Scattering of SV-waves in Fluid-Saturated Medium

Transcription

1 Australian Journal of Basi and Applied Sienes, 4(8): , 00 ISSN Sattering of SV-waves in Fluid-Saturated Medium Dennis Ling C.C., Faisal S.Y., 3 Zainal A.A.,,3 Department Mathematis, Faulty of Siene, University Teknologi Malaysia, Malaysia Department Mathematis, University of Kordofan, Sudan Abstrat: Problem statement: When the medium porosity espeially the saturation degree is asymmetrial, it is expeted to see the variation in SV-waves veloities. Hene, SV-waves propagation in asymmetrial and symmetrial porosity mediums is studied for the influenes of fluid density. The SV-waves propagations and the veloities are studied for the diffusive profiles. Approah: For studying the effets of asymmetrial and symmetrial porosity to the diffusive attribute, SV-waves are propagated in similar diretions and also opposite diretions. The mediums are hosen to fall into two distintive groups: insoluble as well as soluble mediums. Results: In similar diretion of propagation, low density fluid revokes the diffusive harateristis while high density fluid promotes diffusive attribute. However, these diffusive SV-waves are also found in the medium saturated with low density fluid when the medium is asymmetrial in porosity. In the ase of opposite diretion of propagation, the reurring SV-waves are found in the medium saturated with low density fluid. Conlusions: The studies indiate that SV-waves propagation diretion would emerge different diffusive profiles if the SV-waves veloities are different. Key words: Diffusion; Attenuation; Seismi waves; Sattering waves; Saturated-medium INTRODUCTION S-wave is referred to shear wave when an earthquake is disussed and S-wave annot move in liquid beause the liquid is not rigid enough to transmit an S wave. S-waves are divided into horizontal direted (SH) waves and vertial direted (SV) waves. Although SV-waves annot move in liquid yet these SV- waves show the effets on horizontal and vertial motions in a layered soil saturated with water (Yang, J. and T. Sato, 007). Signifiant amplifiation of vertial-omponent motion was observed by a borehole array at Port Island, Kobe, during the Hyogo-Ken Nanbu (KOBE) earthquake of 7 January 995 (Yang, J. and T. Sato, 000). Thus, insoluble and soluble mediums are studied in this researh. Biot's theory has been used in showing the elasti wave propagation in a fluid saturated porous solid (Biot, M.A., 956). The attenuation and dispersion of ompressional waves in fluid-filled porous roks with partial gas saturation was studied in aordane to Biot's theory (Dutta, N.C. and H. Ode, 979). The studies of fluid saturated method were extended through numerial modelling. Finite element modelling of the effetive elasti properties of partially saturated roks was studied (Makarynska, D., 008). Subsequently, this researh will exploit the elasti wave equation to study the diffusion of SV-waves. In investigating the deposited sediment at the reservoir behind the dam, the SV-wave was used for studying the dynami pressure (Wang, J.T., 009). The sensitivity of SV-wave sattering to surfae breaking raks was studied and this inspiration has brought to this researh (Peorari, C., 005). Two setups are done for SV-waves propagation to relate to the sattering of SV-waves (see figure (a) and (b)). In this paper, the aim is to obtain the diffusion profiles depited by SV-wave's propagation in the saturated mediums. The derivation of SV-waves will be shown before the analytial disussion. Methodology: The governing equations: u u u F Boundary onditions for z = 0: t () Corresponding Author: Zainal A.A., Department Mathematis, Faulty of Siene, University Teknologi Malaysia, Malaysia zaa@met.fs.utm.my 3843

2 Aust. J. Basi & Appl. Si., 4(8): , 00 u u u z z x y ux uz xz 0 z x u y u z yz 0 z y z x y 0 () Initial onditions for t = 0: ux, y, z 0 u u x y uz 0 t t t (4) xz In linear elastiity for isotropi medium, and denote the Lame parameters for the stress z, u u u, yz and the displaements x, y and z are ontinuous everywhere. F is the body fore in the diretion of x, y, z respetively and is the density. Provided that is the angular veloity or frequeny, k is the wavenumber, is the wave veloity along with the dispersion relation = k the group veloity of a wave is the veloity with the overall shape of the wave's amplitudes whih is also known as envelope of the waves. In other words, the phase veloity is the average veloity of the omponents, given by V d dk VP given by (Bhatnagar, P.L., 979). g k. The group veloity is veloity of the envelope, When this is applied to the problem of SV-wave's propagation in fluid saturated medium, the group veloity reflets the apparent veloity of the surfae displaement or the overall shape of the SV- wave's amplitude at the boundary of the medium. The envelope is formed by the phase veloity of SV-waves. In this researh, the apparent veloity V app or surfae displaement veloity is measured along the boundary of the similar density medium in aordane to Snell law: Vapp sin f sin e (5) e is the inident angle and f is the refration angle made by the P- and S-waves. However, our objetive is to study S-waves in vertial diretion only. For the ase of fluid-saturated medium, there exists variation in density within the medium (Wang, J.T., 009; Sharma, M.D., 004). There exists slowness indued by refration for fluid-saturated medium (Keith, C.M. and S. Crampin, 977; Sharma, M.D., 007) i.e. the apparent veloity of the displaement at the boundary is slower than phase veloity of the wave in the medium that reads: Vapp In this study, the elasti wave equation will be solved so as to obtain the SV-wave's displaements in the different types of medium that satisfied (5) and (6). By using the divergene operator, 3844 (6)

3 Aust. J. Basi & Appl. Si., 4(8): , 00 u uu (7) equation () is redued to u u uf t (8), (9) and are the veloities for the P-wave and S-wave and equation (8) is the elasti wave equation (Pujol, J.A., 003). Hene, this modeling only valid for elasti medium and it is neessary to redue the right hand side (RHS) of (8) by letting u exp i kxt By inserting seond order derivative of (0) into RHS of (8), the equation reads: u uf u Sine elasti wave onsists of irrotational by P-wave and solenoid by S-wave (Pujol, J.A., 003), the displaement shall be written as: u u u P S For irrotational P-waves, the vortiity u P u P u P u S u S. Thus, the relation () reads:,,,, (3) By inserting () and (3) into (), the equation yields: up kp up us ks us 0 P, S F 0 k k P S, (5) In the following, the equations (6) are the Helmholtz equations (Gang, B., 004) for P- and S-waves: (0) () () (4) u k u 0 P P P, u k u 0 (6) S S S Here, the Helmholtz equations are solved by utilizing Hansen vetor (Pujol, J.A., 003) that gives: lx nz up Alax nazexp i t (7) lx nz usv Bnax laz exp i t (8) 3845

4 Aust. J. Basi & Appl. Si., 4(8): , 00 lx nz ush Cay exp i t ax a a, y, and z are the unit vetors while l and n are the vetor omponents. Only equation (8) will be onsidered in this paper sine the aim is to study the SV-wave only. SV-wave vanishes when the depth go infinity as illustrated in figure (a) and (b). Thus, the modified equation (8) or SV-wave on whih the amplitude redues with depth is: (9) exp usv B ax az ik tx z, (0) is always positive. The veloity of the SV-wave, is measured at the boundary of the medium or z = 0 and it is similar to the apparent veloity V app of the surfae displaement. e is the SV-wave veloity in the medium. For the displaement normal to the propagation diretion by the means of quantity and amplitude B, the equation (0) for vertial diretion of displaements with amplitude redues with depth z reads: exp usv B ax az ik tx z k, () Here, quantity refers to the refrated SV-wave veloity that gives the refrated frequeny k a a x z and the vetors x and z show the displaement is polarized between diretions. The polarization of equation () gives the frequeny equation that reads (Pujol, J.A., 003): B usv expik zexp iktx, tan () exp Consequently, the SV-waves propagate in the positive x-diretion. From wave terminology, the term ik zexp ik t x k in equation () will show that the harmoni wave is diffusive if is omplex (Bhatnagar, P.L., 979). The diffusive waves are assoiated with attenuation of the amplitudes with the time due to ertain dissipation mehanisms present in the system. Next, the roles of the refrated veloity will be shown. For the fluid-saturated medium, there omes the slowness indued by refration. The envelope veloity at medium surfae is different with wave veloity in the medium after the slowness or Vapp. For a partiular ase, the SV-wave veloity in the medium is greater than the envelope's veloity that yields: V, (3) app Here, we propose the relation for another two types of medium onditions suh that: 3846

5 Aust. J. Basi & Appl. Si., 4(8): , 00 V, (4) app V, (5) app Here, we mark that the relations (3) and (5) are meant for the insoluble medium suh that the variation of veloities and is signifiant. When the veloity is similar to, we presume this will explain the soluble medium suh that the fluid mixes well with the medium to give similar veloity. Yang and Tadanobu (000) and Kahraman (007) show that the high density medium promotes high wave's veloity. Hene, the relation (3) will only be present when the low density fluid is saturated in the insoluble medium; the low density fluid will reshuffle the ray veloity or redue the SH-wave veloity. Eventually, the relation (5) is meant for the high density fluid saturated in the insoluble medium. The detail explanations about the tie between medium's solubility and the fluid's density will be disussed next for showing the vital roles play by the relations (3) to (5) espeially the displaements harateristi. When ondition (5) is applied to the quantity in equation (0), a omplex solution will be obtained for that reads: i (6) This indiates an amendment is required for equation () to give (7). Hene, the SV-wave's displaements for three types of mediums read: B usv expk zexp iktx i for (7) B u SV exp ik zexp ikt x B u SV exp ik zexp ikt x for (8) for (9) Equations (7) to (9) show that the SV-waves propagation in x-diretion while the diffusion is in z- diretion. The equations (7) to (9) are arranged suh that the density of the saturated fluid redues from < to >. Next, the equations (7), and (9) will be amended to depit the irumstanes in figures (a) and (b). Two different SV-waves with different diffusive harateristi whih are propagated in similar diretion reads: u B expk z expk z exp iktx, SV, (30) i i usv B expikz expikz exp ik t x,, (3) 3847

6 Aust. J. Basi & Appl. Si., 4(8): , 00 For the ases of opposite diretion of propagation, the equations read: B B usv exp kz+ expkzexp iktx i i, (3) B B usv exp ikz expikzexp iktx, (33), Fig. : The propagation of SV waves. (a) The SV-waves propagation in similar diretion. (b)the SV-waves propagate in opposite diretion Results: Equations (30), (3), (3) and (33) are plotted in aordane to Table for showing the influene of different veloities towards the diffusive harateristis or SV-waves sattering. Disussions: Equations (7) to (9) are derived for the medium with symmetrial density suh that there is no variation in wave s veloities in both the fluid and the medium. If there is variation in saturation degree in the medium, it is expeted to see the variation in SV-waves veloities when the medium porosity and the density are asymmetrial. For this reason, the equations (30) to (33) are derived for studying the diffusive profile of the SV-waves in asymmetrial porosity medium saturated with low and high density. Table : SV-waves' veloities in high density fluid saturated medium and low density fluid saturated medium for SV-waves propagation in similar diretion and opposite diretion. Similar / Opposite diretion > < Initially, equation (9) is referred to a non diffusive wave sine the tern k is in real. For similar diretion of SV-waves propagation, figures (a) and (b) are plotted for diffusive profiles generated by SV-waves propagate in the medium saturated with low density fluid. Figure (a) shows that the SV-waves are non diffusive when both the SV-waves veloities are similar. However, the SV-waves are diffusive when the refrated SV-waves veloities are different signifiantly (see figure (b)). The onsequene implies that the variation of fluid density in insoluble medium is apable of generating different waves veloities and thus the diffusive harateristi for SV-waves propagate in low density fluid in the insoluble medium is formed. 3848

7 Aust. J. Basi & Appl. Si., 4(8): , 00 Fig. : The diffusive profiles generated by SV-waves propagation in the medium saturated with low density fluid. (a) The non-diffusive harateristi by SV-waves propagate in similar diretion. >, -5 x 5, K =, =, =, = 6, = 6. (b) The diffusive harateristi by SV-waves propagate in similar diretion. >, -5 x 5, K =, =, =, = 6, = 6. In ontrast, equation (9) is referred to a diffusive wave sine the term k is in omplex. Figures 3(a) and 3(b) are the diffusive profiles generated by SV-waves propagation in the medium saturated with high density fluid. Apparently, there is no signifiant different between the diffusive profiles when both the SVwaves are propagated in similar diretion with different veloities as illustrated in figures 3(a) and 3(b). The density of fluid higher than the medium is presumed apable in preserving the diffusive harateristi of SVwaves. 3849

8 Aust. J. Basi & Appl. Si., 4(8): , 00 Fig. 3: The diffusive profiles generated by SV-waves propagation in the medium saturated with high density fluid. (a) The diffusive harateristi by SV-waves' propagate in similar diretion. >, -5 x 5, K =, =, =, = 6, = 6. (b) The diffusive harateristi by SV-waves' propagate in similar diretion. >, -5 x 5, K =, =, =, = 6, = 6. For the seond ourrene, the opposite diretion of propagating SV-waves, the figures 4(a) and 4(b) are plotted for showing the SV-waves propagation in the medium saturated with low density fluid. Both the figures 4a and 4b depit a single outome, the reursive SV-waves. From equation (33), the SV-waves suppose to be non-diffusive when both the term k is in real. By means of SV-waves propagating in opposite diretion, these non diffusive waves stak and build the reursive SV-waves as shown in figures 4(a) and 4(b). In other words, the reursive SV-waves in the medium saturated with low density fluid are nondiffusive. 3850

9 Aust. J. Basi & Appl. Si., 4(8): , 00 Fig. 4: The reursive profiles generated by SV-waves propagation in the medium saturated with low density fluid. (a) The reursive harateristi by SV-waves' propagate in opposite diretion. >, -5 x 5, K =, =, =, = 6, = 6. (b) The reursive harateristi by SV-waves' propagate in opposite diretion. >, -5 x 5, K =, =, =, = 6, = 6. Figures 5(a) and 5(b) are plotted for showing SV-waves propagation in the medium saturated with high density fluid. As expeted, the high density fluid will promote the diffusive harateristi (see figures 5(a) and 5(b)) sine equation (3) shows the term k is in omplex. Again, the density of fluid higher than the medium is found apable in preserving the diffusive harateristi of SV-waves. 385

10 Aust. J. Basi & Appl. Si., 4(8): , 00 For soluble medium with, equation (8) is invalid sine 0 gives invalid solution for tan. Consequently, the relation implies that the SV-wave s displaement in the soluble medium does not exist. The fat has shown that the S-waves annot propagate in liquid sine the liquid is not rigid to transmit an S-wave. Fig. 5: The diffusive profiles generated by SV-waves propagation in the medium saturated with high density fluid. (a) The non-diffusive harateristi by SV-waves' propagate in opposite diretion. >, -5 x 5, K =, =, =, = 6, = 6. (b) The diffusive harateristi by SV-waves' propagate in opposite diretion. >, -5 x 5, K =, =, =, = 6, = 6. Conlusion: The studies show that the SV-wave s propagation hold different diffusive harateristis when the medium is fluid saturated. High density fluid enhanes the diffusive SV-wave while low density fluid depits the non- 385

11 Aust. J. Basi & Appl. Si., 4(8): , 00 diffusive SV-wave. However, the variation of SV-waves veloities beause of saturation degree different has indued the diffusive SV-waves in the medium saturated with low density fluid. The reursive waves are found in the asymmetrial porosity medium whih is saturated with low density fluid. The different diretion of wave s propagation has indued dissimilar pae of SV-waves that stak and build the reursive SV-waves. However, the SV-waves are non diffusive in aording to the equations. For the high density fluid, the SV-waves in asymmetrial and symmetrial porosity mediums remain diffusive. As long as the fluid density is higher than the medium, the variations in fluid density and propagation diretion have no effets on diffusive profiles. ACKNOWLEDGMENTS The researh is funded by Ministry of Higher Eduation (MOHE) FRGS no REFERENCE Biot, M.A., 956. Theory of Propagation of Elasti Waves in a Fluid Saturated Porous Solid. J. Aoust. So. Am., 8(): Bhatnagar, P.L., 979. Nonlinear Waves in One-dimensional Dispersive System. India: Thomson Press. Dutta, N.C. and H. Ode, 979. Attenuation and Dispersion of Compressional Waves in Fluid-filled Porous Roks with Partial Gas Saturation (White model) - Part I: Biot theory. Geophysis, 44(): Gang, B., G.W. Wei, Z. Shan, 004. Numerial Solution of the Helmholtz Equations with Wave numbers. Int. J. Numer. Meth. Engng., 59(3): Kahraman, S., 007. The Correlations between the Saturated and Dry P-wave Veloity of Roks. Ultrasoni, 46(4): Keith, C.M. and S. Crampin, 977. Seismi Body Wave in Anisotropi Media: Refletion and Refration at a Plane Interfae. Geophys. J. R. Astr. So., 49(): Makarynska, D., B. Gurevih, R. Ciz and C.H. Arns and A. Mark, 008. Finite Element Modelling of the Effetive Elasti Properties of Partially Saturated Roks. Computers & Geosienes, 34(6): Pujol, J.A., 003. Elasti Wave Propagation and Generation in Seismology. United Kingdom: Cambridge University Press. Peorari, C., 005. A Note on the Sensitivity of SV-wave Sattering to Surfae Breaking Craks. Ultrasoni, 43(7): Sharma, M.D., 007. Effet of Initial Stress on Refletion at the Free Surfae of Anisotropi Medium. J. Earth Syst. Si., 6(6): Sharma, M.D., D wave propagation in a general anisotropi poroelasti medium: refletion and refration at an interfae with fluid, Geophys. J. Int., 57(): Wang, J.T., Z.H. Zhang and F. Jin, 009. Analytial Solutions for Dynami Pressures of Coupling Fluid Porous Medium Solid due to SV Wave Inidene. Int. J. Numer. Anal. Meth. Geomeh., 33(): Yang, J. and T. Sato, 007. Influene of Water Saturation on Horizontal and Vertial Motion at a Porous Soil Interfae Indued by Inident SV wave. Soil Dyn. and Earthquake Eng., 9(5): Yang, J. and T. Sato, 000. Interpretation of Seismi Vertial Amplifiation Observed at an Array Site. Bull. Seismol. So. Amer., 90():