Theoretical Acoustic Anisotropy in Rocks

Transcription

1 Lecture Note Theoretical Acoutic Aniotropy in Rock Rune M. Holt NTNU, Trondheim, 000

2 Table of content 1. Definition and origin of aniotropy Formal decription of aniotropy Aniotropy in layered media (Backu-model) Aniotropy in fractured media (Schoenberg & Douma model) Iolated parallel crack (Hudon theory)...15 Reference and Suggeted reading...18

3 3 1. Definition and origin of aniotropy. By aniotropy we mean that certain material propertie (uch a ound velocitie, permeability, electrical reitivity etc.) are directionally dependant. The origin of aniotropy i heterogeneity characteried by ome degree of tructural order at a length cale much horter than that of the probe (e.g. the wavelength). A an example, the propertie of ingle crytal exhibit macrocopic aniotropy reulting from the ordered lattice of atom at the microcale. learly, at the atomic length cale, crytal are heterogeneou. In rock, the main caue of aniotropy can be claified a follow: Litholoqical (or intrinic) aniotropy: Thi i aniotropy a a reult of the rock texture. It can be caued by lamination or bedding. For example, when a river depoit and grain, there i a tendency for more coare grain to ettle when the flow rate i high, and more fine grain to ettle at low flow rate. Seaonal change in the river flow thu reult in a microlamination of depoited and. Lithological aniotropy may alo be caued by lamination at larger length cale, e.g. meter thick. Individually iotropic bed will contribute to eimic aniotropy, ince the eimic wavelength i 100 m or o. When grain that are nonpherical are depoited, they will tend to depoit with their flat ide preferentially horizontal and facing each other. Inherent aniotropy in the mineral crytallite may alo contribute to the overall lithological aniotropy of rock. The latter mechanim lead to a trong aniotropy in hale, which contain oriented, heet like clay mineral. Figure 1 illutrate ome ource of lithological aniotropy.

4 4 Figure 1: Different lithological factor that may affect aniotropy. Stre-induced (extrinic) aniotropy: External aniotropic tree acting on a material lead to aniotropy for two reaon. Firt, the tre aniotropy lead to an effective aniotropy through the econd order term in the train tenor. Thi o-called direct tre-induced aniotropy i very mall. However, if tree are ufficiently large to generate crack, or to cloe pre-exiting crack, a uually very trong tre- (or crack-) induced aniotropy i induced. Example of tre-induced aniotropy i een in laboratory tet, where velocity aniotropy increae trongly prior to rock failure. It i alo oberved in core pecimen retrieved from the Earth. Then the aniotropy i a reult of the tre releae from in itu to atmopheric condition. Such aniotropy can be removed (or at leat trongly reduced) by reloading the core, which form the bai for a core baed technique to ae in itu tre direction (and potentially alo magnitude). There i evidence for increaed aniotropy prior to Earthquake, and monitoring of eimic aniotropy may therefore be a poible way of predicting earthquake. The poibility for detecting fractured reervoir i an intereting and important application of tre-induced aniotropy.

5 5. Formal decription of aniotropy. Phyical propertie are uually repone function characteriing the effect (e.g. train) of an action (e.g. a tre). The repone function in a tre - train relationhip i an elatic tiffne tenor, relating two econd rank tenor. The tiffne tenor therefore mut be a 4th rank tenor, i.e. Hooke' law may be written σ ε ij ijkl kl kl, (.1) It i convenient here to ue the o-called Eintein convention, which mean that ummation over repeated indice i undertood; i.e. Eq. (.1) read σ ε ij ijkl kl (.) Similarly, the permeability tenor i nd rank, ince it relate two vector (1 t rank tenor) (the flow rate and the preure gradient). The ymmetry of thee tenor are uniquely given by the material ymmetry, which mean that all 4 th rank repone function have the ame ymmetry in the ame material, independent of the phyical propertie that they decribe. Since there are 3 independent patial direction, i, j, k, and 1 may each take 3 different value (1, and 3; equivalent to x, y and z in a arteian coordinate ytem). Thu, the elatic tiffne tenor ha a total of component. Thee are not all independent. Since i and j and alo k and 1 can be interchanged in Eq. (.1) and furthermore ij and kl may be interchanged (may be hown by energy conideration), the total number of independent component in the mot general cae (triclinic ymmetry) i reduced to 1. Thi permit u to repreent ijkl, by a ymmetrical 6x6 matrix IJ where ij I and kl J according to the Voigt notation: 11 1, ; 33 3; 3 4; 13 5; 1 6. Material ymmetry further reduce the number of component. learly, combination of lithological and tre induced aniotropy, or change in tre regime throughout tectonic hitory may lead to complex ymmetrie. However, for practical purpoe uch ymmetrie are of little interet ince a large number of parameter i required in order to apply the theory. If the ymmetry i relatively complex, one hould try to implify and decribe the material by an approximate higher ymmetry. We will therefore focu only on ymmetrie higher than or equal to the orthorhombic. Orthorhombic mean that the x-, y- and z-direction are different, but that mirror ymmetry exit about the origin of the coordinate ytem along each principal direction. Orthorhombic ymmetry may be expected e.g. from a 3D tre tate with 3 independent principal tree. Becaue of invariance of the tre - train relation under ymmetry operation, the following 9 element of the tiffne tenor are non-zero:

6 In edimentary rock, the implet non-iotropic ymmetry i the tranverely iotropic cae, when there i a unique ymmetry axi (ay the z-axi), and everything i iotropic in the xy-plane. Thu, indice 1 and are equivalent in the above cheme, leading to the following 5 independent element of the tiffne tenor in thi cae: ; For an iotropic olid, there are only independent element; e.g. the Lamé coefficient λ and G. Then λ +G; 1 13 λ and G. The wave equation for an aniotropic olid i given by: u σ 1 u u u ρ ( ) i ij k l k ijkl + ijkl t xj xj xl xk xj xl (.3) The latter identity occur becaue k and 1 may be interchanged in ijkl and ince ummation i over k and 1. Aume a propagating wave olution: u i u e ω 0 j( t qixi) i (.4) Here ω i the angular frequency, and q i the wavevector. Then, inerting for the directional coine n i qi q (.5) Eq. (.3) may be rewritten a

7 7 0 ω 0 ijklnjnu l k ρ ui 0 q 0 ijklnjnl ρv δik ùk 0 (.6) δ ik i the Kronecker delta ( 1 when i k; 0 otherwie). The phae velocity v ω /q i inerted. Eq. (.6) i known a the hritophel equation. Wave olution are non-trivial olution of thi equation, i.e. when the determinant of the matrix on the left ide i zero. For the Tranverely Iotropic (TI) medium decribed above, the hritophel equation may be written: 0 11n1 + 66n + 44n3 ρv ( 11 66) nn 1 ( ) nn 1 3 u 1 0 ( 11 66) nn 1 66n1 + 11n + 44n3 ρv ( ) nn3 u 0 0 ( 13 44) nn 1 3 ( 13 44) nn3 44( n1 n ) 33n3 ρv u (.7) For wave propagation parallel to the ymmetry axi n 3 1, n 1 n 0. In thi cae, Eq. (.7) implifie to 0 44 ρv 0 0 u ρv 0 u ρv u3 (.8) Thi equation ha three olution. One i a degenerate olution correponding to v v zx ; zy ; 44 ρ (u x or y; q z) (.9) By ubcript ;zx we indicate that thi i an S-wave, propagating in the z - direction, and with polariation in the x-direction. The third olution i a P-wave (i.e. propagation and polariation in the z-direction): v pz ; 33 ρ (u z; q z) (.10) Wave propagation in the ymmetry plane can be conidered by chooing e.g. n 1 1, and n n 3 0 (any direction in the xy-plane may be conidered, ince they are all equivalent). In thi cae we find three different olution: v px ; 11 ρ (u x; q x) (.11)

8 8 v xz ; ρ 44 (u z; q x) (.1) v xy ; 66 ρ (u y; q x) (.13) The fact that v ;zx v ;xz i a conequence of the underlying ymmetry of the problem. The difference between v ;xy and v ;xz repreent a phenomenon known a hear wave plitting (or birefringence), i.e. that two S-wave propagating the ame path but with different polariation will travel at different peed. If the z-axi i parallel to the vertical direction, then the,xz mode i called an SV -wave, while the,xy mode i termed SH. Thi ymmetry i then referred to a azimuthal aniotropy, or TIV. The angular dependence when conidering a wave propagating at an angle θ to the z-axi i evaluated by inerting n 1 inθ, n 0, and n 3 coθ. The three olution are in thi cae: v in θ + ρ co θ (.14) v 11 in θ + 33 co θ + 44 ± ρ (.15) where ( )in θ ( )co θ + 4[ + ] in θ co θ (.16) Eq. (.14) repreent a hear wave propagating in the xz-plane with a polariation in the y-direction ("SH"-wave), wherea Eq. (.15) give two wave with propagation and polariation direction in the xz-plane. Only along the ymmetry direction do thee wave become pure P- or S-wave. In general, they are quai-p (the + olution) and quai-s (the - olution) wave. Notice in particular that in non-ymmetry direction the phae velocity and the group velocity will have different propagation direction. Thomen introduced three o-called aniotropy parameter, which all are zero in the iotropic cae. Thee are: ε 33 γ ( + ) ( ) δ ( ) (.17) (.18) (.19)

9 9 For cae of weak aniotropy (ε,δ and γ mall), Eq. (.14) and (.15) may be implified to v (.0) 4 p( θ ) vp(0) 1+ δ in θco θ + εin θ v v v (0) v + p (0)[1 ( ε δ)in θco θ] v (0) (.1) v ( ) (0) 1 in (.) h θ v + γ θ One may ee that ε repreent the P-wave aniotropy, γ repreent the S-wave aniotropy, while (δ ε) repreent the deviation from elliptical aniotropy (elliptical aniotropy mean that P-wave-front emanating from a point ource are elliptical).

10 10 3. Aniotropy in layered media (Backu-model). Let u aume that we have a layered olid medium compoed of two et of alternating plane and parallel layer. For implicity, we hall perform our calculation by auming that the two layer material in themelve are iotropic. The layer plane a illutrated in Figure below have normal in the z-direction. Figure : Layer in a periodically layered medium. Layer type 1 ha Lamé coefficient λ 1 and G 1, while layer type have λ and G. The concentration of each type i φ j ; i.e. uch that here φ 1 + φ 1. Furthermore, the total denity i ρ φ 1 ρ 1 + φ ρ. onider firt the cae where a tre σ zz i applied in the z-direction in uch a way that zero lateral train reult. The total reulting train ε zz i then related to the tiffne 33 of the effective medium; i.e. σ zz ε 33 zz (3.1) The two type of layer are both expoed to the ame tre, o that the reulting train alo can be calculated from the Reu average (io-tre model): 1 φ1 φ + λ + G λ + G ( λ1+ G1)( λ + G) φ ( λ + G ) + φ ( λ + G ) (3.) 11 can be found from σ xx ε 11 xx (3.3) by impoing a tre σ xx o that the total lateral train ε yy ε zz 0. It i traightforward to ee that the train in the y-direction mut be zero in both layer. The train in the z-direction will however be different in the two layer, with a total train given by the Reu average; i.e.

11 11 ε φε + φε 0; zz 1 zz1 zz σ σ σ zz zz1 zz (3.4) The train in the x-direction i the ame in both layer, while the tree are different, a for the Voigt (io-train) average: ε xx εxx1 εxx (3.5) σ φσ + φσ xx 1 xx1 xx Uing thee equation, one find: ( G1 G)( λ1+ G1 λ G) 1+ 4φφ ( λ1+ G1)( λ + G) (3.6) By imilar conideration (ee e.g. White' book "Underground Sound") the three remaining tiffne tenor element can be found: 44 1 φ1 φ + G G 1 (3.7) φ G + φ G (3.8) 13 φλ 1 1 φλ + λ + G λ + G φ1 φ + λ + G λ + G 1 1 (3.9) Thi model can be generalied to the cae of everal different layer, which in themelve may be aniotropic. The reult may be written: (3.10) (3.11) (3.1) (3.13) (3.14)

12 1 Hence; it alo follow that (3.15) Notice that the Backu theory aume the wavelength to be much larger than the layer thickne. In the oppoite cae, when the wavelength i much horter than the layer, the wave velocitie perpendicular to the layer are found by adding the travel time in each layer (time average), wherea wave propagating in the layer plane will be plit into individually propagating wave in each type of layer. If a firt arrival i ued to pick velocity, one would meaure the velocity of the fatet layer. Figure 3 how the P- and S-wave velocitie v. angle of incidence for a layered medium; according to the Backu theory Velocitie [m/] vqp vsh vqsv Angle [degree] Figure 3: P- and S-wave velocitie for a periodically layered medium according to the Backu theory. The parameter are λ 1 G 1 5 GPa; ρ 1.5 g/cm 3 ; λ G 1 GPa; ρ.0 g/cm 3.

13 13 4. Aniotropy in fractured media (Schoenberg & Douma model) The elatic propertie of a rock ma with penetrating, parallel fracture can be modelled in a manner quite analogou to the layered media above. Schoenberg & Douma conidered the fracture a compliant layer. The total thickne of the rock ma i H (in the z-direction), out of which a height H fr i contituted by the fracture. The relative fracture thickne i defined a h fr H H fr (4.1) The vertical fracture train i ε fr zz uz 1 H λ + G fr fr fr σ zz (4.) The total train i ε u h z fr Z H λ G σ + σ zz zz N zz fr fr (4.3) where Z N i the normal fracture compliance, defined in the limit when h fr and λ fr +G fr 0. The hear train ε xz fr for the fracture i imilarly given by the hear modulu ε fr xz ux 1 σ H G fr ε u h σ Z σ x fr xz xz T xz H Gfr fr xz and the total train i (4.4) (4.5) Thi define the tranvere fracture compliance Z T. Schoenberg & Douma further introduced the relative compliance E N and E T : E Z ( λ + G ) N N b b (4.6) E Z G T T b (4.7) The ubcript b refer to the background medium (intact rock). Uing the reult of the Backu theory above, the tiffne tenor element of the fractured medium can be written a follow:

14 14 λb EN 11 ( λb + Gb) λ + G 1+ E b b N (4.8) 33 λ b + G 1+ E N b (4.9) 13 λb 1 + E N (4.10) 44 Gb 1 + E T (4.11) G (4.1) 66 b

15 15 5. Iolated parallel crack (Hudon theory). Figure 4: Aligned microcrack. Iolated and oriented crack incluion are a ource of rock aniotropy. Two different non-elf conitent model exit whereby the effect of parallel crack can be calculated. Thee are Hudon' theory, which ue a cattering approach and find the tiffne tenor element a (1 Q ξ + ( ξ ) +...) 0 IJ IJ IJ (5.1) IJ 0 i the tiffne tenor element for the (in principle aniotropic) background material. Here we hall conider the background to be an iotropic olid, with Lamé coefficient λ and G and Poion' ratio ν. Notice that ummation over repeated indice i not performed here. ξ i the crack denity, which for penny-haped thin crack i 3 ξ n a (5.) Here n i the number of crack per unit volume, and a i the crack radiu. <> denote an average value. Garbin & Knopoff calculated the effect of crack uing a tatic approach, in which the crack denity appear a a perturbation to the compliance rather than to the tiffne (a in Hudon' theory): IJ 0 IJ 1 + Q ξ IJ (5.3)

16 16 Notice that to the lowet order in the crack denity the two model give the ame reult. Notice alo that both model aume a patially homogeneou ditribution of crack. In rock that are cloe to failure, localiation of crack will occur, and thee model can not be expected to be valid in that cae. The Q-matrix in Eq. (5.1) and (5.3) i, for crack oriented with their normal in the z-direction: ν ν(1 ν) (1 ν) ν 1 ν 1 ν ν(1 ν) ν (1 ν) ν 1 ν 1 ν (1 ν) (1 ν) (1 ν) Q 1 ν 1 ν 1 ν 3 1 ν ν 1 ν ν (5.4) Thu, the tiffne tenor element in the cae of Hudon' theory can be written a: ν ( λ + G)(1 ξ ) 31 ν (5.5) (1 ν ) ( λ + G)(1 ξ ) 3 1 ν (5.6) (1 ν ) λ(1 ξ ) 3 1 ν (5.7) ν G(1 ξ ) 31 ν (5.8) G 66 (5.9) For low crack denitie, different orientational ditribution may be analyed by imply adding crack et with different orientation. Figure 5 how P- and S-wave velocitie v. angle of incidence according to Hudon theory, for a elected crack denity of 0.10.

17 Velocitie [m/] vsh vqp vqsv Angle [degree] Figure 5: P- and S-wave velocitie for a olid containing parallel crack with their normal in the z-direction (angle of incidence 0), according to the Hudon theory. The parameter are λ G 3 GPa; ρ.0 g/cm 3 and crack denity ξ0.10.

18 18 Reference and Suggeted reading Backu, G.E Long-wave aniotropy produced by horizontal layering. J. Geophy. Re. 66, rampin, S Evidence for aligned crack in the Earth crut. Firt Break 3; pp Garbin, H.D. and Knopoff, L The compreional modulu of a material permeated by a random ditribution of circular crack. Quart. Appl. Math. Jan. ; pp Garbin, H.D. and Knopoff, L The hear modulu of a material permeated by a random ditribution of circular crack. Quart. Appl. Math. Oct. ; pp Holt, R.M., Fjær, E., Raaen, A.M., Ringtad, Influence of tre tate and tre hitory on acoutic wave propagation in edimentary rock. In Shear Wave in Marine Sediment; Ed. J.M. Hovem et al., Kluwer; pp Hudon, J.A Overall propertie of a cracked olid. Math. Proc. amb. Phil. Soc. 88; Hudon, J.A Wave peed and attenuation of elatic wave in material containing crack. Geophy. J. Roy. Atr. Soc. 64; Nur, A. and Simmon, G Stre-induced velocity aniotropy in rock: An experimental tudy. J. Geophy. Re. 74, Rathore, J.S., Fjær, E., Holt, R.M., Renlie, L P- and S-wave aniotropy of a ynthetic andtone with controlled crack geometry. Geophy. Propecting. 43; Schoenberg, M. and Douma, J Elatic wave propagation in media with parallel fracture and aligned crack. Geophy. Propecting 36, Thomen, L Weak elatic aniotropy. Geophyic 51, Thomen, L Elatic aniotropy due to aligned crack in porou rock. Geophy. Propecting 43 White, J.E Underground Sound. Elevier.