Research Express@NCKU - Articles Digest
Research Express@NCKU Volume 13 Issue 8 - April 30, 2010 [ http://research.ncku.edu.tw/re/articles/e/20100430/3.html ]
Propagation and attenuation of Rayleigh waves in a
semi-infinite unsaturated poroelastic medium
Wei-Cheng Lo
Department of Hydraulic and Ocean Engineering, College of Engineering, National Cheng Kung University
lowc@mail.ncku.edu.tw
Advances in Water Resources, 31, 1399-1410, 2008
1.
R
ESEARCH OBJECTIVESThe study of elastic wave propagation and attenuation through a fluid-containing porous medium has been a subject of great interest due to its extensive applications in environmental remediation, hydrocarbon recovery, geophysical exploration, seismology, and biomechanics. Two main groups of elastic waves can be observed to present in such a medium: body and surface waves. The former propagate inside an unbounded domain, such as dilatational (acoustic, compressional) and shear (rotational) waves, whereas the later develop when the medium is stratified or
possesses a free surface. Most common surface waves are Rayleigh and Love waves. Surface waves are significantly different from body waves that induce minor disturbances near the surface of the medium because their energy is dissipated essentially in the interior of the medium. Because of the potential to cause destructive vibrations to structures and buildings, the propagation characteristics of Rayleigh wave through sedimentary materials have been considerably investigated in the disciplines of hydrogeology and geomechanics.
The discovery of Rayleigh wave was pioneered by Rayleigh [1885] in the free surface of a semi-infinite nonporous elastic solid. To elucidate the behavior of body wave propagation and attenuation through an elastic porous medium bearing one single compressible viscous fluid quantitatively, Biot [1956] proposed a solid-fluid coupling model to account for two-phase dynamic mechanical interactions, known as the theory of poroelasticity. Despite substantial progress in applying Biot theory to investigate the propagation and attenuation of Rayleigh wave in a one-fluid system, a systematic study to understand the physics of Rayleigh wave motions in an unsaturated porous medium (i.e. bearing two immiscible fluids, yet none of which fills the pore space completely; e.g. an air-water system) is still lacking. The partially-saturated condition reflects a more realistic scenario since the porous medium right below the ground surface is simultaneously filled by water and air. In the present study, the influence of water saturation and excitation frequency on Rayleigh wave motions was thus investigated theoretically for the first time.
2. APPROACH
Considering a stress-free and permeable boundary for an unsaturated porous medium, we theoretically derive a complex-valued dispersion equation for describing Rayleigh wave motions in presence of viscous dissipation. The derivation was based on the poroelasticity model of Lo et al. [2005] developed for an elastic porous medium saturated by two immiscible, viscous, compressible fluids. The resulting dispersion equation provides an exact solution for the phase speed and attenuation coefficient of Rayleigh waves as an excitation frequency is stipulated.
Columbia fine sandy loam permeated by air and water in pore spaces is numerically studied in a broadband range
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of excitation frequencies (10 Hz - 10 kHz) as an illustrative example. Since the dispersion equation is a cubic polynomial, three different modes of Rayleigh wave with attenuation can be found to travel along the free surface of an elastic porous medium bearing a two-fluid mixture.
3. ACCOMPLISHMENTS
An analytical model for describing the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids is developed in the present study based on the poroelastic equations formulated by Lo et al. [2005]. Our numerical results show that, corresponding to three dilatational waves (P1, P2, and P3), there is also the existence of three different modes of Rayleigh wave in an unsaturated porous medium, which are designated as the R1, R2, and R3 waves in descending order of phase speed, respectively. The phase speed of the R1 wave is non-dispersive (frequency-independent) in the frequency range we examined (10 Hz – 10 kHz) and decreases as water saturation increases, whose magnitude ranges from 20% to 49% of that of the first dilatational wave with respect to water content [Figure 1]. However, it is revealed numerically that the R2 and R3 waves are functions of excitation frequency. Given the same water saturation and excitation frequency, the phase speeds of the R2 and R3 waves are found to be approximately 90% of those of the second and third dilatational waves, respectively [Figures 2 and 3]. The phase speed and attenuation coefficient of the R2 and R3 waves bear the same trend as those of the P2 and P3 waves. The R1 wave has the lowest attenuation coefficient whereas the R3 wave attenuates highest. Due to its very low phase speed and high attenuation, the R3 wave is very difficult to observe in field experiments.
Figure 1 Effect of water saturation on the R1 and P1 waves at four excitation frequencies (a) phase speed (b) attenuation coefficient
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Figure 2 Effect of water saturation on the R2 and P2 waves at four excitation frequencies (a) phase speed (b) attenuation coefficient
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Figure 3 Effect of water saturation on the R3 and P3 waves at four excitation frequencies (a) phase speed (b) attenuation coefficient
4. SIGNIFICANCE OF FINDINGS
Most of the shaking felt from an earthquake is due to Rayleigh wave. The results presented in this study concern the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids. We demonstrated theoretically for the first time that there is the existence of three different types of Rayleigh wave motion, and their propagation and attenuation can be described in terms of relative fluid saturation and excitation frequency. These results are crucial and able to provide the physical basis for better understanding of the mechanism behind earthquake and its effect.
Reference
Lo, W. C. (2008), Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium, Advances in Water Resources, 31, 1399-1410. (SCI) (NSC: 95-2211-E-006-062)
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