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Seismic Velocity Anisotropy in Mica-Rich Rocks: an Inclusion Model

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Seismic Velocity Anisotropy in Mica-Rich Rocks: an Inclusion Model Geophys. J. Int. (2001) 145, 19–32 Seismic velocity anisotropy in mica-rich rocks: an inclusion model O. Nishizawa1 and T. Yoshino2 1 Geological Survey of Japan, 1-1-3 Higashi, Tsukuba Ibaraki 305-8567, Japan. E-mail: [email protected] 2 Department of Civil and Environmental Engineering, Toyo University, 2100 Kujirai Kawagoe, Saitama 350-8585, Japan Accepted 2000 September 11. Received 2000 September 7; in original form 2000 March 7 SUMMARY We calculated seismic wave velocity anisotropy caused by the preferred orientation of mica minerals by using the differential effective medium method (DEM). Spheroidal Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 biotite crystals with their c-axes coinciding with the symmetry axis of the spheroid are embedded in an isotropic matrix up to a volume ratio of 30 per cent. All crystals are aligned with their c-axes parallel to the symmetry axis of the effective homogeneous medium, which shows transverse isotropy. The effect of crystal shape on anisotropy was studied by changing the aspect ratio (the ratio between the minor axis and the major axis of the spheroid) from 0.01 (flat spheroid) to 1 (sphere). The S-velocity anisotropy becomes large as the crystal shape becomes flat, whereas the P-velocity anisotropy shows only small changes with changes of the crystal shape. In particular, biotite generates large S-wave anisotropy, and the anisotropy becomes stronger as the aspect ratio of the biotite crystal becomes smaller. When the volume ratio of the mica mineral is large, the P-wave phase velocity surface shows considerable deviation from the ellipse, and the SV-wave phase velocity surface forms a large bulge and crosses the SH-wave phase velocity surface (singularity) in the plane including the symmetry axis. These results show an interesting contrast when compared with the effect of crack or pore shape on seismic velocity anisotropy: crack (or pore) shape affects the P velocity more than the S velocity. We also calculated Thomsen’s anisotropic parameters, e, c and d as functions of the crystal aspect ratio and the mica volume ratio. Key words: anisotropy, composite media, mica-rich rocks, seismic wave propagation. & Cara 1991; Barrruol & Mainprice 1993), (2) the preferred 1 INTRODUCTION orientation of cracks and fractures (Crampin 1984), and Seismic anisotropy in the crust has been extensively studied in (3) layered structures of different elastic materials (Backus 1962; geophysics and exploration geophysics (Crampin 1987; Leary Helbig 1984). Among the three origins of crustal anisotropy, et al. 1990; Babusˇka & Cara 1991; Thomsen 1986). Seismic the effects of cracks and preferred orientations of minerals have anisotropy is important in reflection seismology because the been studied by many people because there are many variations seismic wavefront affects conventional reflection data process- of anisotropy associated with those origins. Distributions of ing such as moveout corrections (normal moveout, NMO, and crack orientation and mineral orientation control the rock dipping moveout, DMO) and amplitude variation with offset anisotropy, and the crack shape also controls the anisotropy. (AVO) (Thomsen 1988; Tsvankin 1997a). Shear wave splitting Aligned thin cracks produce strong anisotropy (Anderson et al. seen in seismograms of local earthquakes has been reported 1974; Hudson 1981; Nishizawa 1982; Douma 1988). A similar at many observation sites and used for monitoring tectonic effect may occur in rocks containing aligned thin crystals that activity and for the classification of crustal structures (Crampin have strong anisotropy. It is also interesting to compare the et al. 1984; Booth et al. 1990; Crampin et al. 1990; Kaneshima anisotropy caused by the orientation of minerals and aligned 1990). Shear wave splitting is also used for interpreting vertical cracks. seismic profiling data or for characterizing rock formations by Recently, Takanashi et al. (2000) found an interesting velocity well logging (Liu et al. 1989; Douma & Crampin 1990). anisotropy in biotite schist from Hokkaido, Japan. The aniso- A number of experimental measurements and theoretical tropy is characterized by a remarkable bulge for one of the studies have been carried out in order to understand seismic S-wave phase velocity surfaces and a non-elliptic shape of the anisotropy in rocks (Kern & Schenk 1985). The plausible P-wave phase velocity surface in the plane including the direction origins of seismic anisotropy of the crust are (1) the preferred of lineation. This may result from the preferred orientation of orientation of anisotropic rock-forming minerals (e.g. Babusˇka biotite grains because the rock shows a strong concentration # 2001 RAS 19 20 O. Nishizawa and T. Yoshino of the biotite c-axes and a large volume ratio of biotite—over of Eshelby’s tensor following the formulations given by Mura 30 per cent. To study the origin of anisotropy in mica-rich (1968) and Mura & Mori (1976). Results were explicitly shown rocks, we modelled the rocks as composite solids containing in Lin & Mura (1973) and quoted in Nishizawa (1982), but here inclusions of mica-group minerals in an initially isotropic matrix. we describe the process for deriving the resultant formulations. The differential effective medium method (DEM) was employed to calculate elastic constants, where we assume that the composite is an effective medium with transverse isotropy due to the mica-mineral inclusions. This is a slight modification 2.2 Fictitious stress-free strain and Eshelby’s tensor of the method of Nishizawa (1982) for the case that the inclusion We assume a solid composite including spheroids of which the is an anisotropic single crystal. Mineral shapes are assumed to elastic properties are different from the surrounding solid. The be oblate spheroids and the effect of the aspect ratio (the ratio spheroid is called the ‘inclusion’ and the surrounding solid is between the minor and major axes of the spheroid) on elastic called the ‘matrix’. Our goal is to obtain elastic properties of anisotropy was investigated. The effect of oriented cracks on the composite material when we regard the composite as an anisotropy in the mica-bearing rocks was also studied. effective homogeneous medium. This assumption is valid only Here we use the notation P, SH and SV waves instead of when the medium is under a static equilibrium condition, or qP qS qS q the general expressions , 1 and 2 waves, where stands when the static approximation is applicable. The condition is Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 for quasi (Winterstein 1990). Since the anisotropy considered is generally satisfied for field seismic observations since the seismic only transverse, P, SV and SH waves are clearly distinguished. wavelength is much larger than the grain sizes of rock-forming Thus, the abbreviated notation does not cause any confusion. minerals. In laboratory experiments, with the shorter wave- length, the assumption of an effective homogeneous medium may not be strictly valid, but the inclusion model is still a useful 2 COMPOSITE MEDIUM THEORY approximation except for large grain-size rocks. When the composite medium is under stress, its elastic 2.1 Eshelby’s theory of composite media energy differs from that of an inclusion-free matrix material that is under the same stress or strain condition. If we can calcu- We consider a composite material that contains inclusions late the elastic energy of the effective homogeneous medium of mineral grains inside a homogeneous material. In such a of the composite for the same strain or stress that appears in material, stress and strain are not uniform and will fluctuate the inclusion-free matrix, we can obtain the elastic constants randomly from their mean values averaged over the material. (or compliance constants) of the effective homogeneous medium The stress and strain fluctuations can be described by their by differentiating the elastic energy with respect to strain autocorrelation functions, which control the elastic properties (or stress). Therefore, calculation of the elastic energy change of a composite material (Kro¨ner 1967). However, in the present in the composite material is a key to obtaining the elastic treatment we are not concerned about autocorrelation functions properties of the effective homogeneous medium. of stress and strain inside the material, and consider a perfectly Stress and strain inside the composite material are disturbed disordered fluctuation. We regard the composite material as an by an inclusion and the disturbance causes an energy change equivalent homogeneous medium and assume homogeneous from the inclusion-free state. In the following, for simplicity, stress and strain fields in that medium. The homogeneous stress we first assume that stress and strain inside the inclusion are and strain correspond to the effective elastic constants, which A A uniform. sij and eij denote the stress and strain of the inclusion- represent the overall properties of the equivalent homogeneous C C free medium, respectively, and sij and eij , respectively, denote medium. the stress and strain disturbance of the composite material To calculate the effective elastic constants, one should first caused by the inclusion. For the inclusion-free state, we have estimate the energy change between the inclusion-free material and the composite material; the stress–strain relation can then A~ 0 A be derived by the differentiation of the energy with respect to pij cijkl ekl ,(1) homogeneous stress or strain (Eshelby 1957). Thus, we are only 0 concerned about the change in elastic energy of the composite where cijkl are the elastic constants of the inclusion-free matrix A C material and we calculate the energy as a function of the elastic material. In the inclusion, stress and strain are given by sij +sij A C constants of the matrix, the inclusion mineral and the volume and eij +eij , respectively, and ratio of the inclusion.
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