Determination of the Acoustic Coupling Factor of

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Determination of the Acoustic Coupling Factor of Biot's Theory of Elasticity, Using in situ Seismic Measurements Hilmi S. Salem Version of record first published: 10 Nov 2010.

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Determination of the Acoustic Coupling Factor of Biot’s Theory of Elasticity, Using in situ Seismic Measurements HILMI S. SALEM

Downloaded by [Mr Prof. Hilmi S. Salem] at 06:01 30 December 2012

Atlantic Geo-Technology Halifax, Nova Scotia, Canada Besides other elastic, petrophysical, and hydrophysical parameters, the acoustic coupling factor (coupling factor or structure factor, µ) is a critical micro-geometrical parameter in the Biot’s theory of elasticity. The coupling factor exerts a speci®c in¯uence on the mechanisms of propagation and attenuation of vibrations (seismic waves and acoustic signals) in porous media. It is a measure of the internal structure (geometry and shape of pores and pore tubes) of a porous medium, and thus it is a good indicator of the degree of coupling between pore ¯uid and solid grains. It is unity for no coupling and in®nity for perfect coupling. In this study, the coupling factor is obtained for surface soils and shallow sediments, using compressional wave velocity (¸p ) determined from in situ seismic refraction measurements. The investigated soils and underlying sediments (aeration zone and aquifer) exhibit a general µ range of 1.22±3.19 (average = 1.82), corresponding to a general ¸f range of 134±2060 m/s (average = 1134 m/s). The µ values are correlated to the bulk modulus-shear modulus ratio (K/·) and the porosity (¿). A direct linear correlation between µ and K/· and an inverse polynomial correlation between µ and ¿ are obtained, with coe cients of correlation of 0.91 and 0.98, respectively. Also, various items related to the coupling factor, Biot’s theory, attenuation mechanisms, fast and slow compressional waves (®rst and second kinds), and shear waves are discussed. Keywords acoustic coupling factor (structure factor), attenuation mechanisms, Biot’s theory, compressibility, compressional and shear wave velocities, fast and slow compressional waves, frequency, in situ seismic refraction measurements, permeability, porosity, rigidity, shallow sediments (aeration zone and aquifer), surface soils, tortuosity, viscosity

Frenkel (1944), Wood (1949), Gassmann (1951a,b), Biot (1956a,b, 1962a,b), Biot and Willis (1957), and Geertsma and Smit (1961) discussed comprehensively the continuum theory of elasticity of porous media saturated with a compressible viscous ¯uid. In his pioneering studies on theoretical treatment of propagation of vibrations (seismic waves and acoustic signals) through porous media, Biot derived stress±strain relations as functions of dilatation of matrix structure and pore ¯uid, porosity, permeability, viscosity, pore geometry, pore and pore-¯uid pressures, and various elastic moduli.

Received 22 November 2000; accepted 2 February 2001. Address correspondence to Dr. Hilmi S. Salem, Atlantic Geo-Technology, 24 Alton Drive, Suite 307, Halifax, Nova Scotia, Canada B3N 1L9. E-mail: [email protected]

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H. S. Salem


Frequency As indicated in Biot’s theory, the eVect of frequency ( f ) can be evaluated with respect to a reference frequency, known as critical frequency ( fc ), which is a characteristic frequency of an investigated porous medium. The critical frequency is de®ned as ¿²/2º»f K (Wyllie et al., 1961, 1962; Gardner et al., 1964), where ¿ is the fractional porosity; ² is the dynamic viscosity of pore ¯uid; »f is the density of pore ¯uid; and K is the permeability of porous medium. If the employed frequency, f, is lower than 0.15 fc (i.e., f/fc 0.15), it may be assumed that the Poiseuille’s ¯ow no longer occurs in any of the pore spaces, and thus the frequency range is called ``high (or in®nite) frequency.’’ The Poiseuille’s ¯ow implies that the inertia forces of ¯uid, passing through the capillary tubes of a porous medium, are negligible in comparison to the viscosity forces of ¯uid.

Application of Biot’s Theory and Mechanisms of Attenuation In the last 50 years, Biot’s theory has been successfully applied in a diversity of ®elds, including geophysics, seismology, acoustics; petroleum-, water-resources, and civil engineering, as well as other areas of engineering, using a wide range of frequencies, starting from a few Hz or hundreds of Hz (as in the case of seismic exploration) to kHz or MHz (as in the case of laboratory measurements). Some examples are given below for which Biot’s theory has been applied. Stoll and Bryan (1970), Stoll (1974, 1977, 1979, 1980, 1989), and Stoll and Kan (1981) thoroughly investigated the acoustic signal transmission, re¯ection, and attenuation in unconsolidated ocean sediments. They developed theoretical models that include mechanisms of energy losses (also known as wave or signal attenuation, dissipation, or damping) and veri®ed their models experimentally, using parameters such as porosity, permeability, grain size, and eVective stress. The mechanisms of attenuation, incorporated in Biot’s theory, have been discussed, theoretically and experimentally, in several papers (e.g., Wyllie et al. (1961, 1962), Gardner et al. (1964); Stoll and Stoll and coworkers (mentioned above), and other researchers mentioned below). The mechanisms of attenuation can be summarized as follows: (1) Skeletal-frame (intergranular or frictional) mechanism: resulting from friction and/or inelasticity of any bonds existing at the contact points of the composite grains in a porous medium. This mechanism is also known as ``jostling.’’ (2) Viscous (viscoelastic relaxation) mechanism: resulting from movement of the interstitial pore ¯uid relative to the skeletal frame. This mechanism is also known as ``sloshing.’’ (3) Squeeze-®lm (local ¯ow) mechanism: resulting from localized micro-movement of ¯uid in and out of the pore spaces as the pore spaces open and close in a variable stress ®eld. For a typically unconsolidated, porous, and permeable medium saturated with a viscous ¯uid, the three mechanisms of attenuation ( jostling, sloshing, and squeeze ®lm) may arise together because of the high losses at the loose grain contacts


The Acoustic Coupling Factor of Biot’s Theory of Elasticity

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(mechanism #1), the high ¯uid±solid losses due to high permeability (mechanism #2), and the high movement of the pore ¯uid in and out of the pore spaces (mechanism #3). It is also possible that attenuation can be caused by re¯ection of the wave (signal) at the interface between two layers of diVerent lithologies or at the interface between two layers that have similar lithologies but are saturated with diVerent ¯uids. Attenuation is also caused by secondary mechanisms, such as thermoelasticity and scattering. Gardner et al. (1964) investigated the eVects of pressure and partial and total saturation of ¯uids on the elastic wave attenuation in sands. Dutta and OdeÂ (1979a,b) used White’s (1975) model (based on Biot’s theory) to investigate the attenuation and dispersion of compressional waves in sediments saturated partially with gas. Hovem and Ingram (1979) investigated the viscous attenuation of acoustic waves in saturated porous media and concluded that the viscous attenuation may be of signi®cant importance for higher frequency vibrations propagating through sands of high permeability. Berryman (1980, 1981) identi®ed, theoretically, coe cients in Biot’s theory that are important for attenuation in fully and partially consolidated frames. Meissner and Theilen (1983) investigated wave attenuation in shallow sediments and pointed out that wave attenuation depends considerably on permeability of the medium and viscosity of the pore ¯uid. Bedford et al. (1984) and Yavari and Bedford (1991) used the ®nite element method to obtain Biot’s inertial drag coe cient (opposite to viscous) and the virtual mass coe cient for an arbitrary pore geometry. Turgut and Yamamoto (1988) obtained the porosity and permeability for marine sediments by using synthetic seismograms that include the eVects of dispersion and attenuation. Mu and Badiey (1999) examined the eVect of uncertainty of input parameters used in predicting acoustic response in ocean sediments. Wu and Benson (1999) used the Gassmann±Biot±Geertsma and Smit models to obtain elastic moduli of saturated porous media. Biot’s Theory and the Three Kinds of Waves Biot’s theory con®rms the existence of three kinds of body waves (two compressional waves (also known as delatational or primary waves ``P-waves’’) and one shear wave (also known as rotational or secondary wave ``S-wave’’)). According to Biot (1956a, 1962a), the three kinds of waves were originally predicted by Frenkel (1944). The two compressional waves are the fast wave (P1 ) and the slow wave (P 2 ). The P1 -wave and the shear wave (S), which are important in geophysical exploration, propagate with little dispersion. The P2 -wave (also known as the diVusion wave), which is important in acoustical problems involving highly compressible pore ¯uids such as gases, moves very slowly and attenuate s very rapidly. The P1 -wave is described as a true or customary (normal or classical) wave (Biot, 1956a; Geertsma and Smit, 1961; Dutta and OdeÂ, 1979a,b), whereas the P2 -wave is described as an elusive wave (Stoll, 1980). The second kind (slow; P2 ) of the compressional waves is seldom observed in water-saturated sediments because its amplitude is much smaller than the amplitude of the ®rst kind wave (fast; P1 ). Plona (1980) used measurements of ultrasonic frequency (500 kHz) on water-saturate d sediments, with a range of porosity (¿) of 8±28%, and veri®ed the existence of the slow wave (P 2 ) propagatin g at a velocity of 25% of the normal ``fast’’ compressional wave. He also observed that the velocity of the slow wave (P2 ) decreases with decreasing the porosity (unlike the


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velocities of the fast compressional wave (P1 ) and the shear wave (S), which increase with decreasing the porosity). Plona attributed the decrease of the P2 -wave velocity with the decrease in the porosity to the decrease of the wave’s amplitude. Berryman (1980) con®rmed, theoretically, Plona’s (1980) experimental observations that indicated the existence of the P2 -wave. Johnson (1980) conducted experimental measurements on fused glass-bead samples and showed that the fourth-sound speed in super¯uid He, at low temperature, is a special case of the slow compressional wave (P2 ). He observed that the fourth-sound and the P 2 -wave have almost identical velocities, ranging from 137 to 174 m/s. For ¯uid-®lled porous and permeable media, Chandler and Johnson (1981) and Johnson and Plona (1982) indicated that at high frequency range there is always a slow compressional wave (along with the fast compressional wave and the shear wave) and at low frequencies the slow wave is diVusive. For porous media composed of ®ne particles, Turgut and Yamamoto (1988) proposed a theoretical model that predicts higher energy losses through conversion to the slow compressional wave (P2 ). At frequencies below 1 kHz, Chotiros (1995) conducted experiments on unconsolidated water-saturated sands and concluded that the slow wave (P2 ) is diVusive and that it does not contribute to the propagated waves, except as a loss mechanism. Mu and Badiey (1999) demonstrated that the variation coe cients of the fast wave (P1 ) and the shear wave (S) are weakly dependent on frequency, meanwhile they (variation coe cients) are strongly dependent on frequency in the case of the slow wave (P2 ). Study Area The study area, in Schleswig-Holstein, northern Germany, is covered with glacial deposits of the Pleistocene age. The deposits are characterized by a high degree of heterogeneity, both laterally and vertically. They are composed of silts; ®ne, medium, and coarse sands and gravels; and a small fraction of clays ( 1.5%), with a variety of grain sizes and shapes (Schroeter, 1983). Large quantities of the sands and gravels were laid down as outwash material, swept out from the melting glaciers by melt water streams, and also as moraines, and deposited in front of the glaciers. Electric measurements carried out for the same area showed that the surface soils are underlain by an aeration zone that is underlain by an aquifer (Salem, 1999). The aquifer is underlain by an aquiclude composed of glacial clays, known as ``Geschiebemergel ’’ (till). The aquifer is composed of several layers, each of which ranges in thickness from 1 to 54 m. The total thickness of the aquifer ranges from 30 to 70 m, with a water table 5±20 m deep, depending on the altitude of the surface. The permeability of the aquifer ranges from 7.51 10 5 to 2.87 10 3 m/s (average = 8.7 10 4 m/s) (Salem, 1999).

Aim of study In the present study, the acoustic coupling factor (coupling factor or structure factor, µ), as one of the critical micro-geometrical parameters in Biot’s theory of elasticity, has been determined using the compressional wave velocity obtained from in situ seismic refraction measurements for the surface soils and the underlying sediments in the study area. To the author’s knowledge, this is the ®rst time that the coupling factor, which contributes eVectively to the mechanisms of the propagation


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and attenuation of vibrations (seismic waves and acoustic signals) through porous media, is obtained from in situ seismic measurements. In this study, the coupling factor has been investigated statistically, theoretically, analytically, and numerically. Empirical equations, correlating among µ, the ratio of incompressibility (bulk modulus, K ) to rigidity (shear modulus, ·), and the porosity (¿) have been obtained with high coe cients of correlation. These equations enable one to obtain the coupling factor directly from K/· or ¿.

Acoustic Coupling Factor


Background Propagation of the P- and S-vibrations through porous media depends on various in¯uences. These include lithological properties (texture (mineralogy, size, shape, packing, sorting, and distribution of grains, as well as size and shape of pores and pore tubes), kind and amount of clay, degrees of compaction, consolidation and cementation); physical properties (density, porosity, permeability, tortuosity, viscosity, degree of saturation and kind of saturant, pressure, and temperature); and elastic properties (compressibility (inverse of bulk modulus `ìncompressibility’’), rigidity (shear modulus), Young’s (elasticity or extensional) modulus, Lame’s constant, Poisson’s ratio, and acoustic coupling factor (or coupling factor, µ)). Fluid-Motion Parameters The permeability of a porous medium, viscosity of a saturating ¯uid, size of pores, and coupling factor are known as ``¯uid-motion parameters’’ of Biot’s theory. The permeability (K) and viscosity (²) are well-known parameters. The pore-size parameter (ap ), which is twice the hydraulic radius, depends on the size and shape of the pores. The ap (with the dimension of length) has been generally estimated to be in the range of 1/8±1/6 (average 0.145) of the mean diameter of the grains composing a porous medium (Stoll, 1974; Yavari and Bedford, 1991). The coupling factor, µ, which is the main target of the present study, was originally introduced by Zwikker and Kosten (1949) and later used by Biot (1956b) in his theory. Several researchers have obtained the coupling factor either theoretically, experimentally, or numerically. Some researchers (e.g., Stoll (1974, 1979); Ogushwitz (1985a)) suggested that in the case of real materials, µ must be obtained experimentally or empirically. De® nition and Terminology of Coupling Factor The term `àcoustic coupling factor’’ or simply ``coupling factor’’ µ, is used in this study after Geertsma (1961). The coupling factor is a measure of the combined eVects of the sinuosity of pore tubes and the shape of pores, pore tubes, and grains in a porous medium. Thus µ is a good indicator of the degree of coupling between pore ¯uid and solid grains in a porous medium. Biot (1956b) introduced in his theory two micro-geometrical parameters, namely, the ``sinuosity factor’’ (¹), which accounts for the twists and turns encountered in the ¯uid ¯ow through the pore spaces, and the ``structural factor’’ (¯), which accounts for the shape of pores, pore tubes, and grains. The sinuosity factor, ¹, is >1, whereas the structural factor,


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H. S. Salem

¯, is equal to (8.0¹)1/2 for circular pores and to (5.33¹)1/2 for slit-like pores (Biot, 1956b). For circular and slit-like pore shapes, Biot (1956b) chose, arbitrarily, an average value of (8)1/2, ( 2.83), to represent the structure factor, ¯, assuming a value of 1 for the sinuosity factor, ¹. The sinuosity factor, ¹, is widely known in the literature as ``tortuosity, ½.’’ Tortuosity is a measure of the tortuous (wavy or sinuous) nature of the capillary tubes (channels) in a porous system. It describes the eVective (or actual) length of ¯uid ¯ow through a unit length of a porous medium. It is a conceptual dimensionless number, representing the departure of a porous system from being composed of a bundle of straight capillary tubes (Salem and Chilingarian, 2000a,b). Because of the de®nition and the physical impacts of µ, some researchers used the term ``coupling factor’’ or ``coupling coe cient’’ (Geertsma, 1961; Domenico, 1977; Brown and Korringa, 1979; Brown, 1980), and others used the term ``structure factor’’ or ``structure constant’’ (Morse, 1952; Biot, 1956b; Paterson, 1956; Wyllie et al., 1961, 1962; Stoll and Bryan, 1970; Hovem and Ingram, 1979; Stoll, 1980; Johnson and Plona 1982; Meissner and Theilen, 1983; Ogushwitz, 1985a,b). Importance of Coupling Factor The coupling factor, µ, is a function of several in¯uences. For example, an increase in the µ-value indicates a greater pressure and greater cementation, compaction, and consolidation, as well as a greater saturation. The coupling factor is a signi®cant parameter in understanding the mechanisms of propagation and attenuation of vibrations in porous media. Wyllie et al. (1958) pointed out that the degree of ¯uid±solid coupling is aVected by variations in the pore pressure. Berry (1959) indicated that the sediments have rigidity if the grain-to-grai n contact points are somewhat fused or cemented by an acoustically similar material, and hence the grains will be acoustically coupled. Geertsma (1961) pointed out that µ is an important parameter in variations of the compressional and shear wave velocities in porous media. Stoll and Bryan (1970) and Stoll (1974, 1977, 1979) pointed out that µ accounts for the fact that not all of the pore ¯uid moves in the direction of the macroscopic pressure gradient because of the tortuous, multidirectional nature of the pores. Brown (1980) mentioned that the viscous eVect of a ¯uid transmitted through a porous medium provides a degree of coupling between the solid and ¯uid components, and thus one can expect that the viscous eVect limits the motion of ¯uid saturating the pores. Berryman (1980, 1981) described µ as a pure number related to the frame inertia in a ¯uid environment. Ogushwitz (1985a) pointed out that µ accounts for an apparent increase in ¯uid inertia caused by tortuosity of the pores. Wyllie et al. (1962) conducted high-frequency measurements on various samples of sedimentary rocks and pointed out that variations in the µ-value are important for the seismic wave (acoustic signal) attenuation. In his study on highfrequency measurements of acoustic velocities in sand and glass-bead specimens saturated with brine (highly saline water), Domenico (1977) pointed out that µ is one of the most critical parameters in Biot’s theory. Johnson (1980) and Johnson and Plona (1982) conducted high-frequency measurements of acoustic velocities in consolidated and unconsolidated porous media and pointed out that µ is one of the little understood parameters in Biot’s theory. They also pointed out that µ is a crucial parameter for the slow-wave propagation in systems of very stiV frames. In their


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study on low-frequency measurements of seismic wave velocity in shallow sediments, Meissner and Theilen (1983) pointed out that µ is an important parameter in understanding the mechanisms of attenuation.


Coupling Factor and Frequency Several studies have indicated that the coupling factor (structure factor), µ, is an independent parameter of frequency. Biot (1956b) showed that µ remains constant with the change of frequency, but varies with the wave velocity and attenuation. Geertsma (1961) assumed the same µ-value (3.0) in application to measurements of wave velocities at both low and high frequencies. Stoll and Bryan (1970) and Stoll (1977) demonstrated that µ is independent of frequency, unlike the pore-size parameter, ap , and the mobility ratio (viscosity of pore ¯uid divided by permeability of medium, i.e., ²/K ), which are frequency-dependen t quantities. Domenico (1977) pointed out that µ is a purely geometrical quantity independent of frequency. He also suggested that when the coupling between pore ¯uid and grains is perfect or nearly so (i.e., µ Õ ?), the compressional and shear wave velocities are independent of frequency as well. In their theoretical studies on the drag and virtual mass coe cients of Biot’s theory, Bedford et al. (1984) and Yavari and Bedford (1991) indicated that the drag coe cient (de®ned in terms of ¯uid viscosity and pore-size parameter) is frequency dependent, whereas the virtual mass coe cient (de®ned in terms of the coupling factor) has been typically assumed by several researchers to be independent of frequency.

Coupling Factor in Relation to Other Parameters In his study on superleak material (unconsolidated packed powder), Rudnick (1976) found that µ = 2 ± ¿ = n2 , where n is the acoustic index of refraction of super¯uid He. Johnson et al. (1982) found that µ and ¿ are inversely related, whereas µ and n are directly related (µ = ¿ 1/2 = n2 ). Hovem and Ingram (1979) related the coupling factor, µ, to the Kozeny±Carman coe cient (Kcc ), as µ = 1 + (Kcc/6). The Kozeny = Carman coe cient (Kozeny, 1927; Carman, 1937) implies that the ¯uid ¯ow is nonuniform, and thus Kcc can have a wide range of values, depending on the degree of complexity of the passages of ¯uid ¯ow (Salem and Chilingarian, 1999a,b, 2000a,b; Salem, 2000a,b). In his theoretical studies on isolated spherical solid particles oscillating in a ¯uid, Berryman (1980, 1981) found that µ = r (¿ 1 + 1), where r (dimensionless) is the frame’s inertia, which indicates the amount of ¯uid displayed by the particles. For spherical-shaped particles, r = 0.5; for ellipsoidal-shaped particles, r ranges from 0 to 1. Berryman (1980) pointed out that the value of r must be derived from a microscopic model of the solid frame moving in the ¯uid. For unconsolidated and consolidated (fused) glass-bead samples saturated with water, Johnson and Plona (1982) de®ned µ in terms of the hydrodynamic drag parameter (l) as µ = 1/(1 l). The l (dimensionless, with a range of 0±1) is de®ned as a measure of pulling the solid frame through the ¯uid at a constant speed. The de®nitions of both parameters (r in Berryman’s formula and l in Johnson and Plona’s formula) indicate that both parameters have the same physical impact. For example, they (r and l) increase (approach 1) by decreasing the porosity. Yavari and Bedford (1991) used

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the ®nite element method and found that µ and ¿ are inversely related, i.e., µ = 1 + (0.227) [(1 ¿)/¿].


Variations of Coupling Factor Variations in the shape and orientation of the composite grains and pores in a porous medium lead to a wide variation in the value of µ. Zwikker and Kosten (1949) used a µ-value of 3, representing a randomly oriented pore system. Ferrero and Sacerdote (1951) used a µ-value of 4.3 for sands, which was later used by Paterson (1956). For a porous medium composed of closely packed solid particles immersed in a ¯uid, Morse (1952) stated that µ is always >1 and generally ranges from 2.0 to 3.4. Geertsma (1961) pointed out that µ is always >1 and if µ = 1, then there is no coupling between grains and pore ¯uid. He also pointed out that µ, theoretically, is in®nity for perfect coupling. Nolle (1963) pointed out that the fact that µ is always >1 is due to the devious (wavy) paths of ¯uid ¯ow through porous media. As mentioned earlier, Biot (1956b) arbitrarily chose an average µ-value of 2.83, representing circular and slit-like pore shapes. Wyllie et al. (1961) used a µ-value of 3, corresponding to porous media consisting of rigid, immovable, randomly oriented circular tubes. Wyllie et al. (1962) used µ-values ranging from 1 to 3, corresponding to diVerent consolidated sedimentary rocks, with a porosity range of 0.2±0.4. Wyllie et al. (1962) found that the µ-value of 2 gave the best agreement between their theoretical and experimental investigations. Stoll and Bryan (1970) and Stoll (1974) pointed out that for uniform circular pores with axes parallel to the pressure gradient, µ would theoretically have a value of 1, whereas for a random system with all possible orientation of pores, µ would theoretically have a value of 3; for real materials having irregular pores, µ would vary within a wide range of values. For diVerent ocean sediments (sands, silts, clays, and muds with a range of fractional ¿ of 0.34±0.76), Stoll and Bryan (1970) used a µ-range of 1.0±4.3. The value of 1.0 is the minimum value permissible and the value of 4.3 is that given by Ferrero and Sacerdote (1951). In his study on attenuation of acoustic signals in loose marine sediments partially saturated with gas, Stoll (1977) used assumed µ-values of 1.25 and 3. Domenico (1977) observed that the µ-value of 1 (corresponding to low diVerential pressure: the diVerence between matrix pressure and ¯uid pressure) indicates no coupling or a slight coupling between the pore ¯uid and the composing grains. He also observed that by increasing the diVerential pressure, µ increases gradually to values between 2 and 3. Hovem and Ingram (1979) obtained a µ-range of 1.0±1.8, corresponding to water-saturated sands. Dutta and OdeÂ (1979b) investigated the dependence of attenuation on the coupling factor. They used a variety of µ-values (1, 5, 10, and 100) and found that variations of the µ-value produce diVerent levels of attenuation. Johnson (1980) and Johnson et al. (1982) obtained a µ-range of 1.75±3.84, corresponding to a porosity range of 0.11±0.34. Hamdi and Smith (1982a,b) correlated µ to the fractional porosity of unconsolidated marine sediments and found that µ varies from 1 (for single-sized spherical sands) to a value equal to 1/¿ (for clayey sediments), indicating an inverse relationship between µ and ¿. Yavari and Bedford (1991) obtained a µ-range of 1.24±1.64, corresponding to a ¿-range of 0.26±0.48. Chotiros (1995) obtained a µ-range of 1.75±1.89, corresponding to a ¿-range of 0.36±0.40.


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Theory Saturated porous media are aVected by an internal stress produced by the ¯uid(s) saturating the pores and an external stress produced by the overlying sediments. Propagation of vibrations through a saturated medium leads to changes (deformation) in the pore volume, pore-¯uid volume, and solid (grain) volume, which all result in a total (bulk) change in the volume of the medium. The changes of pore volume, ¯uid volume, solid volume, and bulk volume are, respectively, pore compressibility (Cp ), ¯uid compressibility (Cf ), solid (grain) compressibility (Cs ), and bulk compressibility (Cb ). The compressional wave velocity (¸p ), in m/s, can be obtained in terms of the bulk modulus (incompressibility; K = 1/Cb ) and the shear modulus (rigidity; ·), both in N/m 2 ( = 1 Pa = kg/m.s 2 ), as well as the bulk density (»b ), in kg/m 3 , as (Biot, 1956a,b) ¸p = [{K + (4·=3)}=»b ]1=2 :

(1)

If the compressional wave passes through a ¯uid (· is zero), then Equation (1) is ¸p = (K=»b )1=2 :

(2)

The bulk modulus, K, is de®ned as the applied stress divided by strain (the proportional change in volume of a porous material), which can be obtained from Equation (1), i.e., K = ¸2p»b (4·=3):

(3)

The shear modulus, ·, is de®ned as the applied stress divided by the change in shape (shear strain), which can be obtained in terms of the shear wave velocity (¸s ) and »b as · = ¸2s »b :

(4)

Based on Biot’s theory (more details on Biot’s theory are given in the references of Biot and those of many researchers mentioned earlier), Geertsma (1961) de®ned ¸p , in terms of Cs , Cf , Cb , »f , »b , ¿, ·, and µ (all de®ned earlier, except the compressibility factor: = Cs /Cb ), as ¸2p = ((1=Cb ) + (4·=3)) + 1 : »b (1 (»f ¿=»b µ))

(¿ =µ»f ) + (1 )(1 2¿=µ) (1 ¿ )Cs + ¿Cf (5)

The ¸p in Equation (5) represents the velocity of the fast compressional wave (P1 ) at high frequency. Application of Biot’s theory shows that the diVerence between the wave velocity calculated at low ``zero’’ frequency and that at high `ìn®nite’’ frequency is