636 g/cm3 (Simmons, 1964), and 2.646 g/cm3 Brace (1965).

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The sample and the coordinate system settings of spheres are shown in Figure 1.

Preheating was performed at a constant heating rate of 1°C/min, then the desired temperature level was maintained constant for 24 h, and afterward the sample was cooled down at a rate of 1°C/min. All four samples were first heated to 100°C. This is the maximum preheating temperature for sample WG100. After this thermal treatment, WG100 was measured by US at confining pressures up to 400 MPa at incremental pressure values of 0.1, 5, 10, 20, 50, 100, 200, 300, and 400 MPa, to investigate the effect of thermal cracks closure. The other samples, WG200, WG400, and WG600, were measured by US at 5 MPa confining pres-sure, after 100°C preheating. Afterward, these three samples were heated to 200°C, which is the maximum heating temperature for WG200. Following the previously described procedure, WG200 was measured us-ing US at increased pressures up to 400 MPa, but WG400 and WG600 were only measured at 5 MPa. The same routine was repeated for WG400 (maximum preheating temperature of 400°C) and WG600 (600°C). All these temperatures are below expected dihydroxylation region of biotite in WG that is at 650°C–850°C (Blake & Faulkner, 2016), but 600°C is above the transformation temperature of trigonal α-quartz to hexag-onal β-quartz that is at 573°C (Kihara, 1990).

After the US experiments, samples WG100 and WG600 were measured by ND. The protective epoxy layer was removed prior to ND measurements.

The overall layout of thermal treatment and subsequent measurements of different samples is presented in Table 1.

3. Measurements3.1. Mineralogical Analysis

The mineralogical analysis was first performed with a Nikon petro-graphic microscope on a thin section of a cylindrical sample WG400 cut perpendicular to the cylinder axis. A mineralogical and micro-structural analysis on samples WGA and WGB was also performed with a scanning electron microscope MIRA 3 RISE TESCAN with two EDS detectors. The accelerating voltage was 25 keV, emission current 169 µA and absorbed current was about 5.7 µA. Surface scanning of oriented samples was made by the system designed for automatic analysis of mineral grains—TIMA, developed by TESCAN (Gottlieb et  al.,  2015). The TIMA system operates on the principle of energy dispersive X-ray spectroscopy (EDS) combined with the detection of backscattering electrons (BSE). The electron beam continuously scans the total surface of a cylindrical sample and the classical EDS analysis of X-ray radiation starts once the threshold energy of BSE brightness is exceeded. The TIMA TESCAN is able to automatically scan the sur-face area of 25 × 25 mm of the sample pixel by pixel (a pixel size of 4 μm was used for our analysis) and to determine individual miner-

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Sample 100°C Preheating 200°C Preheating 400°C Preheating 600°C Preheating

WG100 (50 mm sphere) US at 0.1–400 MPa + ND

WG200 (50 mm sphere) US at 5 MPa US at 0.1–400 MPa

WG400 (50 mm sphere) US at 5 MPa US at 5 MPa US at 0.1–400 MPa

WG600 (50 mm sphere) US at 5 MPa US at 5 MPa US at 5 MPa US at 0.1–400 MPa + ND

WGA (25 mm disk) TIMA TIMA

WGB (25 mm disk) TIMA

Table 1 Scheme of Thermal Treatment and Measuring Cycles of Different Westerly Granite (WG) Samples

Figure 1. Spherical WG sample (WG100), 5 cm diameter, as mounted on the SKAT neutron diffractometer. Axes of the sample coordinate system (Cartesian, right-handed) XYZ are shown.

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al phases in the sample. In our case, we scanned an area of 25 mm diameter. BSE data were further used for identification and analysis of oriented microcracks.

3.2. Neutron Diffraction

Two spherical WG samples were measured by ND to analyze crystallo-graphic texture and potential changes of mineral grain structures and orientations caused by high temperature treatment.

ND experiments on samples WG100 and WG600 were performed with the SKAT time-of-flight diffractometer (Keppler et al., 2014; Ullemey-er et  al.,  1998) at beamline 7A2 of the IBR-2 pulsed reactor, FLNP JINR, Dubna. This instrument is specifically designed for texture analysis of large rock samples. The combined cold/water modera-tor for the beamline 7A2 was in operation, which increased count-ing statistics at longer neutron wavelengths (Bulavin et  al.,  2016). The standard setup of the diffractometer with 19 3He detectors and 45′ collimators at the scattering angle 2θ = 90° has been used. Dur-ing measurements the sample was rotated in 10° incremental steps around Z (Figure 1) to obtain pole figures with sufficient coverage, resulting in over 600 diffraction spectra for each sample. Minor reac-tor flux variations during the experiment were accounted for using a beam monitor.

All ND spectra obtained from a sample were simultaneously refined using a modified Rietveld method in the MAUD software (Lutterotti et  al.,  1997; Wenk et  al.,  2010). The data preparation for MAUD was made using the SKAT2MAUD program (Wenk et al., 2012). Preliminary analysis of patterns suggested that diffraction peaks of four major phas-es are present: plagioclase, K-feldspar, biotite and quartz. Other miner-als may be present in smaller quantities (∼1 Vol.% or less), but could not be identified from ND patterns. Rietveld refinement requires infor-mation on crystal structures of phases in the sample. We used amcsd

0010721 (Phillips and Colville, 1971) for plagioclase (An16, oligoclase structure), amcsd 0007928 (Tseng et al., 1995) for K-feldspar (orthoclase), amcsd 0001298 (Brigatti & Davoli, 1990) for 1M biotite and amcsd 0006212 (Antao et al., 2008) for quartz. The trigonal quartz structure has been changed to the left-handed space group P3121. For monoclinic orthoclase and biotite, first setting of the crystal coordinate system was used for preferred orientation analysis and microstructure-based modeling of elastic properties, as required by MAUD (Matthies & Wenk, 2009). But through this study, all the relevant plots and labels are in the more conventional second setting.

For refinement in MAUD, the range of lattice spacings was restricted to 1.6–4.7 Å. Phase volume frac-tions, backgrounds for each spectrum (second order polynomial), unit cell parameters of each phase, isotropic temperature factor (one for all atoms of all phases), crystallite size and r.m.s. microstrain (both in isotropic approximation) have been refined, in addition to texture. For texture refinement, a direct E-WIMV (Lutterotti et al., 2014; Matthies, 2002) algorithm has been used with 5° cell size in ori-entation space that reconstructed orientation distribution functions (ODFs) of all four main minerals.

Measured and refined normalized diffraction patterns are shown in Figure 2. A very good agreement is observed, indicating the excellent quality of the refinement. The weighted profile R-factor value, Rwp, is 8.2% for WG100% and 8.3% for WG600 sample refinements, respectively, which can be considered good, taking into account complexity of multiphase samples and relatively simple structure models with a limited number of refined parameters.

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Figure 2. Measured (dots) and refined (solid lines) diffraction spectra of WG100 sample for scattering vector directions aligned with X, Y, and Z axes (a). Some intense diffraction peaks of plagioclase (An16), orthoclase (Or), biotite (Bio) and quartz (Q) are labeled. Stack of all 627 refined (b) and measured (c) WG100 diffraction spectra, darker tone corresponds to higher intensity. Intensity variations of peaks along the vertical axis indicate preferred orientations of minerals.

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3.3. Ultrasonic Sounding

A pulse transmission US on spherical samples was performed in 132 different directions by P-waves using a special setup (Lokajíček & Svitek, 2015; Pros et al., 1998). Transducers with a resonant frequency of 2 MHz were used to emit and receive acoustic pulses. The sensors were made from piezoceramic tablet only, there was no sensor damping. This is due to the fact that mainly the P-wave arrival is detected. The signal from the sensor was preamplified by a low noise preamplifier with the gain 20 or 40 dB. The preamplifier bandwidth was flat in the range 10 kHz–30 MHz. The same gain was preserved during the whole hydrostatic loading cycle to avoid ultrasonic P-wave time arrival discrepancies according to different preamplification sensi-tivity. Preamplified signals were digitized by 100 MHz sampling frequency (10-bit resolution). Recorded signals at each measuring direction were averaged by 8 times to increase signal-to-noise ratio. The surface of sensors is flat, which in combination with spherical surface of samples create a quasi point contact. The spherical sample is placed into the apparatus between two sensors that are supported by a frame. The ar-rangement allows to rotate a sample around by 360°, and tilt the frame up to 75° in 15° incremental steps in both directions. This measuring apparatus is a part of a high-pressure vessel designed to confining loading of samples up to 400 MPa using oil as a pressure medium. Confining pressure increase rate was 5 MPa/min. Hydrostatic pressure vessel is equipped by a calibrated temperature sensor, which continuously measures hydrostatic oil temperature. Temperature increase is about 2°C–3°C due to the adiabatic phenomena. After pressure increase, we waited about 30 min to redistribute oil temperature with pressure vessel (pressure vessel weight is about 350 kg). The pressure vessel is continuously preheated to 25°C to avoid oil viscosity changes by room temperature fluctuations. For every measuring point there is a known inner oil tempera-ture. During one measuring cycle (60 min) the oil temperature drops only by about 1°C.

Waveforms of measured signals are stored and analyzed by a custom made software that allows detecting arrival time of P-waves automatically (Svitek et al., 2010). According to the sample size to sensor size ratio we measured ray velocities (Dellinger & Vernik, 1994; Vestrum & Brown, 1994). After the recalculation to phase velocities (Lipschutz, 1969; Svitek et al., 2014), they serve as input for the inversion process based on iterative calculation scheme, which results in 21 stiffness tensor components of the bulk polycrystalline sample (Svitek et al., 2014).

Dependence of P-wave velocities on pressure may be interpreted in terms of the empirical nonlinear ap-proximation (Nafe & Drake, 1957; Pros et al., 1998)

/ 00 dif· ·10 ,P P

VV P V k P V (1)

where V0—is the velocity at ambient pressure (in an ideal sample without cracks), kV—linear increase of velocity with pressure for a crack free rock, P—confining pressure in MPa, Vdif—crack influence at ambient pressure, P0—pressure at which the crack influence Vdif decreases to 10%. V0 and kV describe solid media and its intrinsic structure, whereas Vdif and P0 are related to properties of voids (cracks, fractures, or pores) in the sample. The V0 interpretation is close to determination of P-wave velocities by ND-based modeling, since ND do not probe cracks and give information only about the crystalline rock (Ivankina & Matthies, 2015; Wenk, 2006).

Figure 3 shows US signal waveforms measured at directions corresponding to the maximum and mini-mum P-wave velocities (only parts close to the first arrival are shown). In the sample WG100, preheated to 100°C, pulse arrival times are ∼13 μs. They decrease slightly with the pressure increase due to closure of microcracks. In the sample WG600, preheated to 600°C, there is a significant delay of P-wave arrival times by about 30–40 μs at normal pressure, meaning that P-wave velocities are much slower in WG600 compared to WG100. The pressure dependence is much more pronounced: P-wave arrival times decrease drastically with pressure in WG600.

Sample WG100 shows a minimal difference between fast and slow direction at 0.1  MPa: 1.1  μs, which is ≈ 10% of the pulse travel time. In case of sample WG600, this difference is 10.6 μs at 0.1 MPa (≈20%–25% of the pulse travel time). The frequency domain of waveforms is also affected by preheating temperature and pressure. The recorded frequency in the maximum velocity direction is 400 kHz for WG100 and 140 kHz for WG600 at 0.1 MPa pressure. At 400 MPa pressure it is 960 kHz for both WG100 and WG600. In the

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minimum velocity direction, the recorded frequency is 360 kHz for WG100 and 130 kHz for WG600 (at 0.1 MPa). It is 860 kHz for both WG100 and WG600 at 400 MPa. A higher frequency content in the maxi-mum velocity direction reflects a slight preferred orientation of cracks.

Regardless of preheating, at 400 MPa pressure all signal waveforms are almost identical in terms of arriv-al time and frequency content. The amplitude of signals in WG600 is lower, which may be due to higher damping in the sample.

4. Results4.1. Optical Analysis

The optical analysis reveals the microstructure of a fine-grained granite composed largely of plagioclase, alkali feldspar, quartz, and biotite (Figure 4). Quartz displays some undulatory extinction (Figure 4b) and is fairly equiaxed. Feldspars, in particular plagioclase, are significantly altered, which is best viewed with

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Figure 3. Onset of ultrasonic signals propagating in samples WG100 (blue) and WG600 (red) at pressures up to 400 MPa in direction of the maximum (top) and the minimum (bottom) P-wave velocity.

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plane polarized light (Figure 4a). Plagioclase reveals compositional zoning (Pz). Orthoclase (O) is trans-forming to microcline, expressed by cross-hatched twinning. Biotite (B) shows a platy habit and is not al-tered. There is some primary muscovite (M) and secondary muscovite associated with feldspar alteration (e.g., brown spots above “Pa,” Figure 4b). There are fractures associated with grain boundaries, but some other fractures cross grains, including quartz (f).

4.2. TIMA

BSE images and mineral phase maps of WG are shown in Figure 5. Grains of main WG mineral components are easily distinguishable: quartz (23.8 wt.%), orthoclase (24.5 wt.%) and plagioclase (40.2 wt.%). There are also secondary phases as chloritized biotite and muscovite. A full list of individual WG mineral phases (with content above 0.1 wt.%) is shown in Table 2 (mean values for WGA and WGB samples). Additionally, BSE images of the sample preheated to 100°C (Figure 5a), display very few cracks that are mainly intergranular. There is a significant increase in quantity and thickness of cracks when the sample is preheated to 600°C (Figure 5b). Cracks are mostly intergranular, but intragranular cracks are also observed.

The WG grain size distribution was also determined. It was found that the maximum grain size of quartz and orthoclase is ≈1.3 mm, and the mean grain size is about 0.8 mm. Plagioclase size distribution has a pronounced peak at 1.1 mm that can be related to twinning. Chloritized biotite has relatively small grains with a well pronounced mass distribution with a maximum at ≈0.4 mm. The majority of mineral grains are larger than 0.1 mm, which indicates that a chosen scanning pixel size of 4 μm was sufficient for the mineral analysis.

Full phase maps of sample WGA (Figure 5) were analyzed by the image analysis program Fiji (Schindelin et al., 2012). Grains of each mineral phase in the size range of 0.04–1.6 mm were interpolated by ellipses, and the orientations of their longest and shortest axes were determined. Orientation distributions of longest axes of grains of four main WG minerals are shown in Figure 6a. Grains of all mineral phases are aligned similar to each other regardless of their size with a preferred orientation of longest axes at an angle ∼35° to the X axis.

A similar analysis procedure was applied to investigate the orientation distribution of cracks of different length (Figure  6b). There is a low number of cracks in WGA sample preheated to 100°C. A significant increase of quantity and length of cracks is observed after the sample was preheated to 600°C. Maximum length of cracks is less than 400 µm. Shorter cracks are aligned more along the Z axis, while longer cracks have more irregular orientations with an orientation maximum at ∼45° to the X axis. It should be noted that some of the observed cracks longer than 400 µm are not straight, and additionally there are interconnected cracks of different length and orientation. Such complex cracks were also interpolated by ellipses, but they were not considered in the subsequent analysis.

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Figure 4. Optical microscopy of sample WG400 with (a) plane polarized and (b) cross-polarized light. B, biotite; M, muscovite; Q, quartz; O, orthoclase (changing to microcline); Pa, altered plagioclase, Pz, zoned plagioclase, f, fractures (in quartz grains).

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Cracks number density (number of cracks per unit area) dependence on crack length for the WGA sample is plotted in Figure  7 in double log-arithmic scale. The sum of all cracks was determined in crack length intervals equal to 4–8, 8–16, 16–24, 24–40, 40–80, 80–160, 160–400 µm, respectively. Maximum length of this interval was used as a crack length up to 400 µm. The sum was divided by the area of 25 mm in diameter of sample under the study. We can see a quasi-linear decrease of the crack number density of WG samples heated to 100°C and 600°C. We can ob-serve a significant increase of cracks number density in WG due to the thermal heating (cf. blue and red curves). The crack number density in-crease due to the thermal heating to 600°C is denoted by dashed lines. In case of heating to 600°C, in comparison with 100°C, we can see an up to 60 times increase in crack number density for the cracks of the length about 100 µm. It is also possible to assess a 2D crack density that is a sum of squared half-lengths of cracks normalized by the studied area (Grif-fiths et al., 2017; Walsh, 1965). For WG100, it is 8 × 10−4, and for WG600 it is 6 × 10−3. These numbers are significantly below the 2D crack densities observed after thermal preheating of a Garibaldi gray granite that is quite similar to WG (Griffiths et al., 2017). We conclude that the applied analy-sis procedure approximating cracks by ellipses, and disregarding too thin, complex and intersecting cracks most likely underestimates 2D crack density. Nevertheless, it allows for comparison of crack distributions in WG samples after different thermal treatment (Figure 7).

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Figure 5. BSE and corresponding phase images of a selected area of sample WGA, preheated to (a) 100°C and to (b) 600°C. Main WG minerals are labeled: Pl, plagioclase; Kfs, orthoclase; Q, quartz; Bi, biotite. Sample coordinate frame XYZ is also shown.

TIMA ND, WG100 ND, WG600

Mineral wt.% wt.% wt.%

Plagioclase 42.9 44.8 44.9

Orthoclase 24.4 25.4 24.8

Quartz 23.8 24.9 25.0

Biotite 5.4 4.9 5.3

Muscovite 1.3

Iron oxide 0.6

Titanite 0.5

Apatite 0.3

Calcite 0.2

Ilmenite 0.2

Rest 0.4

Sum 100.0 100.0 100.0

Note. “Iron Oxide” Denotes Partially Oxidized Iron, Released Mainly from Biotite. Unidentified phases in TIMA analysis did not exceed 0.4 of wt.%.

Table 2 WG Mineral Phases With Wt.% Over 0.1% According to TIMA Analysis (Mean Value of Two Samples WGA and WGB is Given) and ND

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4.3. Neutron Diffraction

ND data refinement provided information on mineral volume fractions and preferred orientations. Volume fractions were recalculated into wt.% for comparison with TIMA results (Table 2).

After the MAUD Rietveld refinement, ODFs of all minerals have been exported into the BEARTEX software (Wenk et al., 1998), and smoothed with a 7.5° Gaussian filter. From these smoothed ODFs, pole figures of all minerals have been recalculated. They are plotted in Figure 8 (WG100) and Figure 9 (WG600) as contour map projections in multiples of random distribution (m.r.d.). Preferred orientations of minerals in both samples are similar documenting homogeneity of the WG block. Small orienta-tion differences are mostly related to uncertainty in initial sample posi-tioning in the SKAT goniometer, which is within 5°.

Oligoclase An16 preferred orientation (Figures 8a and 9a) corresponds to a common texture of plagioclase, in particular, of those of magmatic origin (Satsukawa et al., 2013). Its texture is the strongest among minerals composing WG, with a maximum on (010) pole figure reaching 3 m.r.d. in the WG600 sample. It is close to a (010) fiber (note that it is inclined be-cause the orientation of the original WG cube, from which samples were prepared, is arbitrary). Preferred orientation of alkali feldspar has similar symmetry, but the texture itself is weaker. This is similar to plagioclase

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Figure 6. (a) Orientation distributions of longest axes of grains of different minerals. Projection onto XZ plane. “n” is the number of individual mineral grains considered for the analysis, and “avg” is the mean amount of grains within 10° angular intervals; (b) Orientation distributions of cracks of different length in WG2A sample preheated to 100°C (upper row) and later to 600°C (lower row). Projection onto XZ plane. “n” is the number of individual cracks considered for the analysis, and “avg” is the mean amount of cracks within 10° angular intervals.

Figure 7. Dependence of crack number density on crack length in the WG2A sample preheated to 100°C (blue) and preheated to 600°C (red), as well as relative increase in the crack number density within different length intervals after the preheating to higher temperature (dashed line).

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and alkali feldspar textures observed in orthogneiss (Vasin et  al.,  2017). Biotite shows typical preferred orientation of cleavage planes close to a (001) fiber, but with a large random texture component. Quartz textures are close to random. They are not changed significantly by the preheating to 600°C, which is above α-β transition temperature (Kihara, 1990). All textures are asymmetric with respect to the sample coordi-nate frame {XYZ}.

4.4. P-Wave Velocities at Different Pressures

Figure 10 summarizes P-wave velocities distributions at 5 MPa pressure in samples WG100, WG200, WG400, and WG 600 after respective thermal treatment, displayed in equal-area projection of the upper hemisphere. It is obvious that WG is very homogeneous, since four different samples are almost identical after preheat-ing to 100°C; minor differences are mostly related to small uncertainties in initial orientations of spherical samples in the {XYZ} frame. Obviously, the granite is not isotropic. Although differences between maxi-mum and minimum P-wave velocities at 5 MPa pressure are in most cases around 300 m/s, coefficient of anisotropy systematically increases with rising preheating temperature, which is due to thermal cracks. This is even more obvious, when the US measurements at full pressure range 0.1–400 MPa are considered, and coefficients of anisotropy at different pressure levels are plotted as functions of maximum preheating temperature (Figure 11). The decrease of coefficient of anisotropy with pressure is evident. Samples WG100, WG200, and WG400 show very similar behavior: coefficient of anisotropy at 0.1 MPa increases very rapid-ly with preheating temperature, but applied pressure closes cracks introduced by temperature treatment

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Figure 8. Pole figures of minerals in sample WG100: (a) plagioclase, (b) orthoclase, (c) biotite, and (d) quartz. Equal area projections, linear scale contours in m.r.d. Sample coordinate system {XYZ} and pole figure coverage in the ND experiment are shown.

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relatively quickly. Already at 20 MPa, the majority of cracks in these samples are either closed, or their preferred orientation does not enhance the elastic anisotropy. However, sample WG600 is different. The decrease of coefficient of anisotropy with pressure is moderate and it reduces to the level comparable with other samples only at 100 MPa. This indicates that thermal cracks introduced by preheating above 400°C have different character and are much more difficult to close than those introduced at lower temperatures. The pressure of 400 MPa reduces the anisotropy to around 3% in all samples.

The dependence of coefficient of anisotropy on pressure in all four spherical WG samples is shown in Fig-ure 11 (full 0.1–400 MPa pressure range for each sample). The coefficient of anisotropy A was calculated as A = (Vmax−Vmin)/Vmean × 100%, where Vmax and Vmin are maximum or minimum velocities in any direction and Vmean is an average value of all 132 independent directions (Birch,1961). At 0.1 MPa, the coefficient of anisotropy ranges between 11% and 26%. With the hydrostatic pressure increase, a rapid drop is observed. Above 20 MPa, the majority of cracks induced by thermal treatment should be closed, with the exception of WG600, where the decrease of the coefficient of anisotropy with pressure is slower. At 400 MPa of hydrostatic pressure, the coefficient of anisotropy is close to 4%. The closure of cracks is further support-ed by the P-wave velocities behavior (Figure 12).

Mean P-wave velocities of all heated WG samples, depending on hydrostatic pressure, are shown in Fig-ure 12. Their increase demonstrates the crack closure according to the acting hydrostatic pressure. Mean P-wave velocities in samples WG100, WG200, and WG400 show only minor linear velocity increase at pres-sures above 50 MPa, which can be interpreted as a closure for the majority of cracks (natural, as well as of

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Figure 9. Pole figures of minerals in sample WG600: (a) plagioclase, (b) orthoclase, (c) biotite, and (d) quartz. Equal area projections, linear scale contours in m.r.d. Sample coordinate system {XYZ} and pole figure coverage in ND experiment are shown.

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heating origin). On the contrary, velocities in sample WG600 are gener-ally lower, and display nonlinear increase through the whole pressure interval up to 400 MPa. This can be interpreted that thermal treatment above 400°C has a larger impact on thermal crack development in WG. We can see that the heating of WG from 100°C to 600°C results in rapid mean P-wave velocity drop from 4.36 to 1.17 km/s at 0.1 MPa. By contrast, at 400 MPa the mean P-wave velocity is quite consistent for all samples in the range 6.18–6.33 km/s.

Finally, the effect of cracks on P-wave velocities distribution over sample directions is shown in Figure 13, where velocities measured over the whole pressure range in spherical samples are displayed in equal-area projection of the upper hemisphere. This figure very well illustrates the symmetry of elastic anisotropy, its orientation and changes caused by the increasing hydrostatic pressure. Distributions are asymmetric with respect to {XYZ} coordinates. The elastic aniso-tropy appears to be close to orthorhombic symmetry at all pressure levels and after all thermal treatments. However, while cracks are closing due to the increased hydrostatic pressure, the distribution of P-waves rotates by about 90°. At every pressure level, the distribu-tions are very similar in all samples, with the sample WG600 being only slightly different (cf. Figures for 20 and 100 MPa pressures). It is remarkable, considering that absolute P-wave velocities may differ by

several km/s. This suggests that the thermal cracks have same origin and same preferred orientation patterns through different preheating regimes, yet there may be some additional development after preheating to 600°C. At 400 MPa pressure, P-wave velocity distributions are similar to distributions on pole figures of feldspars and biotite (Figures 8 and 9).

Figure  14 shows results of nonlinear approximation (see Equation  1) calculated for spherical WG samples (Pros et al., 1998). The orientation distribution of parameters V0 and kv is very stable with increasing heating. Since WG is a homogeneous material, properties of the rock without cracks should be same for all specimens and values of V0 are expected to be the same. Indeed, V0 distributions of samples WG100, WG200, and WG400 are similar, with velocity differences around 3%, however sam-ple WG600 exhibits differences more than 400 m/s (7%). This large difference is due to slower closure of cracks with pressure (see Figure  12). The linear part of the VP versus pressure curve is barely

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Figure 10. P-wave velocities measured at 5 MPa pressure in WG samples after different preheating treatment. Equal area projections, scale contour values are in km/s.

Figure 11. Dependencies of coefficient of anisotropy on pressure in samples WG100, WG200, WG400, and WG600.

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reached for WG600, because cracks are not fully closed even at 400 MPa for this sample. Consequent-ly, the linear and nonlinear terms in Equation 1 could not be correctly separated based on experimen-tal data. This also causes deviations in distributions of parameters Vdif and P0. Parameter Vdif describes emerging of new cracks due to the rising temperature in the best way, and the maximum of Vdif may be attributed to the direction normal to the most of the cracks.

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Figure 12. Dependencies of mean P-wave velocity (average of 132 measured values) with increasing hydrostatic pressure in samples WG100, WG200, WG400, and WG600. Displayed bars represent a range of measured velocities at each pressure and preheating temperature.

Figure 13. P-wave velocity distributions measured on samples WG100, WG200, WG400, and WG600 at different pressure levels, equal-area projection of the upper hemisphere, linear scale contours in km/s.

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4.5. US Velocities Inversion

The spherical shape of WG samples allows us to carry out US by P, S1, and S2 (perpendicularly polarized) waves (see Svitek et al, 2014; Lokajicek & Svitek, 2015). In our case we have P, S1, and S2 wave travel times measured in 132 different directions on the spherical sample available for inversion. The acoustic signals are excited and recorded by three piezoceramic sensor pairs with resonant frequency of 2 MHz and 700 kHz for measuring P and S waves, respectively. The wave travel times were determined for WG100 and WG600 samples at 0.1 and 400 MPa only. The calculation is based on solving Christoffel equation with known group velocities in multiple directions. Such an approach enables us to recalculate all 21 components of the elastic tensor from measured group velocities. Elastic parameters calculated for 0.1 and 400 MPa of hydrostatic pressure are shown in Table 3.

5. DiscussionUS measurements on preheated WG sample did show that the granite is elastically anisotropic, and the in-crease of preheating temperature enhances the anisotropy. Even at a pressure of 400 MPa, where P-wave ve-locity versus pressure trends suggest that practically all cracks are closed, a small elastic anisotropy is observed.

There are several factors affecting the elastic anisotropy of WG: preferred orientations of minerals docu-mented with neutron diffraction, preferred orientation of nonspherical grain shapes derived from TIMA analysis, and preferred orientation of thermally induced cracks. The latter is more pronounced at lower pressures, while the former two are essential for the intrinsic elastic anisotropy of WG that manifests at high pressure when the majority of cracks should be closed.

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Figure 14. The comparison of parameters of nonlinear approximation calculated for samples WG100, WG200, WG400, and WG600. V0 ∼ P-wave velocity of ideal sample without cracks at 0.1 MPa, kv ∼ slope of the linear part of approximated curve, Vdif ∼ the difference of measured P-wave velocity at 0.1 MPa and V0; P0 ∼ the pressure at which the ratio between linearly approximated P-wave velocity and the measured one decreased to 1/10 of Vdif. Equal-area projection of the upper hemisphere. RMS shows root mean square error of a fit in all directions calculated by Equation 1 compared to P-wave velocities.

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5.1. Intrinsic Elastic Anisotropy of WG

Let us investigate the degree of the intrinsic elastic anisotropy of WG. We will calculate bulk elastic prop-erties of the granite using measured textures, volume fractions, microstructure features and known single crystal elastic properties of main minerals composing WG: plagioclase, K-feldspar, quartz and biotite.

TIMA and ND results on WG mineral composition are consistent within 1–2 wt.% (Table 1). Our WG sam-ples contain more plagioclase (nearly 43 wt.%) than K-feldspar, and in total, feldspars compose over 66 wt.% of the granite, quartz—about 25 wt.%. Our results are consistent with the “Blue Westerly” granite composi-tion (Dale, 1923) and the G-2 reference WG (Flanagan, 1967).

Elastic properties of main minerals composing the WG are available from the literature. For plagioclase we use elastic properties of An25 oligoclase (Brown et al., 2016), for orthoclase properties of Or93Ab7 or-thoclase (Waeselmann et al., 2016). Elastic properties of biotite are available in a quasi-hexagonal setting (Aleksandrov & Rhyzova, 1961). For quartz we use data of Heyliger et al. (2003).

Single crystal elastic properties are averaged over crystal orientation distributions to obtain bulk elastic properties of WG. We use the GeoMixSelf method (Matthies, 2010). It combines the conventional self-con-sistent method with elements of geometric mean averaging to obtain a unique solution for bulk elastic properties of the material in the “long-wave” approximation. It considers not only single-crystal elastic properties of minerals, volume fractions, and crystallographic preferred orientations, but also shapes of

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GeoMixSelf Model US Velocities Inversion

WG100 WG600 WG100 WG600 WG100 WG600

Biotite and orthoclase grain shape/pressure {1:1:0.2} {1:1:0.2} {1:1:1} {1:1:1} 0.1 MPa 400 MPa 0.1 MPa 400 MPa

C11 102.1 101.2 103.2 102.1 47.6 101.4 3.5 99.5

C12 33.0 32.9 32.4 32.3 16.7 34.8 1.3 32.8

C13 32.7 32.3 32.2 31.7 16.3 34.4 1.3 33.0

C14 0.3 0.4 0.3 0.5 0.1 0.2 0 0.7

C15 0.5 0.3 0.5 0.4 −1.1 0.6 −0.2 −0.2

C16 1.1 0.9 1.3 1.1 −2 0.7 −0.5 0.8

C22 101.2 101.8 102.0 102.9 51.1 100.8 3.4 99.3

C23 32.1 32.4 31.5 31.8 16.8 33.4 1.2 33.2

C24 0.0 0.5 0.0 0.6 0 0.9 −0.1 1.0

C25 0.3 0.5 0.3 0.5 −0.6 0.9 −0.1 0.5

C26 0.9 0.9 1.1 1.2 −2.1 1 −0.4 1.0

C33 101.0 101.1 101.7 101.9 51.7 104.4 3.4 102.3

C34 0.3 0.3 0.4 0.4 0.2 0.9 −0.1 0.6

C35 0.2 0.0 0.3 0.1 −1.8 0.4 −0.2 0

C36 1.0 1.0 1.1 1.1 −1.2 0.4 −0.2 0.7

C44 34.0 34.1 34.7 34.8 16.8 34.1 1.2 33.5

C45 0.3 0.2 0.4 0.3 −0.2 0.1 0 0.1

C46 0.0 0.3 0.1 0.3 −0.2 0.2 0 0.1

C55 34.4 34.3 35.1 35.0 16.8 34.3 1.2 33.5

C56 0.1 0.1 0.1 0.2 0.1 0 0 0.1

C66 34.6 34.6 35.3 35.3 16.8 34.4 1.3 33.4

Note. Results of GeoMixSelf calculations modeling are given assuming spherical grain shapes with axes {1:1:1} for all minerals, and assuming oblate spheroidal grain shapes with axes {1:1:0.2} for biotite and orthoclase. US velocity inversion shows stiffness tensors calculated for WG100 and WG600 spherical samples at 0.1 and 400 MPa.

Table 3 Stiffness Tensors of WG Samples, Values in GPa

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grains (and pores/cracks), approximated by ellipsoids, and shape preferred orientations. Other important properties, peculiarities, and limitations of GeoMixSelf, especially in application to multiphase rocks were discussed elsewhere (e.g., Matthies, 2012, 2015; Vasin et al., 2013, 2017).

To highlight the influence of individual minerals on bulk elastic properties, we modeled elastic properties of all WG mineral aggregates using preferred orientations as in WG100, assuming spherical grain shapes, and calculated group P-wave velocities using mineral density values from same references as single crys-tal elastic tensors. Resulting distributions are shown in Figure 15. The quartz aggregate is slightly aniso-tropic. Biotite is the most elastically anisotropic among WG minerals, and its polycrystalline aggregate is the most anisotropic. Feldspars are less anisotropic, and they act against the anisotropy of biotite: maxi-mum P-wave velocity directions in feldspars correspond to minimum P-wave velocity directions in biotite and vice versa (Figures 15 and 16). The biotite content is very low; thus feldspars should define the overall intrinsic bulk elastic anisotropy of WG.

Consideration of grain shapes may enhance or reduce overall elastic anisotropy, depending on texture strength (Vasin et al., 2014). As demonstrated by optical microscopy and TIMA (Figure 5), grains of all major mineral phases can be roughly approximated by oblate spheroids. Quartz and plagioclase are more or less equiaxed. For orthoclase and biotite there are a few elongated grains with aspect ratios about 0.2, but also there are many grains of almost spherical shape. The TIMA analysis shows that there exists a preferred orientation of elongated grains in the XZ plane (Figure 6). A direct comparison with pole figures that are preferred orientations of lattice planes (Figures 9 and 10) is difficult, because the TIMA analysis shows orientations of grain axes in a 2D XY slice of the material, while pole figures are integral projections of a 3D distribution of crystallographic directions. Grain axes are at right an-gles, while angles between principal crystallographic planes may differ from 90°. Still, there is an evi-

dent similarity between orientations of longer axes of grains and (100) pole figures of biotite and feldspars. Here, shorter axes of grains should be roughly normal to (001) planes of biotite, and (010) and (001) planes of feld-spars, which are cleavage planes. Thus, it is reasonable to assume that cleavage defines grain shapes and 3D grain shape orientation distribu-tion may be derived from ODFs by considering additional rotations of crystal coordinate systems into grain coordinate systems as described by Vasin et al. (2013).

We calculated bulk elastic properties of WG100 assuming spherical shapes of all grains, or oblate spheroidal grains of biotite and orthoclase with axes {1:1:0.2} (Table  3). Adjustment of grain shapes in the model leads to changes in calculated elastic constants that are mostly within 1 GPa range. We can conclude that grain shapes have negligible effect on WG elastic properties. Resulting P-wave velocity distributions in WG100 sample are plotted in Figures 17a and 17b. There is a small decrease in velocity and coefficient of anisotropy associated with oblate spheroidal

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Figure 15. Group P-wave velocity distributions in minerals composing WG with preferred orientations as in WG100 sample. Equal area projections, linear scale contours. Minimum and maximum velocity values are shown in lower left corner, and corresponding coefficient of anisotropy—in lower right corner. Dark blue color corresponds to low velocity values, red - to high values.

Figure 16. Mutual orientation of plagioclase feldspar and biotite crystallographic axes in WG.

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grain shapes of biotite and orthoclase. Additional consideration of the plagioclase cleavage, either (010) or (001), would barely change the velocity distribution.

There is a good agreement in model elastic tensor components and those inverted from measured velocity values at 100 MPa pressure. Differences are within a 4 GPa range, and for most Cij’s they are less than 1 GPa. Minimum and maximum P-wave velocities and coefficients of anisotropy are close. Velocity distributions are quite comparable (cf. Figure 13 at 400 MPa, Figures 14, 17a, and 17b). Additionally, both model Cij’s and those inverted from US experiments suggest that intrinsic VS values in WG are around 3.6 km/s, which is slightly higher than values measured by Simmons (1964): 3.41–3.57 km/s.

Differences in minimum and maximum velocity directions between US experiment and the model are ∼15°, that is, they are comparable to the grid step in US velocity (15°) and ND (10°) measurements. The difference may be related to misorientation of the sample when installed in SKAT compared to when placed into US measurements device, or induced by the modeling assumptions. These include errors in used single crystal stiffness tensors, e.g., quasi-hexagonal elastic properties of biotite have been used instead of a proper mon-oclinic tensor, disregard of small changes of elastic constants with pressure, approximations of all grains with oblate spheroids of the same shape, neglect of local grain interaction and possible stress concentrators, or small residual crack density persisting at high pressure. Nevertheless, we can conclude that texture and a microstructure-based GeoMixSelf approach provides a quantitatively adequate model of WG anisotropy at high pressure.

5.2. Anisotropy Induced by Thermal Cracking

As was discussed in the introduction, there is a large literature about cracks developing in WG during thermal treatment. Heating and cooling of rocks results in increasing number and length of cracks in the material (e.g., Fredrich & Wong, 1986; Griffiths et al., 2017; Vernik et al., 1994). The length and density of cracks depend on rock nature, mineralogical composition, quality of grain boundaries, anisotropic ther-mal expansion of individual minerals, temperature applied, possible phase transitions. Here, we provide experimental evidence that P-wave anisotropy is indeed linked to thermal treatment of samples and that the anisotropy pattern changes at high pressure (400 MPa) and it is linked to intrinsic anisotropy caused by crystal preferred orientation documented by neutron diffraction.

The effect of cracks due to rising preheating temperature is clearly observed in US experiments. There is an expected decrease of P-wave velocities (Figures 12 and 13 at 0.1 MPa) due to increase of number and length of cracks. It is clear that an increase of preheating temperature reduces P-wave velocities more, because of increase of number and length of cracks. There is an increase of number of cracks with length around 100 μm by more than 60 times comparing samples preheated to 100°C and 600°C (Figure 7). This is induced by growth and joining of smaller cracks.

Cracks introduced by heating up to 400°C have similar character and their ability to close is evident in velocity versus pressure trends (Figures 12 and 13). The situation somewhat changes after heating up to

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Figure 17. Group P-wave velocity distributions in different WG GeoMixSelf models: (a) all spherical grains; (b) ellipsoidal grains of biotite and orthoclase with axes {1:1:0.2}; (c) same as model (b), but with 0.119 crack density, assuming cracks parallel to orthoclase cleavage; (d) same as model (b), but with 0.358 crack density, assuming cracks parallel to orthoclase cleavage. Equal area projections, linear scale contours. Minimum and maximum velocity values are shown in lower left corner, and corresponding coefficient of anisotropy in lower right corner. Dark blue color corresponds to low velocity values, red - to high values.

Journal of Geophysical Research: Solid Earth

600°C. It may be related to the α-β transition in quartz that occurs at 573°C and is preceded by fast thermal expansion of α-quartz (Kihara, 1990). Due to significant quartz content, this also causes a visible increase in thermal expansion of the granite (Plevova et al., 2016). The thermal expansion mismatch and the volume effect during the phase transformation produce new cracks on the boundaries between quartz and other minerals, especially feldspars that are the major WG component. The new system of cracks is much less compliant than cracks created at temperatures below 573°C. US measurements indicate that cracks created by heating up to 600°C are not fully closed even at 400 MPa, while cracks created by temperatures below 600°C are closed around 50–100 MPa (Figures 11 and 12).

WG is only weakly anisotropic, thus it is possible to use equations for crack closure stress estimation de-veloped for isotropic rocks (Mavko et  al.,  2009; Walsh,  1965). Considering elastic properties of a “crack free” WG (Table 3), it is possible to calculate that the closing stress for cracks with aspect ratio of 0.001 is ≈ 70 MPa, while cracks closing at pressure over 400 MPa have aspect ratio 0.006 or higher. We also note that the WG600 sample is characterized by lower amplitudes of registered ultrasonic signals, which means higher ultrasonic signal damping at lower values of hydrostatic pressure indicating opening of additional cracks due to the α-β transition. It is possible to conclude that thermal expansion of α-quartz near the struc-tural transition leads to formation of higher aspect ratio cracks in WG.

Additionally, there is an increase of the elastic anisotropy with the increase of the preheating tempera-ture, which is due to preferred orientation of thermal cracks. The preferred orientation of cracks is in good agreement with the orientation of grains (Figure 6b), and as seen in BSE images (Figure 5), rising tem-perature introduces cracks mostly on grain boundaries. This is consistent with observations of stress- and temperature-induced cracks in WG that documented formation of cracks along elongated grains (Moore & Lockner, 1995), especially at boundaries between feldspars and quartz (Fredrich & Wong, 1986). Further, propagation of cracks is facilitated by cleavage of biotite and feldspars (e.g., Moore & Lockner, 1995; Nasseri et al., 2007; Wong, 1982), and stress-induced cracks are formed more readily in feldspars (Moore & Lock-ner, 1995). Distributions of Vdif values (Figure 14) also support the hypothesis that crack preferred orienta-tion is similar to alignment of cleavage planes of feldspars and biotite. The orientation of elongated quartz grains is also similar to feldspars and biotite grains (Figure 6a). Thus, it is reasonable that high aspect ratio cracks formed during heating over quartz α-β transition temperature would form along boundaries between these grains and would have a similar preferred orientation.

We can conclude that the thermally induced microcracks in WG should be mostly oriented along cleavage planes of feldspars and biotite, or parallel to elongated grains, which is also related to cleavage, as discussed in a previous paragraph. There may be also cracks created by pressure cycling over 100 MPa (Tapponnier & Brace, 1976), but the symmetry of VP distributions of WG at 5 MPa pressure is practically independent on preheating temperature and number of pressure cycles (Figure 10). This suggests that the preferred orienta-tion of formed thermal cracks is more or less the same. In addition, it does not depend on the crack length (scale-independent). A similar pattern was observed in Oshima granite (Takemura et al. 2003). But as cracks are closing, the measured P-wave velocity distribution markedly changes (Figure 13). Maximum P-wave velocity direction at low pressures is near minimum velocity direction at high pressure and vice versa.

To study this effect numerically, we used the GeoMixSelf method to model elastic properties of WG with low crack porosities, assuming, that the orientation distribution of these cracks follows the orientation distribution of (001) plagioclase cleavage planes. Cracks may be introduced into GeoMixSelf with shapes approximated by ellipsoids. Their aspect ratios, preferred orientation and volume fraction (crack porosity) are used to calculate effective bulk elastic properties. But for low aspect ratio cracks, the crack porosity parameter is irrelevant for linear elastic properties, and crack density should be used instead (Kachanov & Mishakin, 2019; Vernik, 2016). In the general case, there is no direct relation of crack density and crack porosity (Kachanov & Mishakin, 2019). In the GeoMixSelf model we constrain all the cracks to the same oblate spheroidal shape with axes ratio {1:1:ξ}, and thus, crack density ρ may be simply derived from crack porosity φ as:

3 / 4 · (2)

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In modeling, cracks with ξ = 0.01 were added to the “crack free” elastic model of the WG100 sample. Different crack porosities were tested up to crack porosity value of 0.02, and, according to Equation 2, up to crack density ρ ≈ 0.477. These crack density and crack porosity values are rea-sonable for thermally treated granites (e.g., Griffiths et al., 2017; Wang et al., 2013). The same procedure was also tested for cracks with ξ = 0.001 up to the same crack density value ρ ≈ 0.477. As expected, bulk stiffness tensor components of WG models with different ξ values, but equal crack densities are practically the same.

P-wave velocity distributions for WG models with ρ  =  0.119 and ρ = 0.358 are plotted in Figures 17c and 17d, correspondingly. It is obvi-ous that cracks act against the elastic anisotropy of the “crack free” WG, and effectively “rotate” and smooth the VP distribution. The increase of crack density naturally decreases elastic coefficients and P-wave ve-locities, which reproduces the effect of microcracks opening at lower pressure levels. The interesting effect is that with the increase of crack density in the model, the coefficient of anisotropy A first decreases, and at higher crack densities, it starts to increase gradually (Figure 18).

The elastic anisotropy decrease at low crack density was not observed experimentally, because at high pressures US measurements were made with large pressure increments (100 MPa). But otherwise, our simple model correctly reproduces observed changes of measured P-wave velocity patterns. At lower pressures, the WG elastic anisotropy is defined by oriented microcracks produced mostly during temper-ature cycling.

6. ConclusionsWG samples that have undergone different thermal treatments were investigated by a number of methods: US at increased pressures, time-of-flight neutron diffraction, optical and electron micros-copy. It was shown that WG is a homogenous rock, mainly consisting of four major minerals: plagi-oclase, orthoclase, quartz and biotite. Intrinsic elastic anisotropy of WG is caused by the crystallo-graphic textures of feldspars. Thermal treatment of WG leads to formation of microcracks that reduce elastic wave velocities. Higher preheating temperatures result in higher crack densities, as well as in the appearance of longer cracks. Thermal microcracks induced by preheating up to 400°C are gener-ally soft and are almost completely closed at pressures ≈ 50 MPa, while preheating to 600°C results in formation of some stiffer microcracks that may be related to the rapid thermal expansion of quartz below the α-β transition temperature. The orientation of cracks can be linked to the microstructure and texture of plagioclase and biotite. The majority of thermally induced microcracks are parallel to cleavage planes of biotite and feldspars. Numerical modeling shows that oriented cracks strongly in-fluence the WG elastic anisotropy. It is remarkable that cracks, produced by an isotropic action—tem-perature—invert the rock elastic anisotropy, practically exchanging minimum and maximum velocity directions, and that this effect may be reversed by another isotropic action—pressure, which closes oriented cracks. We can conclude that ND and US results exhibit more or less identical symmetries. As a result, WG studied here exhibits two types of elastic anisotropy, expressed by microcracks at low pressure and by crystal preferred orientation at high pressure. For natural conditions, the extrapo-lation of the experimental results obtained from WG testing implicate: (i) the increasing lithostatic pressure may lead to the change in orientation of anisotropy; (ii) when the rock is not extremely fractured, the majority of cracks close at 50 MPa (lithostatic pressure at approximately 2 km), (iii) for extremely fractured rock, the crack closing pressure may be above 200 MPa, that corresponds to the burial depths greater than 8 km. Consequently, P-wave velocity measurements could be a good indi-cator of the granite deterioration level. It can be used as a nondestructive technique to assess changes in anisotropy with temperature and pressure.

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Figure 18. Coefficient of anisotropy of GeoMixSelf models of WG with different crack densities.

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Data Availability StatementAll data for this study are available in the supplement files (Lokajicek,  2020) - https://doi.org/10.5061/ dryad.9s4mw6mf1, DRYAD repository, as vP velocities: vP_export.xls and preferred orientations derived from ND experiments: 8 files of WG100-biotite.STD, WG100-oligoclase.STD, WG100-orthoclase.STD, WG100-quartz.STD, WG600-biotite.STD, WG600-oligoclase.STD, WG600-orthoclase.STD, WG600-quartz.STD, suitable, for example, for Beartex package. Please do not hesitate to contact us for any additional infor-mation. Tomas Lokajicek:tl@gli.cas.cz for original ultrasonic waveforms data. Roman Vasin: olddragon@mail.ru for original diffraction spectra.

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AcknowledgmentThis study was partially supported by the Czech Science Foundation research grants 16-03950S, 18-08826S, and by the Czech Academy of Sciences project RVO 67985831. Microanalytical meth-ods at ICP has been supported by the projects of the Ministry of the Interior of the Czech Republic VI20152020035. Authors appreciate the access to the SKAT diffractometer at FLNP JINR. The project was partially supported by the JINR theme No. 04-4-1121-2015/2020. H. R Wenk is appreciative for support from NSF (EAR 1343908) and DOE (DE-FG02-05ER15637). We thank two anonymous referees for their valuable comments that helped to improve the manuscript.

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