Study on the Biot-Stoll model for porous marine sediments

Transcription

1 Acoust. Sci. & Tech. 28, 4 (2007) PAPER #2007 The Acoustical Society of Japan Study on the Biot-Stoll model for porous marine sediments Masao Kimura School of Marine Science and Technology, Tokai University, 3 20 Orido, Shimizu-ku, Shizuoka, Japan ( Received 0 April 2006, Accepted for publication 24 October 2006 ) Abstract The Biot-Stoll model is useful for analyzing the acoustic wave propagation in porous marine sediments. However, the practical application of the model is not easy, because 3 physical parameters are required to apply it, and the physical phenomena in this model are difficult to understand. In this study, equivalent circuits for the plane longitudinal wave propagation in porous media are proposed for easy understanding of the physical phenomena. Also, practical approximated equations for the longitudinal wave velocity and attenuation are proposed to simplify practical application of the Biot-Stoll model. Then, simpler models, such as the Wood model, the Gassmann model, and the Biot-Stoll model without frame moduli can be used within limited ranges of frequency and porosity to approximate the exact Biot-Stoll model. Therefore, the ranges of frequency and porosity applicable to some simple acoustic models are obtained to simplify practical application of the Biot-Stoll model. Keywords Biot-Stoll model, Marine sediment, Velocity, Attenuation, Porosity PACS number Ma [doi0.250/ast ]. INTRODUCTION The Biot-Stoll model is extensively used to analyze the acoustic wave propagation in porous marine sediments, and its usefulness has been shown by many researchers [ 9]. Hovem and Ingram showed the viscous attenuation of a longitudinal wave in sand and suspension [,2]. Kimura investigated the velocity and attenuation of a longitudinal wave in high-porosity marine sediment with grain size distribution [3]. Stoll and Kan obtained the reflection characteristics of acoustic waves at a water-sediment interface [4]. Yamamoto illustrated the longitudinal wave velocity and attenuation in marine sediments [5]. Ogushwitz applied the Biot-Stoll model to low-porosity rock and high-porosity suspension and showed the usefulness of the model [6 8]. Holland and Brunson investigated the Biot parameters [9]. As mentioned above, the Biot-Stoll model is useful tool for analyzing the acoustic wave propagation in porous marine sediment. However, it is not easy to apply this model, because 3 physical parameters are needed, and some of the parameters cannot be measured. Also, the physical phenomena in this model are difficult to understand. First, the Biot-Stoll model, the 3 physical parameters mkimura@scc.u-tokai.ac.jp required for applying the model, and the frequency characteristics of the longitudinal wave velocity and attenuation are summarized. In this study, equivalent circuits for the plane longitudinal wave propagation in porous media are proposed to easily understand the physical phenomena. Exact and approximated equivalent circuits are derived. Next, approximated equations for the longitudinal wave velocity and attenuation as functions of porosity and frequency are derived to simplify practical application of the Biot-Stoll model. Moreover, the ranges of frequency and porosity applicable for acoustic models such as the Wood model, the Gassmann model, and the Biot-Stoll model without frame moduli are obtained. 2. ACOUSTIC MODELS In this section, the acoustic models of porous marine sediments are described. The Wood model is the simplest acoustic model in a two-phase media [0]. The Gassmann model is an acoustic model including the frame bulk modulus []. Then, the Biot model was developed as an extensive acoustic model for fluid-saturated porous media [2 5]. Moreover, Stoll applied the Biot model to unconsolidated marine sediments and extended the Biot model by adding the friction loss between the grains [6 20]. Nowadays, this extended Biot model is called the Biot-Stoll model and is widely used in analyzing the acoustic wave propagation in porous marine sediments. In 230

2 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS this section, we summarize the Wood model and the Gassmann model. The Biot-Stoll model will be described in the next section. 2.. The Wood Model In a suspension, where the heterogeneities are small compared with the wavelength, the longitudinal wave velocity c l is expressed as follows [0] sffiffiffiffi K c l ¼ ; ðþ ¼ f þð Þ r ; ð2þ K ¼ þð Þ ; ð3þ K f K r where is the total density of the medium, f and r are the densities of the fluid and the grain, respectively. Here, K is the total bulk modulus, K f and K r are the bulk moduli of the fluid and the grain, respectively, and is the porosity The Gassmann Model Gassmann first introduced the idea of a frame that is an assemblage of grains. In the Gassmann model, the longitudinal wave velocity c l in the two-phase medium is expressed as follows [] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K þ 4 u 3 c l ¼ t ; ð4þ ¼ f þð Þ r ; ð5þ þ K f K r K b K r K ¼ ; ð6þ þ Kr K b K f K r Kb Kr where is the frame shear modulus and K b is the frame bulk modulus. Equation (6) is valid only at lower frequencies, for which there is sufficient time for the pore fluid to flow and eliminate wave-induced pore pressure gradients. 3. THE BIOT-STOLL MODEL 3.. The Biot-Stoll Model Biot developed a comprehensive model of the acoustic wave propagation in fluid-saturated porous media such as marine sediments [2 5]. The Biot model predicts that three kinds of body waves, two longitudinal and one shear, may exist in a fluid-saturated porous medium in the absence of boundaries. One of the longitudinal waves which is called the first kind, and the shear wave are similar to waves found in elastic media. In these waves, the motions of the frame and the pore fluid are nearly in phase, and the attenuation due to viscous losses is relatively small. Grain Pore fluid Frame Fig. Grain, pore fluid, and assemblage of the grains, which is the frame in the Biot-Stoll model. In contrast, the longitudinal wave of the second kind is highly attenuated and the frame and the fluid move largely out of phase. The longitudinal wave of the second kind becomes very important in acoustical problems in gassy sediments, with very compressible pore fluids, such as air [20 22]. However, in the case of marine sediments with no gasses, the longitudinal wave of the first kind is of principal importance. In this study, we will treat only the longitudinal wave of the first kind. Stoll applied the Biot model to unconsolidated marine sediments and extended the Biot model by considering the friction between the grains by making the frame bulk and shear moduli complex [6 20]. Nowadays, this extended Biot model is called the Biot- Stoll model. Porous marine sediment is composed of the assemblage of grains, which is the skeletal frame, and seawater saturating the pores, which is the pore fluid, as shown in Fig.. In the Biot-Stoll model concerning the acoustic wave propagation in porous saturated media such as marine sediment, () fluid loss (viscous loss) incurred by the relative motion of the pore fluid to that of the frame and (2) frame loss incurred by the friction of the grain-to-grain contact are considered to cause the energy loss during the acoustic wave propagation. A summary of the Biot-Stoll model is shown as follows. The wave equations for the longitudinal wave in the porous saturated media derived by Biot are expressed as follows [3] In Eq. (7), r 2 ðhe 2 ðe fþ; r 2 ðce ð fe mþ F e ¼ divðuþ; ðu displacement of the frameþ ð8þ ¼ divðu UÞ; ðu displacement of the pore fluid, porosityþ ð7þ ð9þ 23

3 Acoust. Sci. & Tech. 28, 4 (2007) e dilatation of the element attached to the frame volume of fluid that has flowed in or out of an element of volume attached to the frame H ¼ ðk r K b Þ 2 þ K b þ 4 ; ð0þ D K b 3 C ¼ K rðk r K b Þ ; ðþ D K b M ¼ K r 2 ; ð2þ D K b D ¼ K r þ K r ; ð3þ K f where K f and K r are the bulk moduli of the pore fluid and grain, respectively. Here, K b and are the bulk and shear modului of the frame, respectively and are expressed as follows K b ¼ K br þ jk bi ¼ K br þ j l ¼ r þ j i ¼ r þ j s ; ð4þ ð5þ The imaginary parts in Eqs. (4) and (5) express the loss incurred due to the friction between the grains. Here, l and s are the bulk logarithmic decrement and shear logarithmic decrement, respectively. The density of sediment is as follows ¼ f þð Þ r ; ð6þ where f and r are the densities of the pore fluid and the grain, respectively. The added mass is expressed as follows m ¼ f ; ð7þ where is the structure factor, F=k is the viscous resistance, is the fluid viscosity, and k is the permeability as shown in the next equation, which is called the Kozeny- Carman relation k ¼ d2 3 36k 0 ð Þ 2 ; ð8þ where d denotes the grain diameter, and k 0 is a constant determined by the shape of the pore and the tortuosity. The viscous correction factor is as follows TðÞ FðÞ ¼F r ðþþjf i ðþ ¼ 4 þ j 2TðÞ ; ð9þ where TðÞ ¼ ber0 ðþþjbei 0 ðþ berðþþjbeiðþ ; ð20þ where ber and bei denote the Kelvin functions, and ber 0 and bei 0 are the derivatives of the Kelvin functions, and rffiffiffiffiffiffiffiffi! f ¼ a ; ð2þ where a is the pore size for spherical grains obtained by Hovem and Ingram [,23] as follows a ¼ d 3 ; ð22þ The viscous correction factor can be approximated at lower and higher frequencies using Eqs. (8) and (22) as follows At a lower frequency, FðÞ þ j 2 24 ¼ þ j 5 6 f f r ð23þ And at a higher frequency, FðÞ sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi þ j 5 f 5 f pffiffiffi ¼ þ j ð24þ f r 8 f r In Eqs. (23) and (24), f r is the relaxation frequency in the Biot-Stoll model defined by f r ¼ 2 f k ð25þ The frequency dependence of the real and imaginary parts of the viscous correction factor is shown in Fig. 2. To obtain a frequency equation, solutions for e and of the form and e ¼ A exp½ jð!t k l xþš; ¼ A 2 exp½ jð!t k l xþš; ð26þ ð27þ are considered. In Eqs. (26) and (27), k l ð¼ k lr þ jk li Þ denotes the wave number for the longitudinal wave,! is the angular frequency and x is the propagation distance. Upon transformation to the frequency domain, the following equation results Hk 2 l! 2 f! 2 2 Ck l Ck 2 l f! 2 m! 2 Mk 2 l j F! ¼ 0 ð28þ k The roots of Eq. (28) give the longitudinal wave velocities c l ð¼!=k lr Þ m/s and the attenuation coefficient l ð¼ k li Þ Np/m as a function of frequency for the longitudinal waves of the first and second kinds. If there is no relative motion between the frame and pore fluid, that is u ¼ U, the Biot-Stoll model becomes the Gassmann model. Moreover, in the case that the frame bulk modulus K b can be neglected, that is K b ¼ 0, the Gassmann model is consistent with the Wood model. Therefore, the Biot-Stoll model is a comprehensive model that includes the Gassmann model and the Wood model. 232

4 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS Real part F r Imaginary part F i F r 3.2. Biot Parameters It is necessary to use 3 physical parameters for applying the Biot-Stoll model [24,25]. These parameters can be classified into three categories, that is, parameters for the pore fluid, parameters for the grain, and parameters for the frame. () Parameters for the pore fluid ) Density of pore fluid f The pore fluid is seawater in marine sediment. For example, the density of seawater is,025 kg/m 3 for the temperature of 20 C, the salinity of 35, and the depth of 0 m. In the case of pure water, the density is 998 kg/m 3 for the temperature of 20 C. 2) Bulk modulus of pore fluid K f The bulk modulus of the pore fluid is Pa for the density of,025 kg/m 3 and the sound velocity of,522 m/s (temperature ¼ 20 C, salinity ¼ 35, depth ¼ 0). In the case of pure water, the bulk modulus is Pa for the density of 998 kg/m 3 and the sound velocity of 483 m/s (temperature ¼ 20 C). 3) Viscosity of pore fluid The viscosity of the pore fluid is Pas for = F i = (a) (b) 5 8 f f r F r = Frequency f / f r 5 8 f f r F i 5 = Frequency f / f r Fig. 2 The real part (a) and the imaginary part (b) of the viscous correction factor as a function of normalized frequency. f f r the temperature of 20 C and the salinity of 35. In the case of pure water, the viscosity is Pas for the temperature of 20 C. The above three parameters can be assumed to be almost constant, even if these values change slightly depending on the temperature and salinity. (2) Parameters for the grain ) Density of grain r The density of the grain can be measured. This mean value for sand is about 2,650 kg/m 3. 2) Bulk modulus of grain K r Stoll used Pa for the bulk modulus of the grain [7]. The author also showed that the bulk modulus of the grain is Pa [26]. These two parameters can be assumed to be almost constant. (3) Parameters for the frame ) Porosity Porosity can be measured. The porosity of unconsolidated marine sediment is higher than about Hamilton proposed an empirical equation indicating the relationship between the grain size and the porosity as follows [27] ¼ 0305 þ 00552; ð29þ ¼ log 2 d ðd mmþ 2) Permeability k Permeability can be measured or can be derived from the grain diameter d and the porosity using Eq. (8). 3) Pore size a The pore size can be obtained from the grain size d and the porosity using Eq. (22). 4) Structure factor Stoll used ¼ 25 for sand and ¼ 30 for silty clay [7], and Turgut and Yamamoto used ¼ 25 for silt, medium sand and coarse sand [28]. Berryman obtained the relation ¼ þ r ; ð30þ where r ¼ 05 for isolated spherical particles and lies between 0 and for other ellipsoidal shapes [23,29]. According to the Berryman equation, the structure factor depends on the porosity and the shape of the grains. 5) Real part of the frame bulk modulus K br Hamilton proposed the following equation for expressing the relationship between the porosity and the frame bulk modulus [30] log K br ¼ ; ð3þ where K br 0 8 (Pa). 6) Imaginary part of the frame bulk modulus K bi The imaginary part of the frame bulk modulus is expressed as follows 233

5 Acoust. Sci. & Tech. 28, 4 (2007) Table Values of the Biot physical parameters of marine sediment models. Medium silt Very fine sand Medium sand Grain Diameter d (mm) Density r (kg/m 3 ) Bulk modulus K r (Pa) Pore fluid Density f (kg/m 3 ) Bulk modulus K f (Pa) Viscosity (Pas) Frame Porosity Permeability k (m 2 ) Pore size a (m) Structure factor Bulk modulus K b (Pa) Longitudinal l logarithmic decrement Shear modulus (Pa) Shear s logarithmic decrement K bi ¼ l K br; ð32þ where l denotes the log decrement for the longitudinal wave. The value of 0.5 is used for sand [20]. 7) Real part of the frame shear modulus r The equation for expressing the relationship between the porosity and the frame shear modulus is obtained using the measured results of the shear wave velocity and the density as follows [3] r ¼ c s 2 ; c s ¼ ð33þ 8) Imaginary part of the frame shear modulus i The imaginary part of the frame shear modulus is expressed as follows i ¼ s r; ð34þ where s denotes the log decrement for the shear wave. The value of 0.5 is used for sand [20]. The 3 physical parameters are divided into two groups that can be assumed to be almost constant and one that can be derived from the value of the porosity. 4. LONGITUDINAL WAVE VELOCITY AND ATTENUATION The longitudinal wave velocity and attenuation of the first kind for three typical marine sediments-medium silt, very fine sand and medium sand are calculated using the Biot-Stoll model. The values of the parameters used in these calculations are shown in Table. The calculated results are shown in Figs. 3 and 4. Figure 3 shows the longitudinal wave velocity as a function of frequency. From Fig. 3, it can be seen that the velocities in all sediments are constant at a lower frequency, but start to increase at some frequency and again become constant at a higher frequency. The frequency at which the velocity starts to increase decreases with the grain size. The velocity at a lower frequency is in agreement with the velocity obtained by the Gassmann model. The velocity at a high frequency is in agreement with the velocity in the case of no fluid loss. Figure 4 shows attenuation as a function of frequency. From Fig. 4, it can be seen that the attenuation is approximately proportional to f 2 at a lower frequency without frame loss and approximately proportional to f =2 at a high frequency. The frequency at which the slope of the attenuation begins changing becomes lower with the grain size. The attenuation due to the frame loss increases proportionally to f. The attenuation due to the fluid loss increases proportionally to f 2. The effect of fluid loss on attenuation is predominant except at a lower frequency. In this calculation, the value of.25 for the structure factor was selected for the three typical marine sediments [7,28]. If the structure factor increases, both the longitudinal wave velocity and the attenuation increase slightly at a higher frequency. The relaxation frequency shown in Eq. (25) is transformed using Eq. (8) as follows f c ¼ 2 f k ¼ 8k 0 2 f 2 2ð Þ ð35þ The relaxation frequency as a function of the porosity is shown in Fig. 5. From Fig. 5, it can be seen that the 234

6 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS Velocity c l (m/s) Medium silt No fluid loss Frame loss or no frame loss 400 Gassmann Attenuation α l (db/m) Medium silt Fluid and frame losses No frame loss No fluid loss Velocity c l (m/s) Velocity c l (m/s) No fluid loss Frame loss or no frame loss Very fine sand Gassmann No fluid loss Frame loss or no frame loss Medium sand Gassmann Fig. 3 Longitudinal wave velocity as a function of frequency for medium silt, very fine sand and medium sand. relaxation frequency increases with the porosity up to a porosity of 0.9. It is possible to explain the change in the frequency characteristics of the longitudinal wave velocity and attenuation depending on the different kinds of sediments using the relationship between the relaxation frequency and the porosity. As mentioned above, 8 of the 3 parameters are assumed to be almost constant, and the other 5 parameters can be derived from the value of the porosity. Therefore, the longitudinal wave velocity and attenuation can be obtained from the porosity and frequency. Three-dimensional representations of the longitudinal wave velocity and attenuation in the frequency range of Hz to MHz, and the porosity range of 0.35 to 0.8, are shown in Fig. 6. The Attenuation α l (db/m) Attenuation α l (db/m) Very fine sand Fluid and frame losses No frame loss No fluid loss Medium sand Fluid and frame losses No frame loss No fluid loss Fig. 4 Longitudinal wave attenuation as a function of frequency for medium silt, very fine sand and medium sand. grain density, the grain bulk modulus, the fluid density, the fluid bulk modulus, the fluid viscosity, the structure factor, and the longitudinal and shear logarithmic decrements are shown in Table. The grain diameter is obtained using Eq. (29). The permeability is obtained using Eq. (8), and the pore size using Eq. (22). The frame bulk modulus is obtained using Eq. (3), and the frame shear modulus using Eq. (33). From Fig. 6(a), it is demonstrated that the longitudinal wave velocity increases as the frequency increases or the porosity decreases. Moreover, the variation of the longitudinal wave velocity with frequency changes with the porosity. From Fig. 6(b), it is shown that the attenuation increases with the frequency, and the variation of the 235

7 Acoust. Sci. & Tech. 28, 4 (2007) Relaxation frequency f r (Hz) Fig Porosity β Relaxation frequency as a function of porosity Porosity β Velocity 600 c l (m/s) 500 attenuation with the frequency changes with the porosity. The changes in the characteristics of the longitudinal wave velocity and the attenuation with the frequency and the porosity are clarified in Fig EQUIVALENT CIRCUIT FOR POROUS MARINE SEDIMENTS The most spectacular aspect of the Biot-Stoll model is the prediction of the longitudinal wave of the second kind in addition to that of the first kind. In this study, equivalent circuits for the plane longitudinal wave propagation in porous media are proposed to easily understand the physical phenomena. First, an exact equivalent circuit for the longitudinal waves of the first and second kinds is derived. Next, an approximated equivalent circuit for only the longitudinal wave of the first kind is obtained. Equation (7) may be rewritten in the case of onedimensional longitudinal wave motion for the displacements of the frame u and pore fluid U as follows fh þ ðm þ ðc U ¼ðþ 2 2 m 2 f u ðc þ 2 2 ¼ ð 2 f þ Equation (36) can be rewritten as HM C 2 C HM C 2 F 2 2 ¼ M 2 u 2 þ mc fm If the following substitutions are applied, ¼ fðh CÞ ðc MÞ ðh CÞ Porosity þ ð 2 f U F 2 @ U 2 2 þ F k H C ð þ 2 m 2 f ÞðC MÞ ð f mþðh þ 2 M 2CÞ ðh CÞ @ U @t ; ð36þ ; ð37þ Fig. 6 Three-dimensional representations of the longitudinal wave velocity (a) and attenuation (b) as functions of porosity and frequency. Attenuation α l (db/m) 236

8 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS b ¼ M fc ; ð38þ C E b ¼ HM C2 C ; ð39þ l ¼ fðh CÞ ðc MÞ ; ð40þ ðh CÞ E l ¼ HM C2 H C ; cb ¼ mc fm C c ¼ F k Eq. (37) can be expressed as 2 u E 2 u 2 þ 2 u 2 2 U E 2 U 2 & 2 cl ; cb ¼ ð þ 2 m 2 f ÞðC MÞ ð f mþðh þ 2 M 2CÞ ; ð42þ U U Therefore, the equations of motion of the node number i of the system shown in Fig. 7(a) [32] are obtained by modifying the left-hand side of Eq. (44) using a differential approximation method as 2 u i E b ðu iþ 2u i þ u i Þ¼ þ 2 u i U i þ 2 U i E l ðu iþ 2U i þ U i Þ¼ 2 u i U i ð45þ ; ð4þ ð43þ ð44þ In Fig. 7(a), the frame phase is represented by a network of spring of elastic constant E b and effective density b representing the inertia of the frame. The pore fluid phase is represented by a network of spring of elastic constant E l and effective density l representing the inertia of the pore fluid. In the system, there are two couplings, one is the viscous coupling c ð@u i i =@tþ ( c is represented by a dashpot), the other is the inertia coupling between the frame and the pore fluid & cb ð@ 2 u i =@t 2 U i =@t 2 Þ, & cl ð@ 2 u i 2 U i =@t 2 Þ. The equivalent circuit for the impedance analogy (the force-voltage, and the velocity-current analogies) [33] to the system can be shown in Fig. 7(b). In Fig. 7(b), the viscous coupling is represented by two resisters and an ideal gyrator, and the inertial coupling is represented by a transformer. From Fig. 7(b), the plane longitudinal wave in a porous medium can be shown as composite transmission lines. One line indicates the longitudinal wave in the frame, the other line indicates the longitudinal wave in the pore fluid. The longitudinal wave of the first kind is the wave where the motions of the frame and pore fluid are in phase. The longitudinal wave of the second kind is the wave where the motions of the frame and pore fluid are out of phase. In many marine sediments, the frame modulus is much smaller than the system bulk modulus Kð=K ¼ =K f þ ð Þ=K r Þ, that is K b þð4=3þ K. Under this condition, the approximate relationship C 2 HM ¼ 0 can be used []. Equation (7) may be rewritten in the case of onedimensional longitudinal wave motion as follows & 2 & 2 ; ð46þ 2 & 2 e 2 & F ð47þ Using Eqs. (46) and (47) and the relationship C 2 HM ¼ 0, ðm f ð fm & þ C ¼ 0 ð48þ 2 Using Eqs. (46) and 3 þ 2F k m f 2 H F 2 e m 2 ¼ 0 2 ðmh þ M 2 3 e 2 m 2 ð49þ 237

9 Acoust. Sci. & Tech. 28, 4 (2007) u i E b u i ρ b u i+ (Frame) E a ηc ζ cb ζ cl E l ρ l U i U i U i+ (Pore fluid) Ev η v Viscous coupling Inertial coupling ρ v E b ηc (a) ζ cb ρb (Frame) ρ v (a) E v E l η c η c ζ cl ζ cl ζ cb ρ l (Pore fluid) E a (b) η v Viscous coupling (b) Inertial coupling Fig. 7 Biot-Stoll model (a) and the equivalent circuit (b) for the plane longitudinal wave in porous saturated medium. On the other hand, the equation for the longitudinal wave in the viscoelastic model, as shown in Fig. 8(a) [34], is expressed as 3 u þ vðe a þ E v 2 u E ve 2 u ¼ 0 2 ð50þ In both Eqs. (49) and (50), each coefficient is arbitrary. In this study, the correspondence between the density in the Biot-Stoll model and the density v in the viscoelastic model is adapted to obtain a usual analogy. Comparing Eq. (49) with Eq. (50), these two equations are identical with the following relationships v ¼ ; ð5þ E a ¼ ðmh þ M 2 fcþ ; m 2 f ð52þ HðmH þ M 2 f CÞ E v ¼ ðmh þ M 2 f CÞ Hðm f2 Þ ; ð53þ ðmh þ M 2 f CÞ 2 v ¼ ð54þ F fðmh þ M 2 f CÞ Hðm f2 Þg k The elastic moduli E a and E v are functions of the elastic Fig. 8 Approximated Biot-Stoll model (a) and the approximated equivalent circuit for the plane longitudinal wave in porous saturated medium. moduli of the pore fluid, grain and frame, and also the densities of the pore fluid and grain. The coefficient of viscosity is a function of the elastic moduli, the densities and the viscous resistance F=k. The equivalent circuit for an infinitesimal section of the impedance analogy (the force-voltage, and the velocity-current analogies) [33] to the approximated model can be shown in Fig. 8(b). For the viscoelastic case, v is constant. On the other hand, the term in the Biot-Stoll model corresponding to v is a function of frequency. 6. APROXIMATED EQUATIONS FOR LONGITUDINAL WAVE VELOCITY AND ATTENUATION From Fig. 6, the frequency and porosity dependence of the longitudinal wave velocity can be understood qualitatively. However, the value of the longitudinal wave velocity for a particular porosity and frequency cannot be obtained. The exact solution for the longitudinal wave velocity is not difficult to obtain, but not very practical for actual engineering applications. Therefore, approximate expressions of the longitudinal wave velocity and attenuation are desired for engineering applications. Geertsma and Smit proposed a low- and middle-frequency approximate expression of the longitudinal wave velocity for porous rocks as follows [35] 238

10 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 c 4 l þ c 4 l0 f r c l ¼ u 2 ; ð55þ t f c 2 l þ c 2 l0 f r where c l0 and c l denote the longitudinal wave velocities at the frequencies of zero and infinity in the case of C 2 HM ¼ 0, respectively and are expressed as sffiffiffiffi c l0 ¼ H ; ð56þ c l ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mh þ M 2 f C m 2 f ð57þ The calculated results of the longitudinal wave velocities using Eq. (55) are poor approximations to the calculated results using the exact Biot equations, as shown in Fig. 9. The slopes of the longitudinal wave velocity to the frequency are different for the exact Biot-Stoll model and the Geertsma and Smit model. Thus, the coefficient and the power of the frequency in Eq. (55) are adjusted to obtain good approximated results using a curve-fitting method. The equations obtained are as follows vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 06 c 4 l þ c 4 l0 f r c l ¼ f 2 ; f f u r ; ð58þ t 06 c 2 l þ c 2 l0 f r vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f c 4 l þ c 4 l0 f r c l ¼ f 07 ; f f u r ; ð59þ t 04 c 2 l þ c 2 l0 f r where c l can be used not by Eq. (57), but by the following equation in the case of C 2 HM 6¼ 0 to obtain good approximated results Velocity c l (m/s) Velocity c l (m/s) Velocity c l (m/s) Exact Geertma & smit This study 450 Medium silt Exact Geertsma & Smit This study 550 Very fine sand Exact Geertsma &Smit This study 750 Medium sand Fig. 9 Approximated longitudinal wave velocity as a function of frequency for medium silt, very fine sand and medium sand. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c l ¼ 2ðC 2 HMÞ p ðmh þ M 2 f CÞþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmh þ M 2 f CÞ 2 4ðC 2 HMÞð f2 mþ ð60þ The calculated results of the longitudinal wave velocity versus frequency using the approximated equations (Eqs. (58) and (59)) are shown in Fig. 9. From the figure, it can be seen that the results using the modified Geertsma- Smit model are almost in agreement with the exact Biot results. The longitudinal wave velocity at a particular porosity and frequency can be easily obtained using Eqs. (56) and (58) (60). Next, the attenuation coefficient without the frame loss will be considered. The inverse of the quality factor =Q is defined in terms of the real part k lr and the imaginary part k li of the wave number k l as follows [28] Q ¼ 2 k li ð6þ k lr Supposing the relationship C 2 HM ¼ 0 and high Q values, which occur in almost all marine sediments, are satisfied [,28], Eq. (6) can be approximated as 239

11 Acoust. Sci. & Tech. 28, 4 (2007) 2 Þ Q Imðk l Reðk l2 Þ ¼ H mh þ M 2 f C þ H F i k! m f2 þ F i k! mh þ M 2 f C þ H F i F r k! k! H F þ r m f2 þ F i k! k! where F r and F i are the real and imaginary parts of the viscous correction factor F. At a lower frequency, =Q approaches the following mh þ M 2 f C H m ( 2 f c ) 2 l ¼ ¼ m f 2 k! Q 0 c l0 ð63þ k! m f 2 At a higher frequency, =Q approaches the following mh þ M 2 f C H m 2 f ¼ ¼ ( p Q mh þ M 2 f C 4 ffiffi c ) 2 rffiffiffiffiffiffi l0 a 2 c l m f 2 ð64þ k! H F r k! ; ð62þ The attenuation coefficient can be obtained from l ¼! 2c l Q ð65þ Therefore, the attenuation coefficient at a lower frequency and a higher frequency are expressed as follows l0 ¼ ( c ) 2 l m f 2 k 2c l0 c l0!2 ; ð66þ ( l ¼ p 8 ffiffiffi c ) 2 l0 a pffiffiffiffiffiffiffip ffiffiffi 2 cl c l m f 2 f! ð67þ k From Eqs. (66) and (67), it can be seen that the attenuation coefficient is proportional to! 2 at a lower frequency, and! =2 at a higher frequency. The calculated results of the attenuation versus frequency using the approximated equations at a lower and a higher frequency are shown in Fig. 0. In the calculation of the velocity at a higher frequency c l, Eq. (60) instead of Eq. (57) was used for a better approximation. From Fig. 0, it can be seen that the curves for the approximated cases are consistent with the curves for the no-frame-loss case both at a lower frequency and at a higher frequency. 7. RANGE OF APPLICABILITY FOR ACOUSTIC MODELS The Biot-Stoll model is a comprehensive model for acoustic wave propagation in porous marine sediments. On the other hand, simpler models such as the Wood model, the Gassmann model, and the Biot-Stoll model without frame moduli can be used within limited ranges of frequency and porosity to approximate the exact Biot-Stoll model. One of the objectives of this study is to simplify the practical application of the Biot-Stoll model. Therefore, it is important to obtain the relationship between the Biot- Stoll model and other simpler acoustic models. In this section, the ranges of frequency and porosity applicable for models such as the Wood model, the Gassmann model and the Biot-Stoll model without frame moduli are derived. The calculated results of the longitudinal wave velocity at the frequencies of khz and 00 khz as a function of porosity are shown in Fig. (a). In this figure, the results with and without frame moduli are shown. The values of the parameters used in this calculation are the same as those for the results shown in Fig. 6(a). From Fig. (a), it can be seen that the difference between the longitudinal wave velocity with and without frame moduli increases as the porosity decreases. Assuming that the region in which the difference is less than 3.5% is the well-approximated region, we can obtain the approximated region, as shown in Fig. (b). From this figure, it is indicated that the porosity limit is about 0.79 up to the frequency of 30 khz, and then the porosity limit decreases with the frequency. The relaxation frequency as a function of porosity is shown in Fig. 5. Above about a fifth of the relaxation 240

12 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS Attenuation α l (db/m) Attenuation α l (db/m) Attenuation α l (db/m) Medium silt Medium sand Exact Approx.(Lower freq.) Exact Very fine sand Exact Approx. (Lower freq.) Approx. (Lower freq.) No frame loss No frame loss No frame loss Approx. (Higher freq.) Approx. (Higher freq.) Approx. (Higher freq.) Fig. 0 Approximated longitudinal wave attenuation as a function of frequency for medium silt, very fine sand and medium sand. frequency, the longitudinal wave velocity gradually increases with frequency. Below this frequency, the difference between the specified velocity and the velocity at zero frequency is within 3.5%. Using the relationships shown in Figs. 5 and (b), we can derive the ranges of frequency and porosity for the applicability of the Wood model, the Gassman model and the Biot-Stoll model without frame moduli, as shown in Fig. 2. Figure 2(a) shows the ranges of applicability for the Gassmann model and the Wood model. From this figure, it can be seen that the Gassmann model is applicable for a wide range of porosities at a lower frequency. However, the Wood model is restricted to high porosity. Figure 2(b) shows the range of applicability for (a) Approximated region ( K = µ = 0 ) Fig. Longitudinal wave velocity as a function of porosity (a) and porosity limitation for the Biot-Stoll model without frame moduli as a function of frequency (b). (b) the Biot-Stoll model without frame moduli. From this figure, it can be seen that this model is restricted to high porosity for all frequency ranges. 8. CONCLUSIONS The Biot-Stoll model is a useful tool for analyzing the acoustic wave propagation in porous marine sediments. However, it is not easy to apply this model, because 3 physical parameters are required in this model, and some parameters cannot be measured. Also, it is difficult to understand the physical phenomena in this model. First the Biot-Stoll model, the 3 physical parameters required for applying the model, and the dependence of the viscous loss of the pore fluid and the friction loss of the frame on the frequency characteristics of the longitudinal wave velocity and attenuation for three typical kinds of marine sediments medium silt, very fine sand and medium sand were summarized. In this study, equivalent circuits for the plane longitudinal wave propagation in porous media were proposed for easy understanding of the physical phenomena. Next, practical approximate equations for the longitudinal wave b 24

13 Acoust. Sci. & Tech. 28, 4 (2007) Gassman Gassmann, Wood (a) Biot-Stoll ( K = µ = 0) ) (b) Fig. 2 Ranges of frequency and porosity for applicability of the Gassmann model and the Wood model (a) and for the Biot-Stoll model without frame moduli (b) for analyzing the longitudinal wave velocity. velocity and attenuation as functions of porosity and frequency were proposed. Moreover, the ranges of frequency and porosity applicable for acoustic models such as the Wood model, the Gassmann model, and the Biot-Stoll mode without frame moduli were obtained. Future studies are as follows () Approximate equation of the longitudinal wave velocity and attenuation for the sediments with a grain size distribution [3]. (2) Frequency dependence of the frame bulk modulus using the gap stiffness model, which considers the acoustic relaxation of the elasticity of the fluid between the grains [36]. (3) Depth dependence of the longitudinal wave velocity and attenuation. REFERENCES [] J. M. Hovem and G. D. Ingram, Viscous attenuation of sound in saturated sand, J. Acoust. Soc. Am., 66, (979). [2] J. M. Hovem, Viscous attenuation of sound in suspension and high-porosity marine sediments, J. Acoust. Soc. Am., 67, (980). b [3] M. Kimura, Sound velocity and attenuation constant of high porosity marine sediments, Jpn. J. Appl. Phys., 30, Suppl. 30-, (99). [4] R. D. Stoll and T. K. Kan, Reflection of acoustic waves at a water-sediment interface, J. Acoust. Soc. Am., 70, (98). [5] T. Yamamoto, Acoustic propagation in the ocean with a poroelastic bottom, J. Acoust. Soc. Am., 73, (983). [6] P. R. Ogushwitz, Applicability of the Biot theory.. Lowporosity materials, J. Acoust. Soc. Am., 77, (985). [7] P. R. Ogushwitz, Applicability of the Biot theory. 2. Suspensions, J. Acoust. Soc. Am., 77, (985). [8] P. R. Ogushwitz, Applicability of the Biot theory. 3. Wave speeds versus depth in marine sediments, J. Acoust. Soc. Am., 77, (985). [9] C. W. Holland and B. A. Brunson, The Biot-Stoll sediment model An experimental assessment model, J. Acoust. Soc. Am., 84, (988). [0] G. Mavko, T. Mukeiji and J. Dvorkin, The Rock Physics Handbook (Cambridge University Press, Cambridge, 998), pp [] F. Gassmann, Über die Elastizität poröser Medien, Vierteljahrsschr. Naturforsch. Ges. Zür., 96, 23 (95). [2] M. A. Biot, Theory of elastic waves in a fluid-saturated porous solid.. Low frequency range, J. Acoust. Soc. Am., 28, (956). [3] M. A. Biot, Theory of elastic waves in a fluid-saturated porous solid. 2. Higher frequency range, J. Acoust. Soc. Am., 28, 79 9 (956). [4] M. A. Biot, Mechanics of deformation and acoustic propagation in porous dissipative media, J. Appl. Phys., 33, (962). [5] M. A. Biot, Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Am., 34, (962). [6] R. D. Stoll, Wave attenuation in saturated sediments, J. Acoust. Soc. Am., 47, (970). [7] R. D. Stoll, Acoustic waves in saturated sediments, in Physics of Sound in Marine Sediment, L. Hamilton, Ed. (Plenum, New York, 974), pp [8] R. D. Stoll, Acoustic waves in ocean sediments, Geophysics, 42, (977). [9] R. D. Stoll, Acoustic waves in marine sediments in Ocean Seismo-Acoustics, T. Akal and M. Berkson, Eds. (Plenum, New York, 986), pp [20] R. D. Stoll, Sediment Acoustics (Springer-Verlag, Berlin, 989), pp [2] D. L. Johnson and T. J. Plona, Acoustic slow waves and the consolidation transition, J. Acoust. Soc. Am., 72, (982). [22] T. Bourbie, O. Coussy and B. Zinszner, Acoustics of Porous Media (Editions Technip, Paris, 986), pp [23] B. Yavari and A. Bedford, Comparison of numerical calculations of two Biot coefficients with analytical solutions, J. Acoust. Soc. Am., 90, (99). [24] J. I. Dunlop, Propagation of acoustic waves in marine sediments, a review, Explor. Geophys., 9, (988). [25] M. Kimura and S. Kawashima, Study on physical parameters on the Biot-Stoll marine sediment model, J. Mar. Acoust. Soc. Jpn., 22, (995). [26] M. Kimura, Grain bulk modulus of marine sediment, Jpn. J. Appl. Phys., 39, (2000). [27] E. L. Hamilton, Prediction of deep-sea sediment properties State-of-the-art, in Deep-Sea Sediments, A. L. Inderbitzen, Ed. (Plenum, New York, 974), pp

14 M. KIMURA BIOT-STOLL MODEL FOR POROUS MARINE SEDIMENTS [28] A. Turgut and T. Yamamoto, Measurements of acoustic wave velocities and attenuation in marine sediments, J. Acoust. Soc. Am., 87, (990). [29] J. G. Berryman, Elastic wave propagation in fluid-saturated porous media, J. Acoust. Soc. Am., 69, (98). [30] E. L. Hamilton, Elastic properties of marine sediments, J. Geophys. Res., 76, (97). [3] Y. Hatori, Study on the shear wave velocity in marine sediment model, Master s Thesis, Tokai University, pp (2005). [32] P. N. J. Rasolofosaon, Plane acoustic waves in linear viscoelastic porous media Energy, particle displacement, and physical interpretation, J. Acoust. Soc. Am., 89, (99). [33] M. Konno, Ed., An Introduction to Dynamical Analogy (Corona Pub., Tokyo, 980), pp [34] M. Kimura and H. Shimizu, Equivalent circuit representation for acoustic wave in sediment, IEICE Tech. Rep., US9-3, pp (99). [35] J. Geertsma and D. C. Smit, Some aspects of elastic wave propagation in fluid-saturated porous solids, Geophysics, 26, 69 8 (96). [36] M. Kimura, Frame bulk modulus of porous granular marine sediments, J. Acoust. Soc. Am., 20, (2006). APPENDIX STRESS-STRAIN RELATIONS The stress-strain relations in the Biot-Soll model are as follows [4] xx ¼ He 2ðe y þ e z Þ C& yy ¼ He 2ðe z þ e x Þ C& zz ¼ He 2ðe x þ e y Þ C& xy ¼ z yz ¼ x zx ¼ y p f ¼ M& Ce ¼ M& Cðe x þ e y þ e z Þ; ðaþ Fig. A. p Impervious flexible bag where xx, yy and zz are the components of extensional stress, xy, yz and zx are the components of shear stress, p f is the pressure in the pore fluid, e x, e y and e z are the components of extensional strain, and x, y and z are the components of shear strain. APPENDIX 2 Free draining Jacketed test for deriving the frame bulk modulus. JACKETED TEST There are three bulk moduli, the fluid, grain and frame moduli. The frame bulk modulus is derived from the jacketed test [20]. The frame bulk modulus is defined by the ratio of the externally applied isotropic pressure p to the dilatation e in the jacketed test, as shown in Fig. A. and expressed as follows K b ¼ p e ¼ H 4 C2 ða2þ 3 M In the jacketed test, a saturated porous medium is placed in an impervious, flexible bag, and loaded by means of an external pressure. The pore fluid in the medium is free to flow out of the bag via an upper tube. 243