Propagation Of Rayleigh Waves In Partial Saturated Multiphase Poroelastic Media

The importance of Rayleigh waves can be noted in several fields, as earthquake seismology, ground water, engineering, environmental, geology and material science. Lord Rayleigh (1885) first carried out the theoretical investigations in isotropic half space elastic media. These waves that propagate along a free surface, as the earth and air interface, are generated by couple of P and S wave, with always strong energy in seismograms.Surface wave was treated as ground roll of the most troublesome noise making the useful bulk wave fields in exploration and engineering seismology. But as the surface wave method, especial the Multichannel analysis of surface wave (MASW) has been developing these decades, S wave velocities can be estimated quickly from inversion of Rayleigh wave data. Surface wave is regarded as the available tools to obtain the subsurface information. Surface-wave techniques in the exploration seismic frequency band (up to200Hz) primarily regard surface-wave generating in elastic media. Dynamic elastic theory is the fundamental of surface-wave data gathering, processing, and inversion in the near-surface geophysics community.The earth media, consolidated and unconsolidated, are representative composed of solid with pores, one or multiple fluids are filled in. Multiphase theory can describe this solid of porous media. Biot (1941,1956a,1956b,1962a,1962b) established a poroelastic theory, which can more precisely elucidate the property of wave phenomena in real world media than single-phase elasticity. Two types of P waves, referred as P1waves and P2waves, and one type of S waves were theoretically predicted existing in poroelastic media. Many scientists have widely investigated the wave propagation in poroelastic media, and the poroelastic theory has been progressing and improving to represent the more complex solid and fluid composites.For the existence of the fluid phases in the pores, it generates more than one type of free surfaces of permeable "open-pore", impermeable "close-pore" and partially permeable interfaces (Deresiewicz and Skalak,1963). As the result, the Rayleigh wave, interfered by P and S wave, in poroelastic media has more complex properties, which carry more information about solid media, such as fluid pressure, porosity, permeability, etc. Therefore, to find the propagation characteristics of surface waves in poroelastic media in the exploration seismic frequency band is a key issue to improve prospecting precision of surface wave methods to face real-world conditions. Although, the propagation of Rayleigh wave has been studied by several researchers, in consideration of the more difficulty to describe the surface wave, the propagation of surface wave is not sufficiently analyzed as bulk waves. Mesoscopic loss, an important attenuation mechanism in seismic frequency band has not introduced into the surface wave propagation. Therefore, analysis propagation of surface wave in poroelastic media and establishing an effective simple model to approximate the poroelastic model is meanful to for us to acquire some applicable method to introduce poroelasticity in surface wave method. The main aspects include:(1) as the different free surface interfaces, Rayleigh wave in poroelastic media are effected by this difference. It is necessary to analysis the propagation characteristics under the different surface conditions, especially the partial surface condition, which is lack in explicit analysis;(2) an approximate mathematical single phase model of the poroelasticity can simplify the complexity of the expressions of the surface wave, so introduce a single phase effective media could be feasible; and (3) the approximate method of effective media should be verified in numerical and theoretical experiments.Based on the facts of the studies on the surface wave propagation in poroelastic media, and the problems we face, we studied the characteristics of Rayleigh wave in poroelastic media for different physical conditions and tried to construct a simple approximate model to describe the poroelastic media with the important mesoscopic loss mechanism. In analysis of the propagation, we focused on the following subjects.(1). We investigated velocity dispersion, attenuation and dynamic response characteristics of Rayleigh waves, and gave the dispersion equations and dynamic Green's functions for different free surface conditions.(2). We numerical analyzed the dispersion, attenuation curves, wave field and spectral dynamic responses under different surface conditions, viscous damping, elastic properties and porous fluid flowing conditions in seismic frequency band to figure out the characteristics of Rayleigh wave propagation.(3). We established an effective single phase viscoelastic media to approximate poroelstic media of partial saturation case with the mesoscopic loss mechanism considered. We derived the effective dispersion equation and dynamic Green's functions in a simple expression comparing to poroelastic media.(4). We also analyzed the dispersion, attenuation characteristics and wave field, spectrum responses of Rayleigh waves in the effective media. The results are compared to that of Biot's model to demonstrate the applicable of the effective model.By the analysis of the Rayleigh wave, we conclude that:(1). Different free surfaces of different surface permeabilities for porous fluid at the interface generate the different modes of Rayleigh type surface waves. One mode referred as R1wave exists under all the conditions, as the classical Rayleigh wave in elastic media, the other mode referred as R2wave exists for closed pore and partially permeable surface conditions, the exact propagation characteristics of these two modes of Rayleigh wave are summarized as:i). R1wave This surface-wave mode exists for no matter the surface drainage condition changes. The surface-wave mode propagates with strong energy, as the classical Rayleigh wave in elastic media. At the low frequency, the velocities of surface-wave mode are all asymptotic to a same value of the Gassmann effective Rayleigh wave velocity. Velocity dispersion for partially permeable surface is most (slight more than closed pore condition), and for open pore is least. Large fluid viscosity or low permeability (High couple damping coefficient) make the shift to high frequencies of the main dispersion frequency range, and the attenuation coefficient knee points for the different frequency dependence, which represents the relaxation frequency movement to high frequencies. A rigid solid frame skeleton and low fluid flowing condition of large tortuosities weaken the velocity dispersion for all the surface conditions. At high frequencies in seismic frequency band, R1waves propagate with different limited velocity for the three surface cases, respectively. Because the attenuation coefficient is f1dependence at high frequencies, this surface-wave mode shows constant-Q leaky attenuation. R1waves for the open pore condition is most sensitive to the elastic modulus variation, but least sensitive to tortuosoty variation, which also can be represented in the wave fields and spectra changing to other two cases. The partial surface permeability can generates the non-physical R1waves with negative attenuation at low frequency range, which can be effected by the different physical properties. But it has the no distinct reflection in wave fields and spectra. The relaxation frequency shift makes a most attenuation appear at an intermediate couple damping coefficient. For similarity to the classical Rayleigh wave, we can use this surface-wave mode to describe the real world surface wave data. It is noticed that at very low couple damping coefficient, this surface-wave mode radiates with energy loss into the other surface-wave mode for closed pore and partially permeable surface.ii). R2WaveThis surface-wave mode is only generated by the closed pore and partially permeable free surface. Diffusion at low frequency and a limit velocity at high frequencies in seismic frequency band, this surface-wave mode is similar to the slow P2wave, with slight low velocity and high attenuation. R2wave is sensitive to tortuosity variation as flowing condition changing, high tortuosity diminish the velocity of R2wave velocity with low attenuation. As P2wave in bulk wave seismograms, R2wave is not observed for majority condition. But at very low couple damping coefficient, this surface-wave mode displays. For partially permeable surface, R2wave even bears stronger energy than R1wave, because of radiation of R1wave to the non physical R2wave, the peculiar phenomenon is also reflected in spectra.(2). Partial saturated or unsaturated porous fluid means the multiphase fluids in the pores, for instance, water and gas. Effective mixture fluid can simply describe these fluids, i.e. Reuss average for uniform isostress condition of homogeneous saturation with equilibrium porous pressure, and Voigt average for macroscopic inhomogeneity with non interactions of the multiphase. These averages reflect the microscopic and macroscopic multiphase fluid distribution, but a mesoscopic scale unhomogeneity between these is very important to the attenuation mechanism in seismic frequency band. Patchy saturation model of porous media can describe this mechanism. In the different patches saturated one type of fluids, when longitudinal wave passes through the different patches, the wave induced flow generates the pressure gradient and P2wave between the interfaces among the different patches saturated with different fluids, and diffusion as a mesoscopic loss mechanism. White (White,1975; White et al.,1975) considered a model of two specific patchy geometry sizes to describe this mesoscopic attenuation, referred as White's patchy saturation model, and built on the Biot's theory. As the S wave propagates in the solid frame skeleton, the combination of P wave and S wave in patchy saturation model generates the effective Rayleigh surface wave at the free surface. The viscoelastic approximation of poroelasticity is established with mesoscopic loss considered(3). Results of dispersion, attenuation, wave field and spectrum analyses show that, in the dominate frequency of mesoscopic loss of large fluid viscosity or low solid permeability, opposite to the macroscopic mechanism in Biot's theory, Rayleigh wave, couple of P wave and S wave, has the similar dispersion and attenuation characteristics as P wave, but the mesoscopic loss is weakened by the S wave, which is little effected by mesoscopic unhomogeneity. The patch size also effects the dispersion and attenuation by shifting the relaxation frequency. By comparing to fluid average for microscopic and macroscopic unhomogeneity in Biot's theory, the effective viscoelstic approximation is applicable to reflect the mesoscopic loss in Rayleigh wave. The approximation shows us a simple way for poroelasticity to describe dispersion and attenuation for porous media with mesoscopic attenuation in seismic frequency band.(4). Mesoscopic mechanism is a key factor of effecting the wave propagation in seismic frequency band, which also obviously effects the dispersion and attenuation of surface wave. Introduction the mesoscopic loss into the surface wave method is one of the important perspectives for surface wave technique. The theoretical results of approximation show us a good guide for future work of constructing a simple and applicable guide equation for effective media in seismic modeling and inversion with the mesoscopic unhomogeneity attenuation considered. In order to obtain the velocity dispersion and attenuation of surface wave with the actual fluid distribution scale corresponding the in-situ state of porous media in real world, more exact numerical and physical experiments should be carried on for effectively practical utilizing of poroelasticity in surface wave method.