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Theoretical analysis of observed second bulk compressional wave in a fluid‐saturated porous solid at ultrasonic frequencies
Abstract
We show that the second bulk compressional wave observed recently in an experiment by Plona is consistent with the ’’slow’’ compressional wave as predicted by Biot’s theory. We present dispersion and attenuation curves for all of the body waves for Plona’s samples computed as a function of frequency and Biot’s structure factor S. At high frequencies the velocity of the slow compressional wave is approximately equal to V f / S<sup>1/2</sup>, where V f is the normal compressional‐wave velocity in the pore fluid. Thus, from the measured velocities of slow compressional waves one can obtain S. For Plona’s three samples, the predicted structure factors are 2.1, 2.2, and 3.3.
... Apart from the experimental work, efforts were also dedicated to the analytical investigation of the compressional wave propagation. Dutta [8] and Berryman [3] compared the observed slow wave velocities from Plona [23] with the analytical expressions of the compressional wave velocities derived from Biot's theory [4,5] and showed a good agreement. Boyle and Chotiros [6] carried out experiments with samples made of river sediments and observed the slow wave under ultrasonic-frequency loading. ...
... There are four complex roots for the solution of Eq. (12), as expressed by Eq. (13). These can account for the characteristics of four compressional waves, where their wave velocities and damping ratios can be calculated based on Eq. (8). However, among the obtained expressions of four compressional waves, / 1 and / 3 represent the same type of wave with opposite propagation directions. ...
In geotechnical earthquake engineering, wave propagation plays a fundamental role in engineering applications related to the dynamic response of geotechnical structures and to site response analysis. However, current engineering practice is primarily concentrated on the investigation of shear wave propagation and the corresponding site response only to the horizontal components of the ground motion. Due to the repeated recent observations of strong vertical ground motions and compressional damage of engineering structures, there is an increasing need to carry out a comprehensive investigation of vertical site response and the associated compressional wave propagation, particularly when performing the seismic design for critical structures (e.g. nuclear power plants and high dams). Therefore, in this paper, the compressional wave propagation mechanism in saturated soils is investigated by employing hydro-mechanically (HM) coupled analytical and numerical methods. A HM analytical solution for compressional wave propagation is first studied based on Biot’s theory, which shows the existence of two types of compressional waves (fast and slow waves) and indicates that their characteristics (i.e. wave dispersion and attenuation) are highly dependent on some key geotechnical and seismic parameters (i.e. the permeability, soil stiffness and loading frequency). The subsequent HM Finite Element (FE) study reproduces the duality of compressional waves and identifies the dominant permeability ranges for the existence of the two waves. In particular the existence of the slow compression wave is observed for a range of permeability and loading frequency that is relevant for geotechnical earthquake engineering applications. In order to account for the effects of soil permeability on compressional dynamic soil behaviour and soil properties (i.e. P-wave velocities and damping ratios), the coupled consolidation analysis is therefore recommended as the only tool capable of accurately simulating the dynamic response of geotechnical structures to vertical ground motion at intermediate transient states between undrained and drained conditions.
... His capability to change the angle of incidence of the insonifying pulse allowed for observation of the converted P1, P2, and S modes at a variety of angles; this allowed him to convincingly argue that the P2 arrivals could not be experimental artifacts. This discovery led to a flurry of theoretical analyses of his P2 wave velocities [Berryman, 1980;Dutta, 1980] and transmitted amplitudes [Hovem, 1981] and was followed by further experimental tests of low-frequency behavior [Chandler, 1981;Chandler and Johnson, 1981], and of the influence of the porous frame modulus and tortuosity . Subsequent studies showed the slow wave could exist and propagate in a wide range of air-or water-saturated porous materials including bone [Lakes et al., 1983], sintered metallic filters [Jungman et al., 1989], aluminum foams [Ji et al., 1998], textiles [Gomez Alvarez-Arenas et al., 1994, soils [Nakagawa et al., 1997], xerographic developer mixtures [Stearns, 1992] and anisotropic composites [Castagnede et al., 1998]. ...
... [12] A full description of porous media behavior cannot be understood without knowledge of wave reflection, conversion, and transmission effects at the interface of a porous medium. In a practical sense, such knowledge is necessary to properly interpret laboratory observations [Dutta, 1980;Wu et al., 1990;Santos et al., 1992;Johnson et al., 1994;Aknine et al., 1997;Derible, 2005] particularly if amplitudes and attenuation are considered. The problem is also of obvious consequence to seismic and sonar studies [e.g., Stoll and Kan, 1981;Krebes, 1984;Delacruz et al., 1992;Yang and Sato, 1998;Yang, 1999;Denneman et al., 2002;Carcione and Helle, 2004;Tajuddin and Hussaini, 2005] as the field observations of the variations of seismic amplitudes with angle of incidence become an increasingly important geophysical tool. ...
A new method of probing porous materials has been developed using a large area (10-cm X 7.5-cm) ultrasonic transducer as an acoustic source. This transducer generates a wide, flat beam with central amplitudes remaining stable to up to at least 40-cm in water filled tank. The advantage of such a beam is that geometrical wave-field spreading is small and as such these effects may be neglected. This greatly simplifies measurements that rely on accurate amplitude determination in, for example, the measurement of attenuation or reflectivity. Synthetic water-saturated porous media are characterized by passing the beam through a plate of the material at incidence angles between -50° and 50° in a manner similar to that of earlier experiments by other groups. The initial pulse in the water is converted to fast-P, slow-P, and S-waves at the water-sample interface. Each of these arrivals is detected on the far side of the plate by small near-point receiver ( ~ 1-mm diameter) and is recognized on the basis of easily discriminated travel-time versus angle of incidence images of the recorded waveforms. The velocity of the various modes are measured and modeled. The porosity, permeability, and tortuosity that are measured in separate experiments along with the velocities allow a good characterization of the porous sample. These measurements assist in the interpretation of a series of measurements of acoustic reflectivity from a saturated porous solid. A description of the experimental method and the methodology used in velocity estimates will be given. >http://www-geo.phys.ualberta.ca/ ~ybouzidi
... His capability to change the angle of incidence of the insonifying pulse allowed for observation of the converted P1, P2, and S modes at a variety of angles; this allowed him to convincingly argue that the P2 arrivals could not be experimental artifacts. This discovery led to a flurry of theoretical analyses of his P2 wave velocities [Berryman, 1980;Dutta, 1980] and transmitted amplitudes [Hovem, 1981] and was followed by further experimental tests of low-frequency behavior [Chandler, 1981;Chandler and Johnson, 1981], and of the influence of the porous frame modulus and tortuosity . Subsequent studies showed the slow wave could exist and propagate in a wide range of air-or water-saturated porous materials including bone [Lakes et al., 1983], sintered metallic filters [Jungman et al., 1989], aluminum foams [Ji et al., 1998], textiles [Gomez Alvarez-Arenas et al., 1994, soils [Nakagawa et al., 1997], xerographic developer mixtures [Stearns, 1992] and anisotropic composites [Castagnede et al., 1998]. ...
... [12] A full description of porous media behavior cannot be understood without knowledge of wave reflection, conversion, and transmission effects at the interface of a porous medium. In a practical sense, such knowledge is necessary to properly interpret laboratory observations [Dutta, 1980;Wu et al., 1990;Santos et al., 1992;Johnson et al., 1994;Aknine et al., 1997;Derible, 2005] particularly if amplitudes and attenuation are considered. The problem is also of obvious consequence to seismic and sonar studies [e.g., Stoll and Kan, 1981;Krebes, 1984;Delacruz et al., 1992;Yang and Sato, 1998;Yang, 1999;Denneman et al., 2002;Carcione and Helle, 2004;Tajuddin and Hussaini, 2005] as the field observations of the variations of seismic amplitudes with angle of incidence become an increasingly important geophysical tool. ...
Two compressional wave modes, a fast P1 and a slow P2, propagate through fluidsaturated porous and permeable media. This contribution focuses on new experimental tests of existing theories describing wave propagation in such media. Updated observations of this P2 mode are obtained through a water-loaded, porous sintered glass bead plate with a novel pair of ultrasonic transducers consisting of a large transmitter and a near-point receiver. The properties of the porous plate are measured in independent laboratory experiments. Waveforms are acquired as a function of the angle of incidence over the range from -50° to +50° with respect to the normal. The porous plate is fully characterized, and the physical properties are used to calculate the wave speeds and attenuations of the P1, the P2, and the shear S waves. Comparisons of theory and observation are further facilitated by numerically modeling the observed waveforms. This modeling method incorporates the frequency and angle of incidence-dependent reflectivity, transmissivity, and transducer edge effects; the modeled waveforms match well those observed. Taken together, this study provides further support for existing poroelastic bulk wave propagation and boundary condition theory. However, observed transmitted P1 and S mode amplitudes could not be adequately described unless the attenuation of the medium's frame was also included. The observed P2 amplitudes could be explained without any knowledge of the solid frame attenuation.
... Biot [25][26][27][28][29] conducted pioneering study on fluid-saturated porous materials and developed an effective theory to study elastic waves in such materials. This theory successfully predicted the existence of the slow P-wave, which was subsequently confirmed by Plone [30] and Dutta [31]. On the basis of Biot theory, numerous scholars have investigated the propagation of elastic waves in porous materials, including the characteristics of wave propagation in single-layer [32] and multilayer porous materials [33], as well as the behavior of wave reflection and transmission in thermoelastic porous materials [34][35][36][37]. ...
... It is utilized broadly in several fields, such as geophysics, seismology, soil dynamics, material science, hydrogeology, etc. Poroelasticity theory was originally presented by Biot [1,2], who predicted three waves propagate in poroelastic media, including an Swave and two P-waves, named fast P-wave and slow P-wave. The existence of slow Pwave was validated by laboratory and field experiments [3][4][5][6][7][8][9][10][11]. ...
Actual sedimentary rocks are generally homogeneous porous media saturated by fluid, which are described by poroelastic media. Wave velocity and dissipation factor in poroelastic media provide essential information about rock and fluid properties and are widely used in many fields. The exact wave velocity and dissipation factor can be obtained from wavenumbers determined by dispersion equation through complex mathematic operations. Reported approximate formulas are most given under the assumption that the characteristic angular frequency (ωc) is far larger than the angular frequency (ω) and becomes invalid for high frequency. Thus, this paper derived phase velocity and dissipation factor approximation of the two P-waves in poroelastic media for the whole low-frequency range. Based on the Geertsma–Smit approximation, we modified the infinite-frequency limiting velocity for fast P-wave approximation and removed the ω/ωc<<1 assumption for slow P-wave approximation. Numerical analysis shows that the proposed approximate formulas for P-waves in porous media can reveal the exact variations of phase velocity and dissipation factor in the low-frequency range, which is from zero frequency for isotropic media to the characteristic frequency. Phase velocity approximations of the two P-waves in poroelastic media are very close to exact values, with maximum relative errors being far less than 1%.
... Most of these studies directly extend the modeling algorithm of solid elastic wave propagation to the poroelastic case. The dramatic velocity difference between the slow P wave and the two other waves (i.e., fast P wave and S wave), which is presented in real-world observations (Plona, 1980;Dutta, 1980), is of most concern compared with the solid elastic wave simulations (Masson et al., 2006;Alkhimenkov et al., 2021b). ...
For poroelastic media, the existence of a slow P-wave mode, next to the standard fast P and S waves, hinders efficient numerical implementations to propagate poroelastic waves through arbitrary seismic models. The slow P-wave speed can be an order of magnitude smaller than the fast P-wave speed. Hence, a stable and accurate simulation that can capture the slow P wave requires a fine grid and a small time step, which increases the overall computation cost greatly. To decrease the computation cost, we propose a poroelastic finite-difference simulation method that combines a discontinuous curvilinear collocated-grid method with a nonuniform time step Runge-Kutta (NUTS-RK) scheme. The fine grid and small time step are only used for areas near interfaces, where the contribution of the slow P wave is nonnegligible. The NUTS-RK scheme is derived from a Taylor expansion and it can circumvent the need for interpolations or extrapolations otherwise required by communications between different time levels. The accuracy and efficiency of the proposed method are verified by numerical tests. Compared with the curvilinear collocated-grid finite-difference method that uses a globally uniform space grid as well as a uniform time step, the proposed method requires fewer computing resources and can reduce the computing time greatly.
... In Equation (27), the equations of the potential functions of the P1 and P2 waves have the forms of wave equations. ...
The permeability of saturated soils has great influence on the velocities and attenuation characteristics of fast compressional wave P1, low compressional wave P2, and shear wave S in saturated soils, respectively. In three different cases, namely zero, finite, and infinite permeability, the wave equations and theoretical velocities of P1, P2, and S wave in saturated soils are given based on the u-w-p equation, respectively. According to the solutions of the wave equations, the real velocities and attenuation coefficients of three waves are redefined, respectively. In different saturated soils, the influences of the permeability and the loading frequency on the wave velocities and attenuation are discussed, respectively. Moreover, the suitable application scope of the u-p equation is discussed based on different permeabilities and loading frequencies.
... In 1956, Biot presented a phenomenological homogeneization theory for wave propagation in two-phase FSP media [9,10] and suggested the existence of two longitudinal and two transverse waves, in contrast to elastic media that support only one longitudinal wave and two transverse waves. Later, the slow longitudinal wave was experimentally unveiled by Plona [14] and Dutta [15]. Researchers from different disciplines further discussed wave propagation in FSP media [16][17][18][19][20][21]. ...
Wave propagation in a two-dimensional periodic fluid-saturated porous metamaterial (FSPM) is investigated. The constitutive relation considered for fluid-saturated porous materials is based on Biot's homogeneization theory. Such media generally support two shear and two longitudinal elastic waves. Anisotropic wave propagation results both from anisotropy of the solid matrix and from the periodic structure of the metamaterial. Special attention is devoted to the dispersion and attenuation of evanescent Bloch waves in a FSPM whose matrix is transversely isotropic in which case in-plane elastic waves are a superposition of one shear wave and of two longitudinal waves. Bloch waves, complex band structures, and transmission properties are obtained numerically using finite element analysis. The case of homogeneous fluid-saturated porous media is considered since numerical simulations can be compared directly with analytic results in this case. The effects of material anisotropy and of fluid viscosity on wave propagation in two-dimensional FSPM are then discussed. It is found that wave-number band gaps appear in the complex band structure of the lossless FSPM due to the interference of waves with different but nonorthogonal polarizations. Wave-number band gaps are connected continuously by complex-frequency bands, hence, defining stop bands in the time domain. No complete frequency band gaps are found in two-dimensional lossless FSPM, in contrast to the one-dimensional case [Y.-F. Wang et al. Phys. Rev. B 99, 134304 (2019)] due to the presence of the additional quasishear wave accompanying the two quasilongitudinal waves. Both the complex band structure and the transmission properties are affected by the anisotropy of the solid matrix. Wide transmission dips come up when viscosity is introduced as a result of the strong attenuation and the coupling of all wave polarizations. Concurrently, wave-number band gaps are washed out by viscosity. This theoretical paper has relevance to practical applications of fluid-saturated porous metamaterials, e.g., in concrete structures and geological soils.
... It follows from this theory that two longitudinal and two transverse waves exist. Plona [33] and Dutta [34] later proved this fact: the speed of the slow compressional wave is smaller than the speed of sound in the fluid. Wave propagation in FSP media has since then been investigated from different viewpoints and by using different methods [35][36][37][38][39]. ...
Fluid-saturated porous metamaterials described following Biot's theory support two longitudinal elastic waves. The phase velocity and attenuation of these waves depend nonlinearly on porosity and viscosity of the fluid. Furthermore, when two fluid-saturated porous metamaterials are arranged to form a periodic composite, different band gaps are opened for the two longitudinal waves and these couple to form anticrossings in the dispersion relation. The complex band structure of one-dimensional composites is derived and compared with numerical transmission through a finite sample obtained by the finite element method. It is found that the anticrossings disappear rapidly as viscosity increases, while attenuation band gaps become dominated by the fastest of the two longitudinal waves. Increasing porosity further leads to wider and lower-frequency band gaps. These results are relevant to practical applications of fluid-saturated porous metamaterials, e.g., to engineered soils.
... Plona (1980) observed the second type of compressional wave in his liquid/porous-solid liquid experiment at ultrasonic frequencies. Berryman (1980) and Dutta (1980) demonstrated quantitatively agreement between Biot's theory and Plona's measurements. Teng (1990) numerically simulated the slow compressional wave predicted in Biot's theory using Finite-Element method at ultrasonic frequencies. ...
... In 1980, Plona [7] reported an important observation of a second bulk compressional wave in water saturated sintered glass spheres by experiment. Soon afterwards, Dutta [8] confirmed that the second bulk compressional wave observed in Plona's experiment was consistent with the "slow" compressional wave predicted by Biot theory. Since then, the wave characteristics of saturated porous materials are intensively investigated in many engineering fields, such as seismology [9][10][11], civil engineering [12], bioengineering [13] and other engineering fields. ...
A nonlocal Biot theory is developed by combing Biot theory and nonlocal elasticity theory for fluid saturated porous material. The nonlocal parameter is introduced as an independent variable for describing wave propagation characteristics in poroelastic material. A physical insight on nonlocal term demonstrates that the nonlocal term is a superposition of two effects, one is inertia force effect generated by fluctuation of porosity and the other is pore size effect inherited from nonlocal constitutive relation. Models for situations of excluding fluid nonlocal effect and including fluid nonlocal effect are proposed. Comparison with experiment confirms that model without fluid nonlocal effect is more reasonable for predicting wave characteristics in saturated porous materials. The negative dispersion is observed theoretically which agrees well with the published experimental data. Both wave velocities and quality factors as functions of frequency and nonlocal parameter are examined in practical cases. A few new physical phenomena such as backward propagation and disappearance of slow wave when exceeding critical frequency and disappearing shear wave in high frequency range, which were not predicted by Biot theory, are demonstrated.
... where f is the frequency, and vc is the Biot's reference velocity [1]; the σij nondimensional parameters determine the elastic response of the media (σ11: the solid, σ22: the fluid, and σ12: the mechanical coupling between both), while the γij nondimensional parameters define the dynamic properties of the media (γ11: the solid, γ22: the fluid, and γ12: the inertial coupling between them). In addition , under the experimental conditions given in this work, it can be assumed that: σ12 = γ12 = 0. Experimental verification of this theoretical approach was first made in terms of the velocity of these waves [18]–[20]. However, many porous materials, either air-or water-saturated (e.g., aerogels [21] and marine sediments [22] ) exhibit a linear variation of the attenuation coefficient over a very wide frequency range, whereas the corresponding theoretical predictions provide a dependency on the square root of the frequency. ...
The propagation of ultrasonic waves in the cylindrical micro-pores (pore diam. <1 μm) of ion-track membranes (ITMs) is studied. This membrane fabrication technique provides unique possibilities to obtain cylindrical micro-pores with a very high degree of accuracy in pore shape, size, and orientation. Several ITMs were specially produced having the same pore diameter, orientation, and geometry, but different thickness. Porosity, pore diameter, and shape were determined using scanning electron microscopy, and then the coefficient of ultrasound transmission was measured using air coupling and spectral analysis. These experimental conditions permit us to eliminate the influence of the boundary conditions and to achieve a strong decoupling between the fluid filling the pores and the solid constituent of the membrane. Hence, the velocity and the attenuation coefficient for ultrasound propagation in the pores can be measured. These parameters are compared with the predictions made by conventional theories for sound propagation in porous media and in cylindrical channels. The conclusions of this work provide a better understanding of wave propagation in micro-pores and establish the basis of an ultrasonic porometry technique for ITMs.
... As the frequency increases, the slow compressional wave takes on the character of a propagating wave. The nature of these waves was discussed recently by Dutta and Odd (1979) and Dutta (1980). Biot' s theory has been applied to study the reflection at a stressfree boundary (Deresicwicz and Rice, 1960: Kosachevskii, 1959), surface waves (Kosachevskii, 1959; Dcresiewicz, 1962 ...
Using the Biot theory, we have performed calculations to show that viscous fluid flow affects seismic wave amplitudes and reflection coefficients at a gas-water boundary in a porous sand reservoir. Our formulation of the boundary value problem for fluid-filled porous rocks is applicable at all angles and follows parallel to the classical reflection and transmission problem solved by Knott, Zoeppritz, and others for two elastic media in perfect contact. It is pointed out that there are two major differences between the two: 1) in the present case, we deal with the propagation of inhomogeneous waves, and 2) there are interference fluxes of energy between various types of waves. The latter exists only at an oblique incidence.-from Authors
... Porous media cause the propagation of two compressional waves, known as fast and slow waves. 1 Observation of the slow wave (second compressional wave) was first reported and experimentally confirmed in the 1980s. [1][2][3][4] About 20 years later, it was found that this phenomenon may play a significant role in wave propagation in bones. 5,6 Two-wave phenomena in bovine and human cancellous bones (porous media) were observed experimentally using a water-immersion ultrasound technique. ...
Ultrasound propagation in cancellous bone (porous media) under the condition of closed pore boundaries was investigated. A cancellous bone and two plate-like cortical bones obtained from a racehorse were prepared. A water-immersion ultrasound technique in the MHz range and a three-dimensional elastic finite-difference time-domain (FDTD) method were used to investigate the waves. The experiments and simulations showed a clear separation of the incident longitudinal wave into fast and slow waves. The findings advance the evaluation of bones based on the two-wave phenomenon for in vivo assessment.
... where f is the frequency, and vc is the Biot's reference velocity [1]; the σij nondimensional parameters determine the elastic response of the media (σ11: the solid, σ22: the fluid, and σ12: the mechanical coupling between both), while the γij nondimensional parameters define the dynamic properties of the media (γ11: the solid, γ22: the fluid, and γ12: the inertial coupling between them). In addition , under the experimental conditions given in this work, it can be assumed that: σ12 = γ12 = 0. Experimental verification of this theoretical approach was first made in terms of the velocity of these waves [18]–[20]. However, many porous materials, either air-or water-saturated (e.g., aerogels [21] and marine sediments [22] ) exhibit a linear variation of the attenuation coefficient over a very wide frequency range, whereas the corresponding theoretical predictions provide a dependency on the square root of the frequency. ...
The propagation of ultrasonic waves in the cylindrical micro-pores (pore diam. 1 microm) of ion-track membranes (ITMs) is studied. This membrane fabrication technique provides unique possibilities to obtain cylindrical micro-pores with a very high degree of accuracy in pore shape, size, and orientation. Several ITMs were specially produced having the same pore diameter, orientation, and geometry, but different thickness. Porosity, pore diameter, and shape were determined using scanning electron microscopy, and then the coefficient of ultrasound transmission was measured using air coupling and spectral analysis. These experimental conditions permit us to eliminate the influence of the boundary conditions and to achieve a strong decoupling between the fluid filling the pores and the solid constituent of the membrane. Hence, the velocity and the attenuation coefficient for ultrasound propagation in the pores can be measured. These parameters are compared with the predictions made by conventional theories for sound propagation in porous media and in cylindrical channels. The conclusions of this work provide a better understanding of wave propagation in micro-pores and establish the basis of an ultrasonic porometry technique for ITMs.
The hydromechanical coupling effect between soil skeleton and pore fluid phases, that is, dissipation of dynamically induced excess pore water pressure, can be a critical factor in determining the dynamic response of geotechnical structures in a certain permeability range.A theoretical study for investigating this certain permeability range is the propagation mechanismof compressional waves in geotechnical media, which has not been thoroughly investigated, especially for its dual behavior (i.e., the fast wave and slow wave) in the concerned frequency range for geotechnical engineering problems (0.1-20 Hz). Therefore, this paper numerically studies the propagation mechanism of compressional waves in saturated geotechnical media under soil dynamics loads. Investigations reveal the transition mechanism between the two kinds of compressional waves considering the effects of loading frequency, soil stiffness, and permeability. More importantly, based on the numerical results, this paper proposes empirical transition lines between the fast wave and slow wave, specified by loading frequency and material permeability. Practically, the proposed empirical lines can be used as the boundaries defining the applicable ranges of one-phase and two-phase coupled analysis for predicting the dynamic response of saturated geotechnical structures.
A new theory of propagation of elastic waves in porous materials was developed for pore characterization. The idea is based on the arrival probability of a direct ray, reflecting ray, and creeping ray. The theory gives a good account of the nature of elastic waves propagated in porous solids not only qualitatively, but also quantitatively. The authors show a rather good correlation between pore characteristics (porosity and pore size) and ultrasonic characteristics (waveform and arrival time), and propose a nondestructive technique based on the presented theory for evaluating the porosity and pore size.
The use of a pulsed laser to generate well characterised acoustic sources has certain advantages over traditional methods in that it is reproducible and results in a wide bandwidth. Further, it is not necessary to couple the source transducer to the sample by immersion techniques or under pressure, as is frequently the case when the attenuation properties of porous solids are studied. An investigation into the propagation of ultrasonic transients within porous solids using the pulsed laser technique is described. The mechanisms leading to ultrasonic generation by irradiation of solids with pulsed lasers is reviewed in Section 1.1, followed by a brief outline of elastic wave propagation through porous media. A description of the apparatus used is given in section 2 and the results obtained by laser irradiation are compared to those obtained using PZT transducers in section 3.
In this paper, we present a probabilistic theory of propagation of ultrasonic waves in porous ceramics, and propose a new method of ultrasonic inspection for nondestructive pore characterization. The idea is based upon the arrival probability of ultrasonic rays. When incident rays impinge on a pore, they travel around the pore surface and increase the propagation time. We studied this process probabilistically, and found that the propagated waveforms can be expressed as Gaussian functions. The Gaussian waveform is determined by the porosity, pore size and pore shape. This new finding led to the following important laws. (1) The delay time of an ultrasonic wave passing through porous ceramics is proportional to the porosity. (2) The pulse width of the wave increases with increasing mean pore size. (3) The amplitude of the wave decreases with mean pore size. (4) The delay time and pulse width of the wave increase as the mean pore perimeter increases. Formulae for these relationships between ultrasonic and pore characteristics were derived, and an ultrasonic method for evaluating porosity and pore size was proposed.
The transmission coefficient for sound at an interface between a fluid and a porous medium has been derived. Numerical values are compared with experimental values obtained by Plona [Appl. Phys. Lett. 36, 259 (1980)] who transmitted a short ultrasonic pulse through a porous disk. The good agreement between theory and experiment supports the conclusion that a second compressional wave has been observed.
The two-wave phenomenon, the wave separation of a single ultrasonic pulse in cancellous bone, is expected to be a useful tool for the diagnosis of osteoporosis. However, because actual bone has a complicated structure, precise studies on the effect of transition conditions between cortical and cancellous parts are required. This study investigated how the transition condition influenced the two-wave generation using three-dimensional X-ray CT images of an equine radius and a three-dimensional simulation technique. As a result, any changes in the boundary between cortical part and trabecular part, which gives the actual complex structure of bone, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. The results led us to the possibility of using the two-wave phenomenon in a diagnostic system for osteoporosis in cases of a complex boundary.
Our interest is in dynamic filtration through periodic, porous, saturated media. More precisely, here we develop a three-dimensional numerical model, based on boundary element methods, to compute the dynamic permeability over a wide range of such media. This generalized Darcy coefficient is obtained by the homogenization process applied to a periodic, deformable, porous medium under dynamic solicitations. An unusual choice of Green functions is made. A simple numerical procedure is used for the treatment of the periodic boundary conditions. Recent advances to treat singular integrals are employed and extended to our case. The method is tested on simple examples where theoretical results are available. In the static case results are compared with many previous results on periodic arrays of spheres. New results are given in the dynamic case. The scaling behavior for dynamic permeability in porous media is checked and discussed.
Recently, a new method of deriving the coefficients in the strain energy functional of Biot’s theory for elastic waves in fluid‐saturated porous media was proposed. The results of this approach have been criticized as being unphysical in the limit K */K f → 0, where K * and K f are the frame and fluid bulk moduli. We show that an error arose due to the literal interpretation of the term e f as the fluid dilatation whereas this term actually differs substantially from the fluid dilatation in some situations. We also show that by lumping part of the solid together with the fluid (conceptually) to form a fictitious fluid suspension, the corresponding fictitious e f may be correctly interpreted as the dilatation of the fluid suspension. This trick leads to a transformation which takes the recently published formulas into standard form. The standard formulas are found to be invariant under the same transformation. New insight into the significance of the standard formulas for the coefficients is obtained.
Biot's linear model of stress-wave propagation in a fluid-saturated elastic framework is combined with a linear theoretical description of an inelastic frame to describe fluid-saturated media in terms of a composite model. The composite model, the Constant Q (CQ) model, assumes an inelastic frame with frequency-dependent complex elastic moduli and results in a frame response that is causal with Q exactly independent of frequency. The influence of frame inelasticity on the composite-model Type I (compression), Type II, and shear-wave attenuation response is found to be greatest for high and low frequencies, considering a frequency range of 10-10⁷ Hz. -from Author
It is accepted widely that the Biot theory predicts only one shear wave representing the in-phase/unison shear motions of the solid and fluid constituent phases (fast S-wave). The Biot theory also contains a shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow S-wave). From the outset of the development of the Biot framework, the existence of this mode has remained unnoticed because of an oversight in decoupling its system of two coupled equations governing shear processes. Moreover, in the absence of the fluid strain-rate term in the Biot constitutive relation, the velocity of this mode is zero. Once the Biot constitutive relation is corrected for the missing fluid strain-rate term (i.e., fluid viscosity), this mode turns out to be, in the inertial regime, a diffusive process akin to a viscous wave in a Newtonian fluid. In the viscous regime, it degenerates to a process governed by a diffusion equation with a damping term. Although this mode is damped so heavily that it dies off rapidly near its source, overlooking its existence ignores a mechanism to draw energy from seismic waves (fast P- and S-waves) via mode conversion at interfaces and at other material discontinuities and inhomogeneities. To illustrate the consequence of generating this mode at an interface, I examine the case of a horizontally polarized fast S-wave normal incident upon a planar air-water interface in a porous medium. Contrary to the classical Biot framework, which suggests that the incident wave should be transmitted practically unchanged through such an interface, the viscosity-corrected Biot framework predicts a strong, fast S-wave reflection because of the slow S-wave generated at the interface. © 2008 Society of Exploration Geophysicists. All rights reserved.
We perform wave propagation simulations in porous media on microscale in which a slow compressional wave can be observed. Since the theory of dynamic poroelasticity was developed by Biot (1956), the existence of the type II or Biot's slow compressional wave (SCW) remains the most controversial of its predictions. However, this prediction was confirmed experimentally in ultrasonic experiments. The purpose of this paper is to observe the SCW by applying a recently developed viscoelastic displacement-stress rotated staggered finite-difference (FD) grid technique to solve the elastodynamic wave equation. To our knowledge this is the first time that the slow compressional wave is simulated on first principles.
The poroviscoelastic model, which is asynthesis of the well-known viscoelasticmodel and Biot’s poroelastic model, is presented in the context of the propagation of plane acoustic waves in linear viscoelasticporous media. Except for the propagation of a second compressional P 2 wave in such media, results concerning particle displacement, maximum attenuation, and direction of maximum energy flow are very similar to that obtained by Borcherdt [J. Geophys. Res. 78, 2442–2453 (1973)] in simple viscoelastic media. The energy conservation relation is given explicitly in the case of time harmonic radiation field. From this, expressions of the energy flux, energy densities, dissipated energy, and Q −1 are derived. Furthermore it is demonstrated that the total attenuation Q total −1 is equal to the sum of the viscoelasticattenuation Q visco −1 and Biot’s poroelastic attenuation Q poro −1 . This allows a direct comparison between energy dissipated by viscoelasticity and that dissipated by Biot mechanism. For propagation of acoustic waves in infinite natural porous media the latter is always negligible compared to the former. Computed curves and physical interpretations are proposed to illustrate the theoretical derivations.
This paper presents in detail the resonant bar technique used to measure acoustic properties of materials in the sonic frequency range (congruent-to 5-20 kHz). Measurements are corrected for the effects of added mass and jacketing; extrinsic effects such as the sample diameter or the dispersion at higher frequencies can mask intrinsic properties of material and are analyzed here. When this technique is used on porous saturated media such as rocks, it is important to avoid the "Biot-Gardner-White" effect; this intrinsic effect can lead to erroneous high attenuations in unjacketed saturated samples. Experimental evidence of its occurrence on water-saturated rods is presented. Experimental results of velocity and attenuation obtained on various rock types such as limestones and sandstones show the drastic sensitivity of these measurements to effective pressure. Hertz's theory can be applied to describe the behavior of under-consolidated sandstones. Under high confining pressure, sonic attenuation in water-saturated limestone samples is found to be very low (Q greater-than-or-equal-to 100); attenuation in sandstones is always low (Q greater-than-or-equal-to 50), except for some shaly sandstones.
The homogenization process applied to fine periodic deformable saturated porous medium under dynamic solicitations leads to the macroscopic description. This method enables us to perform a complete calculation of the effective parameters. The main fact is that the formulation so obtained—similar to Biot’s results—exhibits a generalized example of Darcy’s law which contains all the dynamic couplings between the two phases. After recalling the main facts of the subject this work presents some properties of the generalized Darcy coefficient and an experimental checking. An agreement between experimental and numerical results using the homogenization process is obtained.
Some recent mathematical results are used to study the propagation features of the high-frequency vibrational energy density in three-dimensional fluid-saturated, isotropic poro-visco-elastic media. The theory of high-frequency asymptotics of the solutions of hyperbolic partial differential equations shows that their energy satisfies Liouville-type transport or radiative transfer equations for randomly heterogeneous materials. For long propagation times these equations can be approached by diffusion equations. The corresponding diffusion parameters-mean-free paths and diffusion constants-associated with Biot's linear model for such media, are derived. The analysis accounts for the thermal and viscous memory effects of the fluid phase, and the viscous memory effect of the solid phase through time convolution operators. In this respect it also extends the existing mathematical results.
The time harmonic Green function for a point load in an unbounded fluid‐saturated porous solid is derived in the context of Biot’s theory. The solution contains the two compressional waves and one transverse wave that are predicted by the theory and have been observed in experiments. At low frequency, the slow compressional wave is diffusive and only the fast compressional and transverse waves radiate energy. At high frequency, the slow wave radiates, but with a decay radius which is on the order of cm in rocks. The general problem of scattering by an obstacle is considered. The point load solution may be used to obtain scattered fields in terms of the fields on the obstacle. Explicit expressions are presented for the scattering amplitudes of the three waves. Simple reciprocity relations between the scattering amplitudes for plane‐wave incidence are also given. These hold under the interchange of incident and observation directions and are completely general results. Finally, the point source solution is Fourier transformed to get the solution for a load which is a delta function in time as well as space. We obtain a closed form expression when there is no damping. The three waves radiate from the source as distinct delta function pulses. With damping present, asymptotic approximations show the slow wave to be purely diffusive. The fast and transverse waves propagate as pulses. The pulses are Gaussian‐shaped, which broaden with increasing time or radial distance.
The experimental measurements of tortuosity of porous structures using either the acoustic index of refraction of superfluid 4He or the electrical conductivity are shown to agree with each other. This and other measured parameters are used to calculate directly the acoustic speeds of water-saturated, fused-glass-bead samples; there are no adjustable parameters and agreement with experiment is excellent. The dependence of tortuosity on pore volume fraction, varphi, is discussed.
The propagation of ultrasonic waves in the cylindrical
micro-pores (pore diam. <1 μm) of ion-track membranes
(ITMs) is studied. This membrane fabrication technique provides
unique possibilities to obtain cylindrical micro-pores
with a very high degree of accuracy in pore shape, size, and
orientation. Several ITMs were specially produced having the
same pore diameter, orientation, and geometry, but different
thickness. Porosity, pore diameter, and shape were determined
using scanning electron microscopy, and then the coefficient
of ultrasound transmission was measured using air coupling
and spectral analysis. These experimental conditions permit us
to eliminate the influence of the boundary conditions and to
achieve a strong decoupling between the fluid filling the pores
and the solid constituent of the membrane. Hence, the velocity
and the attenuation coefficient for ultrasound propagation in
the pores can be measured. These parameters are compared
with the predictions made by conventional theories for sound
propagation in porous media and in cylindrical channels. The
conclusions of this work provide a better understanding of
wave propagation in micro-pores and establish the basis of an
ultrasonic porometry technique for ITMs.
A straightforward nondestructive method based on the probabilistic theory of ultrasonic wave propagation [JSME Int. J., Ser. A, Mech. Mater. Eng. 39, 266 (1996)] was developed to quantitatively evaluate porosities, pore shapes, and pore sizes in advanced porous ceramics merely by measuring the ultrasonic delay time and pulse width. The extensive ultrasonic measurements and image microanalyses were conducted in advanced porous alumina, sialon, and zirconia with different porosities. A universal equation was established for porous ceramics, clarifying the intrinsic relationships between ultrasonic characteristics (propagation time and pulse width) and pore distribution (porosity, pore shape, and pore size). The critical volume fraction porosity were estimated separately as approximately 0.06, 0.11, and 0.10 in these ceramics using image microanalysis techniques, at which the transition from the continuous to discontinuous pore phase takes place during sintering. An excellent agreement of two useful corollaries with the data on the above nondestructive and destructive examinations validates the quantitative applicability of the probabilistic theory to pore characterization of advanced ceramics, metals, and their combinations. © 1999 American Institute of Physics.
Digital rock methodology combines modern microscopic im-aging with advanced numerical simulations of the physical prop-erties of rocks. Modeling of elastic-wave propagation directly from rock microstructure is integral to this technology. We sur-vey recent development of the rotated staggered grid RSG fi-nite-difference FD method for pore-scale simulation of elastic wave propagation in digital rock samples, including the dynamic elastic properties of rocks saturated with a viscous fluid. Exami-nation of the accuracy of this algorithm on models with known analytical solutions provide an additional accuracy condition for numerical modeling on the microscale. We use both the elastic and viscoelastic versions of the RSG algorithm to study gas hy-drates and to simulate propagation of Biot's slow wave. We apply RSG method ology to examine the effect of gas hydrate distribu-tions in the pore space of a rock. We compare resulting P-wave velocities with experimentally measured data, as a basis for building an effective-medium model for rocks containing gas hy-drates. We then perform numerical simulations of Biot's slow wave in a realistic 3D digital rock model, fully saturated with a nonviscous fluid corresponding to the high-frequency limit of poroelasticity, and placed inside a bulk fluid. The model clearly demonstrates Biot's slow curve when the interface is open be-tween the slab and bulk fluid. We demonstrate slow wave propa-gation in a porous medium saturated with a viscous fluid by ana-lyzing an idealized 2D porous medium represented alternating solid and viscous fluid layers. Comparison of simulation results with the exact solution for this layered system shows good agree-ment over a broad frequency range.
We perform numerical simulation of ultrasonic experiments on poroelastic samples, in which Biot's slow compressional wave had been observed. The simulation is performed using OASES modeling code, which allows to compute elastic wave fields in layered poroelastic media. Modeled were the experiments of Plona (1980), Rasolofosaon (1988), and our own measurements. In all the three situations, a good agreement between experiment and simulations has been observed. This further confirms the fact that Biot's theory of poroelasticity, on which the simulations were based, adequately describes the behavior of the porous materials under investigations at ultrasonic frequencies.
The porous materials studied are those described as two-component
composites where both the fluid phase and the solid phase are continuous
and interpenetrating. The bulk acoustic wave propagation in these
materials is shown both theoretically and experimentally to include two
types of compressional waves. The fast or normal compressional wave
corresponds to an in-phase motion of the fluid and the solid, whereas
the slow compressional wave corresponds to an in-phase motion.
Highlights of the Biot theory are described as well as some recent
velocity and attenuation measurements in one particular kind of porous
medium
First Page of the Article
Biot’s theory [J. Acoust. Soc. Am. 28, 168 (1956)] is investigated by the finite‐element method at ultrasonic frequencies. The numerical results for an acoustic/poroelastic/acoustic system are presented by using the double‐node layers technique of Teng and Kuo [Comput. Acoust. Int. Assoc. Mathe. Comput. Simul., 239 (1988)]. This work is motivated by the work of Plona [Appl. Phys. Lett. 36, 259 (1980], Johnson [Appl. Phys. Lett. 37, 1065 (1980)], and Johnson and Plona [J. Acoust. Soc. Am. 72, 556 (1982)], in which they have experimentally and analytically investigated the problem of wave propagation in a fluid/fluid‐saturated‐porous‐solid/fluid system and fourth sound in a porous ‘‘superleak’’ saturated with superfluid helium. Given suitable physical parameters in the finite‐element calculations, if the open‐boundary conditions at the fluid/porous‐solid interfaces are used, excellent agreement with Plona’s measurements has been obtained. The finite‐element results with the sealed interfaces for the same model, showing a diminishing of the second type of compressional wave, also agree with the experimental work performed by Rasolofosaon [Appl. Phys. Lett. 52, 780 (1988)].
Formulas for the scattering from an inhomogeneous sphere in a fluid‐saturated porous medium are used to construct a self‐consistent effective medium approximation for the coefficients in Biot’s equations of poroelasticity [J. Acoust. Soc. Am. 28, 168 (1956)] when the material constituting the porous solid frame is not homogeneous on the microscopic scale. The discussion is restricted to porous materials exhibiting both macroscopic and microscopic isotropy. Brown and Korringa [Geophysics 40, 608 (1975)] have previously found the general form of these coefficients. The present results give explicit estimates of all the coefficients in terms of the moduli of the solid constituents. The results are also shown to be completely consistent with the well‐known results of Gassmann and of Biot and Willis, as well as those of Brown and Korringa.
The propagation of an ultrasonic wave through a porous material, when the wavelength is large compared with the size of the inhomogeneities, can be studied in terms of Biot's theory. The fundamental prediction of this theory is the existence of a second bulk compressional wave which propagates with a velocity lower than the velocity of propagation of the wave in the fluid. This second bulk compressional wave has only been observed in a few experimental cases. In this paper the observation of a second bulk compressional wave, in a quite different experimental situation, which can be related to Biot's predictions, will be described. Samples are not fully water-saturated, as was required, but include small air bubbles. The influence of air bubbles in the generation of the second bulk compressional wave will be analysed. It is also possible to make some hypotheses on the influence of material properties on slow-wave generation. Since the measurements include strong dispersive phenomena, specific spectral techniques to measure the phase velocity of both propagating modes have been employed.
A method for estimating the strength of the slow wave in the modes propagating in porous layers is presented. It is based upon expansions on transition terms which are linear combinations of the reflection and transmission coefficients. Suitable forms of these coefficients are needed and it is shown how they can be obtained. Both open pore and sealed pore boundary conditions are investigated. It is shown that the zeroth-order and the first-order terms of the expansions suffice to describe accurately the modes and to estimate the strength of the slow wave. Approximations of the absorption coefficient by the porous layer can be deduced. Angles of incidence above and below the critical angle of the shear wave are considered. Comparisons between theory and experiments for the two types of boundary conditions are presented at normal incidence for the transition terms.
Supervised by M. Nafi Tokshoz and Chuen H. Cheng. Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 1990. Includes bibliographical references (leaves 202-210).
A theory is developed for the soundvelocity and attenuation in a medium composed of closely packed solid particles immersed in a fluid. The absorption mechanism considered is that of viscous motion of fluid between the particles, and the macroscopic point of view is taken. For high enough frequencies the attenuation is proportional to the square root of the product of frequency and static flow resistance of the medium. Comparisons are made with data reported previously by others, and the agreement with the theoretical results is good for cases involving particles of nearly uniform size. Still another frequency effect, dependent on the size distribution, is evident for unsorted granular substance.
In this investigation, Biot's (1962) theory for wave propagation in porous solids is applied to study the velocity and attenuation of compressional seismic waves in partially gas-saturated porous rocks. The physical model, proposed by White (1975), is solved rigorously by using Biot's equations which describe the coupled solid-fluid motion of a porous medium in a systematic way. The quantiative results presented here are based on the theory described in Dutta and Ode' (1979, this issue). We removed several of White's questioned approximations and examined their effects on the quantitative results. We studied the variation of the attenuation coefficient with frequency, gas saturation, and size of gas inclusions in an otherwise brine-filled rock. Anomalously large absorption (as large as 8 dB/cycle) at the exploration seismic frequency band is predicted by this model for young, unconsolidated sandstones. For a given size of the gas pockets and their spacing, the attenuation coefficient (in dB/cycle) increases almost linearly with frequency f to a maximum value and then decreases approximately as l/..sqrt..f. A sizable velocity dispersion (of the order of 30 percent) is also predicted by this model. A low gas saturation (4 to 6 percent) is found to yield high absorption and dispersion.
An exact theory of attenuation and dispersion of seismic waves in porous rocks containing spherical gas pockets (White model) is presented using the coupled equations of motion given by Biot. Assumptions made are (1) the acoustic wavelength is long with respect to the distance between gas pockets and their size, and (2) the gas pockets do not interact. Thus, the present theory essentially is quite similar to that proposed by White (1975), but the problem of the radially oscillating gas pocket is solved in a more rigorous manner by means of Biot's theory (1962). The solid-fluid coupling is automatically included, and the model is solved as a boundary value problem requiring all radial stresses and displacements to be continuous at the gas-brine interface. Thus, we do not require any assumed fluid-pressure discontinuity at the gas-water contact, such as the one employed by White (1975). We have also presented an analysis of all of the field variables in terms of Biot's type I (the classical compressional) wave and type II (the diffusion) wave. Our quantitative results are presented in Dutta and Ode (1979, this issue).
A unified treatment of the mechanics of deformation and acoustic propagation in porous media is presented, and some new results and generalizations are derived. The writer's earlier theory of deformation of porous media derived from general principles of nonequilibrium thermodynamics is applied. The fluid‐solid medium is treated as a complex physical‐chemical system with resultant relaxation and viscoelastic properties of a very general nature. Specific relaxation models are discussed, and the general applicability of a correspondence principle is further emphasized. The theory of acoustic propagation is extended to include anisotropic media, solid dissipation, and other relaxation effects. Some typical examples of sources of dissipation other than fluid viscosity are considered.
A second bulk compressional wave has been observed in a water‐saturated porous solid composed of sintered glass spheres using an ultrasonic mode conversion technique. The speed of this second compressional wave was measured to be 1040 m/sec in a sample with 18.5% porosity. The theory of Biot, which predicts two bulk compressional waves in porous media, provides a qualitative explanation of the observations. To the author’s knowledge, this type of bulk wave has not been observed at ultrasonic frequencies.
A theory is developed for the propagation of stress waves in a porous elastic solid containing compressible viscous fluid. The emphasis of the present treatment is on materials where fluid and solid are of comparable densities as for instance in the case of water‐saturated rock. The paper denoted here as Part I is restricted to the lower frequency range where the assumption of Poiseuille flow is valid. The extension to the higher frequencies will be treated in Part II. It is found that the material may be described by four nondimensional parameters and a characteristic frequency. There are two dilatational waves and one rotational wave. The physical interpretation of the result is clarified by treating first the case where the fluid is frictionless. The case of a material containing viscous fluid is then developed and discussed numerically. Phase velocity dispersion curves and attenuation coefficients for the three types of waves are plotted as a function of the frequency for various combinations of the characteristic parameters.