Tilted Transverse Isotropy

Transcription

1 NAFA-GAZ listopa ROK LXVII Anej Kostecki Institute of Oil an Gas, Kakow ilte ansvese Isotop Intouction In seismic pospecting, the tansvese isotop (I) moel, i.e. a moel of thinl statifie (laminate) meium, is the most common one. he moel has been popose b Postma [], an assumes that bounaies sepaating isotopic laes ae paallel planes. In the case when the ais of smmet is vetical, the moel is efee to as the vetical tansvese isotop (VI) moel, while if the ais of smmet an the vetical ais fom an angle θ, this is a monoclinal meium escibe b the tilte tansvese isotop (I) moel. his is a vesatile moel of anisotop, which can be use to obtain fomulas escibing moels with hoiontal aes of smmet, i.e. hoiontal tansvese isotop (HI) moels as well as compose moels [7]. ntil now, stuies of the mathematical esciption of the I meium [, ] ealt with one of two possible cases of monoclinal meia (fo eve vaiant). his aticle pesents esciptions of both cases fo eve vaiant of the moel. Basic elationships Knowlege of tansvese isotop paametes is ve useful fo analsis of othe tpes of anisotop. Geneall in the case of anisotopic meia escibe b the moels of tansvese isotop, elasticit matices, which pesent the components of the elasticit tenso, epen on the spatial oientation of I moels. Figue pesents a ight-hane cooinate sstem,, an a cooinate sstem of,, associate with ipping paallel stata which coespon with the I meium. he spatial aangement of the meium in elation to the cooinate sstem,, is efine b an angle θ between planes an, i.e. its inclination. he elation between the stess an stain tensos in the,, cooinate sstem ae the same as in the VI meium. In oe to etemine the pinciples of wave popagation in the,, cooinate sstem, otation of the,, sstem to the,, sstem shoul be one using the mati of cosines of angles between these two sstems (angles ae measue in the clockwise iection). he geometical situation pesente in Figue is escibe b the following mati of iection cosines: Fig.. Dawing of monoclinal stata ipping at an angle θ (between the - an -aes) of tansvesel isotopic meium 3 3 cos, cos, cos cos ' ' o, ' cos9 cos, cos, cos, cos cos ' ' ' cos o, ' cos7, ', ' cos sin sin () 769

2 NAFA-GAZ B using Bon s law [, 3, ], we can eive the geneal elationship between the mati D of elastic mouli, ecoe in the,, sstem, an the mati of known tenso elements in the,, sstem, whee the plane foms an angle θ with the plane: whee the R mati is as follows: D = R R () R (3) while the mati R is the tanspose of the mati R. B using equation (), we obtain Bon s mati R (+) (otation aoun the -ais, inclination oiente towas the positive -ais): R cos sin sin cos sin cos sin cos cos sin sin sin cos sin cos () while the mati escibing the elationship between stess an stain in the I/VI meium is as follows: (5) hus, b using fomulas () an (5), we obtain fom elationship () the elasticit mati D (+) ( inicates otation aoun the -ais, while (+) inicates the inclination of the plane). D (6) he elements of the mati D (+) ae: ` cos cos sin sin sin cos sin 77 n /

3 atkuł sin cos sin cos sin cos sin sin cos sin cos sin cos cos sin cos sin sin cos sin cos cos sin cos 6 sin cos sin cos cos sin sin sin sin cos cos (7) Let us now consie the same vesion of the I moel, i.e. otation aoun the -ais, but in a situation whee the plane is ipping towas the negative -ais (Figue ). In such a case, the mati of iection cosines is epesse as: 3 3 cos sin sin cos (8) while Bon s mati is epesse as: R cos sin sin cos sin cos sin cos cos sin sin sin cos sin cos (9) Fig.. Geometical epesentation of coincience of the - an -aes. he -ais is incline towas the stata inclination i.e. negative -ais. Aows inicate the clockwise iection of measuing angles B using matices (5) an (9), the elasticit mati D (-) is obtaine an it epesents a monoclinal meium incline towas the negative -ais: D () n / 77

4 NAFA-GAZ whee elements ae efine b equations (7). A compaison of matices D (+) an D (-) shows that the matices iffe onl b signs fo elements 5, 5, 35, 6 an fo the smmetical elements. Let us now consie the secon vesion of the I meium which is a esult of otating the isotop plane aoun the -ais. We will analse the case whee the lamination plane of the meium is oiente towas the positive -ais an foms an angle θ with the ecoing suface (Figue 3). Fig. 3. Ovelap of the - an -aes. he -ais is oiente towas the positive -ais an in the same iection as the inclination of the stata In such a situation, the mati of iection cosines is: cos sin sin cos () while the elasticit mati D (+) calculate in a simila wa as in the pevious cases is: D () whee the elements of the mati ae: ` sin cos cos sin sin 3 3 sin cos cos sin sin cos sin cos sin cos sin sin cos cos sin cos sin sin cos cos 65 cos sin cos cos sin sin cos sin sin sin cos cos sin cos sin cos sin () 77 n /

5 atkuł Let us consie anothe case of otating the isotop plane aoun the -ais whee the ipping -ais is oiente towas the negative -ais (Figue ). Fig.. he geomet of the,, an,, cooinate sstems. he -ais is oiente towas the stata inclination, i.e. the negative -ais In such a case the mati of iection cosines is: cos sin sin cos () Fom elationships () an (3), the elasticit mati D (-) is obtaine: D (5) whee elements of the mati ae epesse b equations (). A compaison of matices () an (5) shows that iffeent esults ae obtaine epening on the oientation of the incline plane also in this vesion of a monoclinal meium. he elasticit matices D an D, compose of elements which ae tenso components D ijkl (using the shotene Voigt notation), ae veifie hee both in tems of using the cooinate sstem,, oiente in an iection in elation to the cooinate sstem,, an in tems of the metho of calculating the tenso D ijkl. he veification can be one using elationship [6]: D ijkl = ii jj kk ll i j k l (6) It gives the same esults (7) an () fo the elements of mati D in both cases of the monoclinal meium, i.e. the two-imensional wavefiel ecoe up-ip an ecoe own-ip, as well as in the case when acquisition is caie out along the etent of the stuctue fo both tpes of ipping stata. B using the basic elationship between the tenso of stess ij an the tenso of stain E ik (Hooke s law) the following mati is obtaine: 3 E E E E,, E, D E,,,, (7) n / 773

6 NAFA-GAZ he elationships between the components of the elasticit tenso an the eivatives of paticle movement in the meium, an ae escibe in the two-imensional case, i.e. the eivatives of the wavefiel in elation to the -ais ae equal to eo. Stating with the law of motion (ignoing etenal foces) fo each component, the following equations ae obtaine:,,,,3 3.3,3 ρ t ρ t ρ t 3 (8) whee ρ is ensit of stata an t is time. Analsing the case of the I stata, whee the smmet ais is locate on the plane, i.e. fo mati D (+, -), in both cases we get the following wave equations: 5, 5,, 53, ρ (9) t,, 6, ρ () t,, () t 5, 35,,,, 35, In fomulas (9-), the sign (+) efes to acquisition along the -ais moving in the positive iection of the ais, i.e. own-ip (Figue ), while the sign ( ) efes to the up-ip iection. he above elationships inicate that the coss-line isplacement is neithe inclue in fomula (9) no in fomula (). he shea wave SH is escibe sepaatel b equation (). In the case of low angles of inclination (θ ), we can assume that the elements of the elasticit mati 5 = 5, 53 = 35, 6 an thus the influence of the ip iections of the isotop plane on the wave equation can be ignoe. Fom fomulas (9-) with θ = o, the following wave equations fo the VI moel with a vetical ais of smmet ae obtaine:,,, () t,, (3) t () t,,, while the following ae obtaine fo the HI moel with the smmet ais oiente paallel to the -ais when the angle of inclination is θ = 9 o :,,, (5) t,, (6) t 77 n /

7 atkuł (7) t,,, Let us now consie the secon case of the I moel, whee the ecoing is caie out in the stike iection of the statifie meium, i.e. the smmet ais is paallel to the -ais. In that case, the following is obtaine fom equations (), (5) an (8):,,,, 56, (8) t, 65, 65, 3, (9) t,,, 56, 3, (3) t Fomulas (8-3) impl that the iection of the isotop plane incline at an angle θ has a iect influence on the chaacte of the wave equation. his influence isappeas when the angle θ is elativel low, while in the eteme case, when the angle is θ = o, the epecte wavefiel equations fo the VI moel, i.e. equations (-), ae obtaine. It is eas to notice that in the case of the wavefiel ecoe along the etent of the stuctue (the -ais) thee is no sepaation of the longituinal an shea SH waves which eist both in equations (8) an (3) espite the assumption of vanishing eivatives of movement i, =. In the case of vetical isotop plane (o factues, iscontinuities) with the angle of θ = 9 o, fom (8-3) we obtain equations escibing the wavefiel in the HI meium, i.e. with a hoiontal ais of smmet pepenicula to the ecoing iection along the -ais:,,, () t,, (3) t,,, () t he above equations inicate that the elationship fo the shea SH wave is the sepaate fomula (3), while thee is no component in equations () an (). he above motion equations fo each tpe of the anisotopic meia fom a basis fo calculations of ispesive equations. hese ae necessa fo analsis of the popagation of all tpes of waves in the wavenumbe-fequenc omain. When equations (9) an () ae use togethe with the Fouie tansfom ( k, t ω), an elements 5, 53, 6 ae ignoe ue to the low angle of inclination, the following mati equation is epesse: k k k k k k k k (3) whee k an k ae the hoiontal an vetical wavenumbes, an ω is the angle fequenc. When we assume that the velocit of the shea SV wave is eo fo low angles θ, i.e. similal as fo the VI meium [, 5], an we assume that: cos cos cos (35) n / 775

8 NAFA-GAZ the ispesive equation fo the vetical wavenumbe k I in the I-tpe anisotopic meium is obtaine: k I S p S p qi k cos (36) S p I k whee: q S I I p q VI VI V pp cos cos (37) while V pp is the velocit of longituinal wave in the vetical iection, i.e. V pp = V p^ cosθ, whee V p^ is the longituinal wave velocit in the iection pepenicula to the statification in the I meium, while q VI = + ε, η VI = (ε δ) ae homsen s paametes [] fo the hoiontall statifie meium (VI) with the vetical ais of smmet. In a simila wa, ispesive equations can be obtaine fo othe oientations of the I stata. Such equations ae essential fo solving ual-omain migation algoithms pefome in the fequenc-space an fequenc-wavenumbe omains [7, 8, ]. he aticle was sent to the Eitoial Section on Accepte fo pinting on Liteatue [] Alkhalifah.: Acoustic appoimation fo pocessing in tansvesel isotopic, inhomogeneous meia. Geophsics, 63, 63-6, 998. [] Aul B.: Acoustic fiel an waves in soli. Kiege Publishing ompan, vol., 99. [3] Bansal R., Sen M.: Finite iffeence moelling of S-wave splitting in anisotopic meia. Geophsical Pospecting, 56, 93-, 8. [] Danek., Leśniak A., Pięta A.: Numeical moeling of seismic wave popagation in selecte anisotopic meia. Stuia, Ropaw, Monogafie n 6, Insttut Gospoaki Suowcami Minealnmi i Enegią, PAN,. [5] Han Q., Wu R.: A one wa ual omain popagato fo scala qp waves in VI meia. Geophsics, vol. 7, D9 D7, 5. [6] Helbig K.: Founations of anisotop fo eploation seismics. Seismic eploation, vol., Pegamon, 99. [7] Kostecki A.: Algoithm of migation MG(F-K) in othohombic meium. Nafta-Ga n, 5-5,. [8] Kostecki A.: Algotm migacji MG(F-K) la aniotopowego ośoka tpu HI (Hoiontal ansvese Isotop). Nafta-Ga n, 8-8,. Reviewe: Anna Półchłopek [9] Kostecki A.: Algotm głębokościowej migacji w aniotopowm ośoku VI. Nafta-Ga n, -7, 7. [] Kostecki A.: he algoithm of migation MG(F-K) in monoclinal anisotopic meium (moel I). Nafta-Ga n,. [] Postma G.: Wave popagation in statifie meium. Geophsics, vol., 78-86, 9. [] homsen L.: Weak elastic anisotop. Geophsics, vol. 5, 95-9, 986. [] Zhu I., Doman I.: wo-imensional thee component wave popagation in a tansvesel isotopic meium, with abita oientation finite element moeling. Geophsics, 65, no. 3, 93-9,. Anej Kostecki Pofesso of geophsics. he main subject of inteest electomagnetic an seismic wave popagation, epouction of eep geological stuctues b means of seismic migation, the analsis of migation velocities, seismic anisotop. he autho of publications. 776 n /