Intrinsic attenuation from inhomogeneous waves in a dissipative anisotropic poroelastic medium
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 21 August 2009
Accepted: 13 December 2010
Published: 28 February 2011
A procedure is suggested to translate the quality factor of a plane harmonic attenuating wave in a general anisotropic elastic medium into its phase velocity and two finite non-dimensional attenuation parameters. A chosen value of the quality factor of an attenuated wave in the dissipative medium is used to specify its complex slowness vector for a general direction of propagation. In this specification, one of the attenuation parameter identifies the component of intrinsic attenuation along the direction of propagation of wave, i.e., homogeneous propagation of wave. Another parameter represents the component of attenuation in the direction orthogonal to propagation direction. It measures the deviation from homogeneous propagation and is termed as the inhomogeneity strength of the attenuated wave. These attenuation parameters alongwith phase velocity are used to calculate the rate of decay of amplitude of the attenuated wave along any given direction in propagationattenuation plane. Biot’s theory is used to study the propagation of four attenuating waves in an anisotropic poroviscoelastic medium in the presence of initial stress. For each wave, the specification of complex slowness vector is obtained in terms of its phase velocity and two attenuation parameters. Numerical results show that the attenuation contribution from the homogeneous propagation (of any of the three faster waves) is only a little in the total attenuation of any of these attenuated waves. The effects of the changes in anisotropy-type, initial-stress, frequency, fluid-viscosity, viscous characteristic length, and anelasticity of porous frame on the attenuation are also studied. It is found that though they affect the phenomenon of wave propagation and wave characteristics, major contribution to total intrinsic attenuation comes from the inhomogeneous propagation of the attenuated wave.
The accurate analysis of observed seismic attenuation is important for the advancement in knowing the structure of the earth. This is particularly true in exploration industry and in the investigation of tectonic stress and failure where small scale fracturing and flow of fluid into the fractures is important. The confined stresses and shale anisotropy helps in predicting flow path for improved oil recovery and designing hydraulic fracturing schemes. The flow mechanism to equilibrate fluid pressure produces a great deal of seismic attenuation at high frequency. The total attenuation inferred from the seismograms is the sum of intrinsic attenuation and scattering attenuation. Major contribution comes from the intrinsic attenuation (Sams et al., 1997), which represents the internal frictions between the adjacent material grains not elastically bonded. The seismic data from sedimentary regions exhibit more intrinsic attenuation that can be explained through existing theoretical models (Pride et al., 2004). Analysis of seismic data suggests that hydrocarbon deposits are often associated with higher than usual values of attenuation, but this is generally ignored during amplitude-versus-offset (AVO) analysis. These larger attenuations are, generally, translated into the stronger sources of attenuation, i.e., intrinsic physical processes, such as the interactions between the solid and the fluid, grain friction etc. Chapman et al. (2006) studied the reflections from the interface between the medium based on squirt flow concept and an elastic overburden. They found that reflection coefficient varies with frequency and the impact of this variation depends on the AVO behaviour at the interface. Liu et al. (2007) approved the use of difference in velocity and attenuation anisotropy to understand the mechanisms and to extract the additional information about subsurface fracture systems.
Whatever be the sources of attenuation of seismic waves in sedimentary rocks, the mathematical models are often required to explain the (rate of) decay of amplitude of waves propagating away from the source. This demands a much deeper insight into the process of wave propagation in realistic models of sedimentary rocks in the crust. These rocks can, more closely, be modeled as fluid-saturated porous solids pervaded by aligned cracks. The fluid-saturated microcracks are highly compliant and crustal rocks respond immediately to the small changes in differential stress (i.e., difference between confined tectonic stress and fluid pressure). Hence, a pre-stressed anisotropic porous solid makes a much realistic geophysical model to be used for seismic characterization of sedimentary or reservoir rocks. In particular, such a composite physical model facilitates the parametric studies of the influence of various measurable physical properties of the medium (i.e., porosity, permeability, pore-fluid viscosity, frame anelasticity, initial-stress, elastic/hydraulic anisotropy, pore-size, etc.).
Biot (1956) used Lagrange’s equations to derive a set of coupled differential equations that governs the propagation of motions of solid and fluid particles in a fluidsaturated porous medium. Biot (1962a, b) extended the acoustic propagation theory in the wider context of the mechanics of porous media and developed the new and simplified derivations of the fundamental equations of poroelastic propagation. A little later, Biot (1963) introduced the effect of initial stress in the mechanics of porous media and developed the basic equations. Stoll and Bryan (1970) used Biot’s theory (1962a, b) to study the wave attenuation in saturated sediments. In the sedimentary rocks, the anelastic nature alone may not be able to explain the observed attenuation. The confining stresses and wave-induced flow of viscous-fluid in pores and cracks are another important factors. In an initially-stressed porous medium of transverse isotropy, Sharma and Gogna (1991, 1993) studied the propagation of SH waves. Sinha et al. (1995) considered inhomogeneous pre-stresses in the porous medium resulting from the pressurized fluid in borehole. Sams et al. (1997) measured the large intrinsic attenuation in the stratified sequence of water-saturated porous sediments. Attenuation anisotropy in reservoir intervals is generally stronger than in overburdens. Numerical examples studied by Sharma (2005a) suggest that effect of initial-stress may be much more on attenuation as compared to velocities. Shapiro and Kaselow (2005) developed a formalism that describes the elastic moduli, anisotropy and porosity of rocks as functions of confining stress and pore pressure.
An earlier study of author (Sharma, 2005b) shows that, compared to velocity, the attenuation is more sensitive to the inhomogeneity of waves propagating in dissipative anisotropic poroelastic medium. Much earlier, Winterstein (1987) related the quality factor (Q) variations in a multilayered medium to the inhomogeneity (angle between propagation direction and direction of maximum attenuation) of the attenuating waves. In another study, Carcione (1999) suggested that the differences in amplitude variation with offset (AVO) of waves transmitted at ocean bottom depend not only on the properties of the medium but also on the inhomogeneity of the wave. This implies that the propagation of inhomogeneous waves may be able to explain, mathematically, the larger attenuation of seismic waves in sedimentary regions. Importance of inhomogeneous waves in single-phase viscoelastic media are found in Borcherdt (1977, 1982) and Cerveny and Psencik (2005a, b). Experimental results (Borcherdt et al., 1986; Hosten et al., 1987) confirm the generation and existence of inhomogeneous body waves and the differences in their physical characteristics from elastic body waves. Carcione (2006) studied the Rayleigh-window effect to explain the role of inhomogeneous waves in amplitude reduction of the reflection coefficient of the ocean bottom. The theory about inhomogeneous body waves in porous media has been given in Carcione (2007).
A mathematical model of wave propagation should explain the attenuation through the decay of amplitude away from source, i.e., AVO. The amplitude decay of a plane harmonic wave in dissipative media is derived from the specification of its complex slowness vector. A general analysis of attenuation requires the complex slowness vector to represent the propagation of inhomogeneous waves. The work presented proposes a procedure to relate the quality factor of attenuation to the inhomogeneity of the attenuated wave in a dissipative medium. The specification of complex slowness vector of the attenuated elastic wave provides its phase velocity, homogeneous attenuation and inhomogeneity strength, which can be used to calculate the decayrate of amplitudes of displacements of material particles. The procedure proposed is used to study the propagation of inhomogeneous waves in an anisotropic, initially-stressed, anelastic porous solid medium saturated with viscous fluid. The numerical examples considered certifies the direct relationship between quality factor of attenuation and inhomogeneity strength of an attenuated wave.
2. Inhomogeneous PlaneWaves
The general plane waves propagating in a dissipative medium are inhomogeneous waves (Borcherdt, 1982). These waves exhibit physical properties those are different from the body waves in elastic media. According to Caviglia et al. (1990), a plane wave is said to be inhomogeneous if its slowness vector is complex valued. The real and imaginary parts of complex slowness vector are termed as propagation vector and attenuation vector, respectively. In general, the inhomogeneity of an attenuating wave is represented through the difference in the directions of its propagation vector and attenuation vector. In other words, an angle between equi-amplitude plane and equi-phase plane of an attenuating plane wave represent its inhomogeneous character.
Finally, we have a system of equations to calculate the unknowns γ, δ, υ and inhomogeneity angle γ from a given value of Q for an attenuating wave in a dissipative medium. In other words, the procedure explained above is capable of translating the attenuation quality factor for a wave in any dissipative anisotropic elastic medium into its propagation and attenuation characteristics as inhomogeneous wave. However, this procedure requires an algebraic system for the medium so as to calculate the complex function h (= V2) corresponding to a given complex vector N. In the next section, such a system is derived for a general and realistic dissipative (pre-stressed anisotropic poroelastic) medium.
3. Pre-Stressed Anisotropic Porous Solid
3.1 Fundamental equations
3.2 Harmonic plane waves
4. Numerical Examples
The general mathematical model explained in previous sections provides a procedure to specify the complex slowness vectors for the propagation of four inhomogeneous waves in a saturated poroelastic solid under initial stress. The complex slowness vector of each inhomogeneous wave is constituted by the values of phase velocity (υ) and attenuation parameters (β, δ). Numerical examples are studied to analyze the variations of υ, β and δ with propagation direction for chosen values of Q, η, ω/ωc involved in the mathematical model. The numerical model is considered for North-Sea sandstone, a general anisotropic porous reservoir rock of density ρ = 2216 kg/m3 and porosity (Rasolofosaon and Zinszner, 2002). The interconnected pores of tortuosity and elastic constant R = 5 GPa, are filled with a viscous fluid of density 1000 kg/m3. The elastic constants for this saturated sandstone are defined in Appendix.
For a general direction (θ, φ), in three-dimensional space, the propagation direction is defined by (sin θ cos φ, sin θ sin φ, cos θ). A vertical plane, fixed by , is considered the propagation-attenuation plane for this study. In this plane, the orthogonal vector is considered along and θ, varying from 0 to 90°, represents propagation direction. Fixed values are assumed for frequency ω = θc and parameter η = 1.
In the numerical results discussed above it is noticed that, for three faster (qP1, qS1, qS2) waves, the value of β is much smaller than δ. Moreover, this smaller values of β is not the property of the physical parameters chosen for the elastic model used in numerical computation. To ascertain this, the numerical results were calculated for a numerical example with different physical parameters. Dolomite reservoir rock (Rasolofosaon and Zinszner, 2002) of porosity , tortuosity , density 2423 kg/m3 and R = 10 GPa was considered as an anisotropic porous solid saturated with water. The elastic tensor and permeability tensor for dolomite are given in Appendix. The variations of β and δ with θ were calculated for all the four waves with different values of Q. The values of β/δ for three faster waves in dolomite were not much different from those observed for sandstone in Figs. 1 to 3.
5. Concluding Remarks
It is generally believed that the phenomena associated with viscosity of pore fluid is the main cause of intrinsic attenuation of elastic waves in reservoir rocks and other fluidsaturated porous materials. In the present work, alongwith pore-fluid viscosity, the effects of pre-stress, anisotropy, frequency, anelasticity of porous frame on intrinsic attenuation are also studied. This study provides a procedure to relate the quality factor of an attenuating wave to its phase velocity and two finite non-dimensional attenuation parameters. These parameters represents an attenuating wave in a dissipative medium as a general inhomogeneous wave. The homogeneous attenuating wave is then obtained as a special case in this representation. When supported with a real/synthetic data, the suggested specification of the complex slowness vector of an attenuating wave can be used in the simulation studies on exploration seismology. For example, with this specification, it is possible to estimate the rate of radial decay of amplitudes of a wave in different directions in propagation-attenuation plane. Apart from attenuation parameters, this decay rate varies with phase velocity as well as frequency of the attenuating wave.
The intrinsic attenuation calculated for the propagation of a homogeneous wave in a dissipative medium is found to be much smaller as compared to the attenuation observed across the seismic range of frequencies. This implies that a larger intrinsic attenuation of seismic waves in sedimentary rocks may not be along the directions near to their propagation direction. More strictly, it should be in a direction, which is nearly normal to the direction of propagation. In other words, propagation of nearly evanescent waves may be able to explain the larger intrinsic attenuation observed in any anelastic or dissipative material. Two recent papers (Chapman et al., 2006; Liu et al., 2007) summarize the application of attenuation and anisotropy in the context of exploration geophysics. Hence, the ultimate applications of this study are geophysical, whether for hydrocarbon exploration, earthquake and structural engineering, or to the exploration of the solid earth. However, a pre-stressed dissipative anisotropic porous solid makes a realistic elastic model to study attenuation in composite materials for shock/sound absorbing properties.
- Albert, D. G., A comparison between wave propagation in water-saturated and air-saturated porous materials, J. Appl. Phys., 73, 28–36, 1993.View ArticleGoogle Scholar
- Biot, M. A., Non-linear theory of elasticity and the linearized case for a body under initial stress, Phil. Mag., 27, 468–489, 1939.View ArticleGoogle Scholar
- Biot, M. A., The theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low-frequency range, II. Higher frequency range, J. Acoust. Soc. Am., 28, 168–191, 1956.View ArticleGoogle Scholar
- Biot, M. A., Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys., 33, 1482–1498, 1962a.View ArticleGoogle Scholar
- Biot, M. A., Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Am., 34, 1254–1264, 1962b.View ArticleGoogle Scholar
- Biot, M. A., Theory of stability and consolidation of a porous medium under initial stress, J. Math. Mech., 12, 521–544, 1963.Google Scholar
- Borcherdt, R. D., Reflection and refraction of type-II S waves in elastic and anelastic media, Bull. Seismol. Soc. Am., 67, 43–67, 1977.Google Scholar
- Borcherdt, R. D., Reflection-refraction of general P and type-I S waves in elastic and anelastic solids, Geophys. J. R. Astron. Soc., 70, 621–638, 1982.View ArticleGoogle Scholar
- Borcherdt, R. D., G. Glassmoyer, and L. Wennerberg, Influence of welded boundaries in anelastic media on energy flow and characteristics P, S-I and S-II waves: Observational evidence for inhomogeneous body waves in low-loss solids, J. Geophys. Res., 91, 11503–11518, 1986.View ArticleGoogle Scholar
- Carcione, J. M., The effects of vector attenuation on AVO of off-shore reflections, Geophysics, 64, 815–819, 1999.View ArticleGoogle Scholar
- Carcione, J. M., Vector attenuation: elliptical polarization, raypaths and the Rayleigh-window effect, Geophys. Prosp., 54, 399–407, 2006.View ArticleGoogle Scholar
- Carcione, J. M., Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media, Elsevier, Amsterdam, 2007.Google Scholar
- Caviglia, G., A. Morro, and E. Pagani, Inhomogeneous waves in viscoelastic media, Wave Motion, 12, 143–159, 1990.View ArticleGoogle Scholar
- Cerveny, V. and I. Psencik, Plane waves in viscoelastic anisotropic media. Part 1: Theory, Geophys. J. Int., 161, 197–212, 2005a.Google Scholar
- Cerveny, V. and I. Psencik, Plane waves in viscoelastic anisotropic media. Part 2: Numerical examples, Geophys. J. Int., 161, 213–229, 2005b.View ArticleGoogle Scholar
- Chapman, M., E. Liu, and X. Y. Li, The influence of fluid-sensitive dispersion and attenuation on AVO analysis, Geophys. J. Int., 167, 89–105, 2006.View ArticleGoogle Scholar
- Crampin, S., Suggestions for a consistent terminology for seismic anisotropy, Geophys. Prospect., 37, 753–770, 1989.View ArticleGoogle Scholar
- Hosten, B. M., M. Deschamps, and B. R. Tittmann, Inhomogeneous wave generation and propagation in lossy anisotropic solid. Application to the characterisation of viscoelastic composite materials, J. Acoust. Soc. Am., 82, 1763–1770, 1987.View ArticleGoogle Scholar
- Johnson, D. L., J. Koplik, and R. Dashen, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid Mech., 176, 379–402, 1987.View ArticleGoogle Scholar
- Liu, E., M. Chapman, I. Varela, J. H. Queen, and H. B. Lynn, Velocity and attenuation anisotropy: implication of seismic characterisation of fractures, The Leading Edge, 26, 1171–1174, 2007.Google Scholar
- Pride, S. R., J. G. Berryman, and J. M. Harris, Seismic attenuation due to wave-induced flow, J. Geophys. Res., 109, B01201, 2004.Google Scholar
- Rasolofosaon, P. N. J. and B. E. Zinszner, Comparison between permeability anisotropy and elasticity anisotropy of reservoir rocks, Geophysics, 67, 230–240, 2002.View ArticleGoogle Scholar
- Sams, M. S., J. P. Neep, M. H. Worthington, and M. S. King, The measurements of velocity dispersion and frequency-dependent intrinsic attenuation in sedimentary rocks, Geophysics, 62, 1456–1464, 1997.View ArticleGoogle Scholar
- Shapiro, S. A. and A. Kaselow, Stress and pore pressure dependent anisotropy of elastic waves in porous structures, 3rd Biot Conference, Norman, Oklahoma, 2005.Google Scholar
- Sharma, M. D., Effect of initial stress on the propagation of plane waves in a general anisotropic poroelastic medium, J. Geophys. Res., 110, B11307, 2005a.View ArticleGoogle Scholar
- Sharma, M. D., Propagation of inhomogeneous plane waves in dissipative anisotropic poroelastic solids, Geophys. J. Int., 163, 981–990, 2005b.View ArticleGoogle Scholar
- Sharma, M. D., Propagation of inhomogeneous plane waves in viscoelastic anisotropic media, Acta Mech., 200, 145–154, 2008a.View ArticleGoogle Scholar
- Sharma, M. D., Existence of transverse waves in anisotropic poroelastic media, Geophys. J. Int., 174, 971–977, 2008b.View ArticleGoogle Scholar
- Sharma, M. D., Existence of longitudinal waves in anisotropic poroelastic media, Acta Mech., 208, 269–280, 2009.View ArticleGoogle Scholar
- Sharma, M. D. and M. L. Gogna, Propagation of Love waves in an initially stressed medium consisting of slow elastic layer lying over a liquid-saturated porous solid half-space, J. Acoust. Soc. Am., 89, 2584–2588, 1991.View ArticleGoogle Scholar
- Sharma, M. D. and M. L. Gogna, Reflection and transmission of SH waves in an initially stressed medium consisting of sandy layer lying over a fluid-saturated porous solid, Pure Appl. Geophys., 140, 613–628, 1993.View ArticleGoogle Scholar
- Sinha, B. K., S. Kostek, and A. N. Norris, Stoneley and flexural modes in pressurised boreholes, J. Geophys. Res., 100, 22375–22381, 1995.View ArticleGoogle Scholar
- Stoll, R. D. and G. M. Bryan, Wave attenuation in saturated sediments, J. Acoust. Soc. Am., 47, 1440–1447, 1970.View ArticleGoogle Scholar
- Winterstein, D. F., Vector attenuation: Some implications for plane waves in anelastic layered media, Geophysics, 52, 810–814, 1987.View ArticleGoogle Scholar