Gyrotropy and Anisotropy of Rocks: Similarities and Differences

GYROTROPY AND ANISOTROPY OF ROCKS: SIMILARITIES AND DIFFERENCES T. I. CHICHININA and I. R. OBOLENTSEVA GYROTROPIE ET ANISOTROPIE DES ROCHES : SIMILITUDES ET DIFFƒRENCES Institute of Geophysics of Russian Academy of Sciences Les principales caractŽristiques de la propagation des ondes dans (Siberian Branch)1 les milieux gyrotropes sont comparŽes avec la propagation des ondes dans les milieux azimutalement anisotropes. Les rŽsultats dÕune modŽlisation numŽrique sont prŽsentŽs pour trois mod•les caractŽristiques dÕexploration sismique. Les deux premiers mod•les sont des milieux anisotropes (de symŽtrie orthorhombique, groupe 2m) avec et sans gyrotropie. Le troisi•me mod•le est un milieu gyrotrope transverse isotrope avec un axe de symŽtrie vertical. Ces calculs ont ŽtŽ rŽalisŽs pour la propagation des ondes transversales le long de lÕaxe de symŽtrie vertical. Pour des trajets sismiques suffisamment courts (pour nos mod•les, moins de 400 m), les sismogrammes ˆ deux composantes (x, y) sont similaires pour les trois mod•les. Pour des trajets plus longs, la forme et la durŽe du signal diff•rent sensiblement pour les mod•les 1 et 3. Ceci a pour but de montrer (ˆ lÕaide des donnŽes expŽrimentales et dÕun micromod•le) que la gyrotropie dans les roches existe, ou, tout au moins, peut exister. GYROTROPY AND ANISOTROPY OF ROCKS: SIMILARITIES AND DIFFERENCES The main features of wave propagation in gyrotropic media are compared with wave propagation in anisotropic media. The results of numerical modelling are presented for three typical seismic exploration models. The first two models are azimuthally anisotropic media (of orthorombic symmetry system, group 2m) without and with gyration. The third model is a gyrotropic transversely isotropic medium with a vertical symmetry axis. The computations have been made for propagation of shear waves along the vertical symmetry axis. For sufficiently short wave paths (in our models less than 400 m) the two-component (x, y) seismograms are similar for all three models. For longer paths both signal shape and signal duration for the first and the third model differ noticeably. Some evidence (experimental data and a micromodel) is given to show that the gyrotropy of rocks does exist or, at least, can exist. GIROTROPêA Y ANISOTROPêA DE LAS ROCAS : SEMEJANZAS Y DIFERENCIAS Las principales caracter’sticas de la propagaci—n de ondas en los medios girotr—picos se comparan con la propagaci—n de ondas en medios anisotr—picos. Los resultados de la construcci—n de (1) Academician Koptug pr., 3, Novosibirsk, 630090 - Russia modelos numŽricos se presentan a partir de tres modelos t’picos REVUE DE L’INSTITUT FRANÇAIS DU PÉTROLE VOL. 53, N° 5, SEPTEMBRE-OCTOBRE 1998 655 GYROTROPY AND ANISOTROPY OF ROCKS: SIMILARITIES AND DIFFERENCES de exploraci—n s’smica. Los dos primeros modelos son medios INTRODUCTION azimutalmente anisotr—picos (de sistemas de simetr’a ortorr—mbica, grupo 2m), sin y con giro. El tercer modelo es un The polarization of shear waves is now a common medio girotr—pico transversalmente isotr—pico con un eje de simetr’a vertical. Los c‡lculos han sido hechos para la propagaci—n tool in seismic prospecting for cracked hydrocarbon de ondas de cizallamiento a lo largo del eje de simetr’a vertical. reservoirs. While in isotropic media the velocity of Para trayectos de onda suficientemente cortos (en nuestro shear wave Vs is independent of polarization, in modelo, menos de 400 m), los dos sismogramas componentes anisotropic media there are two shear waves (S1, S2) (x, y) son similares en los tres modelos. Para trayectorias m‡s with mutually perpendicular polarization and generally largas, tanto la forma de la se–al como su duraci—n difieren different velocity. Azimuthal anisotropy of at least of notablemente entre el primero y el tercer modelo. monoclinic symmetry is caused by vertically oriented Se entregan algunas evidencias (datos experimentales y un cracks. In such media one displacement vector u lies micromodelo) para mostrar que existe, o que al menos puede s1 existir, la girotrop’a de las rocas. in the symmetry plane (parallel to the cracks) and the u other displacement vector s2 is normal to this plane. If there are two systems of mutually perpendicular vertical cracks, the medium is orthorhombic and the displacement vectors lie in the two symmetry planes. For general acquisition geometries (i.e., the source- receiver direction does not coincide with the normal to a plane of symmetry or with one of the two normals if they are two), each of the three components (x, y, z) is the superposition of S1 and S2 waves propagating with velocitiesV andV . In this paper, we show that a s1 s2 similar two-component seismogram may correspond to a gyrotropic wave propagation. The concept of seismic gyrotropy has recently been introduced [12-15]. Here we present briefly the main features of elastic wave propagation in an anisotropic gyrotropic geological medium and compare them with the features of wave propagation in purely anisotropic media. 1 ON PHENOMENOLOGICAL THEORY OF GYROTROPY Optical gyrotropy (see, for example, [9, 5, 8 and 7]) is known since 1811 when F. Arago observed a rotation of the polarization plane of light propagating along an optical axis of quartz. By now the publications on optical gyrotropy run into the hundreds, and the optical gyrotropy is a powerful instrument in studying the fine structure of matter in many fields, e.g., physics of crystals, stereochemistry, biophysics, and biochemistry. Acoustical gyrotropy [1, 19-21, 4 and 23], the analog of optical gyrotropy, has been investigated much later, since the sixties of our century; the first were, to our knowledge, [1, 21, 19 and 20] and some others ([24, 25 and 6]), see [10]. Up to this point, the publications in acoustic gyrotropy are not as numerous as they are in optical gyrotropy. Nevertheless, the phenomenon REVUE DE L’INSTITUT FRANÇAIS DU PÉTROLE VOL. 53, N° 5, SEPTEMBRE-OCTOBRE 1998 656 GYROTROPY AND ANISOTROPY OF ROCKS: SIMILARITIES AND DIFFERENCES attracts much attention and is interesting both from the whereas the inner symmetry of the tensor c is: physical viewpoint and for its practical application. cijkl = cjikl = cijlk = cjilk and cijkl = cklij Seismic gyrotropy can be regarded [12-15] as an extension of acoustic gyrotropy to geological media. The dispersive term ibijklm (w) km in Eq. (2) Practically all the features of shear-wave polarizations describes the effects which are called gyrotropy. observed in many seismic field experiments can be Hooke's law in a gyrotropic medium is: explained by a combination of two concepts: anisotropy ¶e cb kl and gyrotropy. seij =+ijkl kl ijklm (3) ¶xn For electromagnetic and acoustic waves, gyrotropy of crystals is known as an exhibition of first-order where ekl = (1/2) (¶uk/¶xi + ¶ui/¶xk). The equations of 2 2 spatial dispersion, i.e., non-local response of a crystal to motion ¶sij/¶xj = r¶ ui/¶t become: a wave. Non-locality implies that in material equations ¶23u ¶ u ¶2u D = E E, s = Ce that relate the electric displacement D c k b k i ,,,.i 123 ijkl +==ijklm r 2 (4) to the electric field E and the stress s to the strain e, ¶¶xxj l ¶¶¶xxxj l m ¶t respectively, D and s at a given point depend, For plane waves u (r, t) = u A exp [i (nr/V – t)] respectively, on the E and e not only at that point, but at 0 w neighbouring points as well. This dependence is propagating with phase velocity V in the direction of the wave normal n and polarized along a unit vector A, expressed as a functional dependence E (w, k) for Equations (4) are: electromagnetic waves and C (w, k) for the acoustical analog. Here w is circular frequency, and k is the wave cnn iVb-12 nnnA VAi , 123,,(5) vector of a plane harmonic wave with wave normal n []ijkl j l + wrijklm j l m k ==i and phase velocity V (k = w n/V). The above functions E (w, k) and C (w, k), are general in the sense that they or account for both time frequency and spatial dispersion. VA2 0123, k ,, , For seismic waves, we also accept, by analogy with []LDik +-ik dik k == (6) acoustic waves, that C = C (w, k). This assumption has been verified by numerous experimental data on -1 -1 -1 where Lik = cijkl r nj nl, (Dik = iw V r bijklm nj ni nm). polarizations of shear waves. As for causes of gyrotropy The Equation (6) is the Christoffel equations for an on a microlevel, one concrete micromodel of rock anisotropic gyrotropic medium, and the tensor (L + D) imitating sandy deposits of dissymmetric is the appropriate Christoffel tensor. microstructure will be demonstrated below. The Equation (6) is a system of uniform equations If the magnitude of the non-local part of the elastic which has non-trivial solution if its determinant is equal stiffness C is small, the stiffness may be expanded in a to zero: power series in k: 2 (7) det(LDik +- ik V dik ) = 0 Cijkl (w,k )=+ cijkl (w) ibijklm (w) km + dijklmn (w) kmn k + ... (1) 2 In further considerations, we shall truncate this The determinant det (Lik + Dik – V dik) = 0 is a 2 2 expansion after the second term: polynomial of the third degree in V if the terms LikD kl are disregarded. Equation (7) has three (positive) roots, C (w, k) = c (w) + ib (w)k (2) the eigenvalues of the matrix. Then the solution of ijkl ijkl ijklm m Equation (6) are three vectors A for three values (three c =(c ) is the well known tensor of elastic stiffnesses eigenvalues and three eigenvectors of the matrix ijkl ( + – V2 ), i.e., there are three waves as in an and b =(b ) is the gyration tensor.

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