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Confirmation of Biot's Theory
Abstract
Plona’s recent measurements of elastic‐wave speeds in a water‐saturated porous structure of sintered glass beads are compared quantitatively to the predictions of Biot’s theory. The theoretical predictions lie within the bounds of experimental error (3%) for the fast compressional wave and for the shear wave in all cases. For the slow compressional wave, the theoretically predicted speeds lie within about 10% of the experimental values and increase with increase in porosity as observed. Our model achieves this agreement with no significant free parameters. The frame moduli are estimated using a recently developed self‐consistent theory of composite materials. The induced mass of the frame in a water environment is also estimated theoretically.
... These models are now incorporated in a number of standard software packages including COMSOL Multiphysics Ò and AlphaCell. The parameters resulting from the inversion are compared to results from nonacoustical models of Umnova et al. [22], Berryman [23], Revil, Glover, Pezard and Zamora (RGPZ) [9] and Kozeny and Carman [24] which relate key properties of granular media with the grain size. To the best of our knowledge, this work has never been done systematically and it is a main novelty of our paper. ...
... There are a number of models that can relate to some of the above pore parameters. The work by Berryman [23] suggests that the tortuosity of a stack of identical, spherical solid particles can be expressed via porosity only: ...
... The NUPSD [18], JCAL [27] and Miki's [20] models have been used to invert the non-acoustical parameters of these media via an optimisation algorithm. Some of these non-acoustical parameters have been estimated using Berryman's [23], Umnova's [22], KC [24] and RGPZ [9] analytical models. An important outcome of this work is a better understanding of the accuracy of popular models to predict or invert key parameters of porous media which can be of interest in areas of science and engineering other than acoustics. ...
There is a general lack of understanding on the accuracy of theoretical and empirical models which can be used to predict the acoustical properties of unsaturated granular media. It is also unclear how well these models are suited for parameter inversion in application to these media. In this work three popular prediction models available in commercial software packages, e.g. COMSOL Multiphysics, are used to invert key non-acoustical parameters of two types of glass beads and two fractions of silica sand for which acoustical data are obtained with a standard acoustic impedance tube setup. These results are compared against predictions made with four established non-acoustical analytical models which relate these characteristics to the effective particle size and other pore characteristics. A discrepancy between the two model classes (acoustical and non-acoustical models) and measured data is quantified. This work also quantifies the accuracy of acoustical characterisation methods used to estimate key pore characteristics of granular media with a relatively inexpensive laboratory setup and rapid inversion method based on optimisation.
... Bourbi é et al. 1987 ). Note that in fact, quantities ρ a and β are the ab initio macroscopic material properties, and ρ f and a are derived from them (Berryman 1980 ). Parametrization (3) with seven material properties shows that Biot's model is identified by zero viscosity ( η K = 0 ) and the inverse mobility ϑ = η f / κ (in matrix d B ) only acting on the relative movement of the pore fluid. ...
... First, porosity φ and densities ρ and ρ f are the key macroscopic observables which are relati vel y readil y measurable. The tortuosity a and the above relation for the inertial parameter β in eq. ( 3 ) are constrained by considering a linear 'global' pore flow through the rock frame (Berryman 1980 ). For elastic properties, the corresponding experiments are well known and consist of measuring the drained (at zero pore pressure) bulk modulus K D , the undrained (at zero pore flow) modulus K U = K D + α 2 M, and the moduli of the solid grains K s and pore fluid K f . ...
In a large body of rock-physics research, seismic wave velocity dispersion and attenuation in fluid-saturated porous rock are studied by constructing analytical or numerical models for time- or frequency-dependent dynamic (effective, or viscoelastic) moduli. A key and broadly used model of such kind is the Zener's, or the standard linear solid (SLS). This model is qualitatively successful in explaining many field and laboratory observations and serves as the key element of many generalizations such as the Burgers model for plastic deformations or the generalized SLS explaining band-limited or near-constant seismic attenuation. However, as a physical model of fluid-saturated porous rock, the SLS has several major limitations: disregard of inertial effects, absence of secondary wave modes and lack of key physical parameters such as porosity and Skempton coefficients. Grainy and porous rock is an unconsolidated material in which the effective density is frequency-dependent, and its effects on wave velocities may exceed those of the dynamic modulus. To overcome these limitations of the empirical SLS, we propose a rigorous rheologic model based on classical continuum mechanics and called the extended SLS, or eSLS. This rheology explains the available attenuation and dispersion observations equally well, but it is also close to Biot's model, honours all poroelastic relations, includes inertial effects, and reveals several new physical properties of the material. Detailed comparison of the eSLS and Biot's models gives a physical-mechanism-based classification of wave-induced fluid flow (WIFF) phenomena. In this classification, the so-called ‘global-scale’ flows occur in Biot's type structures within the material, whereas the ‘local-scale’ WIFF occurs in eSLS-type structures. Combining Biot's and eSLS models gives a broad class of rheologies for linear anelastic phenomena within rock with a single type of porosity. The model can be readily generalized to multiple porosities and different types of internal variables, such as describing squirt flows, wetting or thermoelastic effects. Modelling is conducted with relatively little effort, using a single matrix equation similar to a mechanical form of the standard SLS. By combining the eSLS and Biot's models, observations of dynamic-modulus dispersion and attenuation can be inverted for macroscopic mechanical properties of porous materials.
... On the other side, the natural earth foundation can be divided into entirely and partially saturated states. The wave propagation behavior in partially saturated soil is significantly more complex than that in completely saturated soil because the capillary pressure and coupling effect will affect the motion behaviors of elastic waves in unsaturated soil [1,2]. ...
... Undoubtedly, the emergence of Biot theory [3,4] and mixture theory [5] not only give the sound basis for the research of wave theory but accelerate the research process. For the bulk and Rayleigh waves propagation in saturated soil, researchers including Jones [6], Plona [7], Berryman [1], Berryman [8,9], Zhou and Ma [10], Straughan et al. [11], Rohan et al. [12], Tung [13], and Wang et al. [14] implemented several theoretical and experimental works. They drew that the longitudinal, transverse, and Rayleigh waves depend on not only the frequency but the soil parameters such as permeability and soil mass types. ...
The flow-independent viscosity of the soil skeleton has significant influence on the elastic wave propagation in soils. This work studied the bulk and Rayleigh waves propagation in three-phase viscoelastic soil by considering the contribution of the flow-independent viscosity from the soil skeleton. Firstly, the viscoelastic dynamic equations of three-phase unsaturated soil are developed with theoretical derivation. Secondly, the explicit characteristic equations of bulk and Rayleigh waves in three-phase viscoelastic soil are yielded theoretically by implementing Helmholtz resolution for the displacement vectors. Finally, the variations of the motion behavior for bulk and Rayleigh waves with physical parameters such as relaxation time, saturation, frequency, and intrinsic permeability are discussed by utilizing calculation examples and parametric analysis. The results reveal that the influence of soil flow-independent viscosity on the wave speed and attenuation coefficient of bulk and Rayleigh waves is significantly related to physical parameters such as saturation, intrinsic permeability, and frequency.
... which can be defined by [42]. ...
... where n is porosity; ρ g is the density of the solid material; ρ f is the density of the fluid; γ is the coefficient of induced inertia by solid-fluid interaction, which depends on the shape of the solid particles. When the solid skeletons are modeled as spherical particles, γ = 0.5 [42]. The elastic moduli have been correlated by Biot and Wills [44] based on experimental measurements, which is supposed to be ...
An analytical solution for the scattering of harmonic P1 and SV waves in a poroelastic half-plane with a shallow lined tunnel is obtained using the plane complex theory in elastodynamics. In light of the wave function expansions, the wave fields of the poroelastic medium and the liner with unknown coefficients are obtained based on Biot's theory and Helmholtz decomposition. Complex-valued expressions of the effective stresses, the fluid stress, and the displacements of the poroelastic medium and the liner are expressed by the complex variable function method and the conformal transformation technique. With the boundary conditions and the continuity of the medium-liner interface, the boundary value problem results in a series of algebraic equations. The unknown coefficients in the infinite set of algebraic equations can be solved numerically by truncating the series number. A parametric study for the incident SV waves is performed to investigate dynamic stress concentrations and fluid stress of the medium and the liner. Numerical results show that the embedment depth of the tunnel, the incident angle of the excitations, and the porosity of the medium have considerable influence on the dynamic responses of the medium and the liner. The shielding effect of the tunnel on the incident SV waves is obvious. For the big embedment depth of the tunnel, the scattered waves contribute little to the displacements and dynamic stress concentration of the medium and the liner. For a high porosity close to the critical value, the response of the medium-liner system to the incident waves is great.
... The coefficient ρ 12 is attributed to the tortuosity a. A typical simplified relation was proposed by Berryman in 1980 [24] and has the form ...
... The theory of Biot found many applications, especially for the description of processes in geomaterials, e.g., Rice and Cleary [105], Pecker and Deresiewicz [100] who also considered thermal effects or Schrefler [112], relied on this model. Biot's theory has been verified by Plona in 1980 conducting experiments with sintered glass [24,101]. The main difference of the Biot model (27) and the model (17)- (22) is the occurrence of the relative accelerations and the static coupling parameter Q. ...
First, different porous media theories are presented. Some approaches are based on the classical mixture theory for fluids introduced in the 1960s by Truesdell and Coworkers. One of the first researchers who extended the theory to porous media (thus mixtures containing at least one solid constituent) and also accounting for chemical reactions was Bowen. Another important branch of porous media theory goes back to Biot. In the beginning, he dealt with classical geotechnical problems and set up his model empirically. Mathematicians often use reaction–diffusion equations which are limited in comparison with continuum models by several restrictive assumptions and very often only applicable to special problems. In this paper, the focus lies on approaches based on the mixture theory which incorporate chemical reactions. Different strategies to describe the chemical potential for mixtures are presented, and different opinions about the exploitation of the second law of thermodynamics for mixtures are put forward. Finally, several works of different types including chemical reactions in porous media are summarized.
... According to [53], λ, μ, P, and Q are the four elastic moduli in Biot's wave equations, which can be determined by the critical porosity n cr , critical bulk modulus of saturated soil Kcr, bulk modulus of soil particles Kg, and volume modulus of fluid Kf. b is the dissipation coefficient. ρ11, ρ12, and ρ22 are the dynamic mass coefficients that can be described by [54]. ...
River valley terrain can lead to local amplification effects of site seismic responses and significant spatial variation of strong ground motions. This paper presented a semi-analytical solution for the scattering of incident plane P-waves by a saturated river valley with arbitrary shapes containing water, using the moment method and the wave function expansion method. The influence of factors such as valley geometry shapes, soil porosity, wave incident angle and frequency on the seismic response of valley topography is further investigated. Parameter analysis shows that these factors have a significant impact on the site seismic response. Soil porosity has a greater impact on the horizontal response of the site than on the vertical response. Moreover, the soil porosity is directly related to the amplitude of the pore pressure, while the incidence angle and frequency of the incident wave determine the distribution position of the pore pressure amplitude. This semi-analytical solution can be seen as an attempt to solve the scattering problem of incident P wave by saturated river valleys of arbitrary shapes containing water. The purpose of this paper is to provide a reference and a possible way for subsequent scholars to study the analytical solution to this problem. The conclusions drawn from the parameter analysis in this paper can also serve as a theoretical basis for seismic fortification in saturated valley areas.
... To accurately describe seismic dispersion and attenuation caused by global flow, Biot (1956a , b ) proposed a poro-elastic theory based on analytical mechanics principles, specifically on Lagrange's equations, which allowed for the derivation of a dynamic system. Prediction of a new type of compressional wave (the slow P wave) made by Biot's theory had been confirmed through laboratory observations (Plona 1980, Berryman 1980, Bouzidi and Schmitt 2009. Thus, Biot's theory has gained widespread acceptance and was believed to provide a comprehensive framework for quantitatively modeling the macroscopic wave propagation in porous media (Brutsaert 1964, Pride 2005 in the past few decades. ...
Characterizing seismic wave propagation in fluid-saturated porous media well enhances the precision of interpreting seismic data, bringing benefits to understanding reservoir properties better. Some important indicators, including wave dispersion and attenuation, along with wavefield, are widely used for interpreting the reservoir, and they can be obtained from a rock physics model. In existing models, some of them are limited in scope due to their complexity, for example, numerical solutions are difficult or costly. In view of this, this study proposes an approach of establishing equivalent dynamic equations of existing models. First, the framework of the equivalent model is derived based on Biot's theory, while the elastic coefficients are set as unknown factors. The next step is to use deep neural networks (DNNs) to predict these coefficients, and surrogate models of unknowns are established after training DNNs. The training data is naturally generated from the original model. The simplicity of the equations form, compared to the original complex model and some other equivalent manners such as viscoelastic model, enables the framework to perform wavefield simulation easier. Numerical examples show that the established equivalent model can not only predict similar dispersion and attenuation, but also obtain wavefields with small differences. This also indicates that it may be sufficient to establish an equivalent model only according to dispersion and attenuation, and the cost of generating such data is very small compared to simulating the wavefield. Therefore, the proposed approach is expected to effectively improve the computational difficulty of some existing models.
... To account for the presence of pores within non-clay inorganic framework, we utilize the isotropic self-consistent approximation (SCA) to derive the equivalent elastic moduli of the non-clay inorganic framework, denoted as K * non− clay and μ * non− clay . The isotropic SCA model can be written as follows (Berryman James, 1980;Berryman, 1995): ...
... In this regard, inclusion-based rock physics models appear to be suitable for modeling the velocity of carbonate rocks, as they allow for the incorporation of the effects of variable pore structure (Agersborg et al. 2009). The DEM outline approximates the porosity of a two-phase compound by gradually incorporating a small number of pores (phase 2) into the matrix (phase 1) (Berryman 1980a(Berryman , 1980b. By solving a coupled system of ordinary differential equations with the initial conditions, this scheme simulates the high-frequency elastic response of porous rocks (effective bulk moduli and effective shear moduli). ...
... The slow compressional wave is one of the two fundamental longitudinal waves propagating in a fluid-saturated porous medium. Although it has been predicted as early as 1956 by Biot (1956aBiot ( , 1956b, its existence was not confirmed until the observation of Plona (1980) in a water-saturated synthetic sample, which was compared with Biot's theory (Berryman, 1980). Subsequently, more experimental evidences of the slow compressional wave were reported for air-saturated materials (Van der Grinten et al., 1985;Nagy et al., 1990) and water-saturated materials (Van der Grinten et al., 1987;Rasolofosaon, 1988;Kelder and Smeulders, 1995). ...
Theoretical studies predict that a slow compressional wave propagating in a fluid-saturated porous medium can produce a coseismic electric field due to the electrokinetic effect, but the experimental proof is still lacking. Laboratory experiments are conducted to measure such a seismoelectric conversion inside a synthetic rock. Fluid pressure signals are recorded by using mini hydrophones 1.6 mm in diameter, and then electric field signals generated at the liquid-solid interface and inside the rock sample based on the seismoelectric effect are recorded by electrode arrays, respectively. The seismoelectric waves induced by fast and slow compressional waves can be clearly identified in the recorded electric signals and their attenuation property are analyzed at an ultrasonic frequency, which confirms that the seismoelectric signals induced by fast/slow compressional waves are measurable in the experiments. To support our explanation of the experimental observation, theoretical simulations are conducted according to the experimental model, and then compared with the recorded experimental data. The results find that the simulated wavefields are in excellent agreement with those signals measured in the measurements, which proves the theoretical prediction of the seismoelectric signal accompanying the slow compressional wave and suggests a feasible way for detecting the slow compressional wave property with seismoelectric conversions in field measurements.
... Material properties inferred from general observations with sandstones are labeled "6" in Table 1. Among these parameters, we assume that the physical properties of the rock frame are constant within the temperature range of 23°C-37.5°C and that the tortuosity a can be approximated from porosity as a ¼ ð1 þ ϕ −1 Þ=2 (Berryman, 1980). The properties of glycerol require the most attention because they contain the strongest contrasts in physical properties and the greatest sensitivity to temperature variations. ...
Rocks can be viewed as composites of solid minerals and pores or cracks filled with softer material such as pore fluids, kerogen, bitumen, and other organic matters. Rigidities and particularly viscosities of these soft phases are highly sensitive to the ambient temperature, which can significantly influence both static and dynamic properties of the composite rock. Temperature-dependent viscosity causes scaling of the characteristic relaxation frequencies, affects coupling between the solid rock frame and soft components, and may lead to nonlinear wave attenuation and dispersion. Based on a continuum-mechanics formulation with temperature-dependent properties of soft material, we propose a temperature-dependent visco-poroelastic model of porous rock. In this model, the classical frequency-dependent microscopic squirt flow and mesoscopic WIFF models are elegantly unified by using physically meaningful, real-valued, and time- and frequency-independent material properties. The model is illustrated by explaining the broad attenuation peaks and Young’s modulus dispersion and attenuation observed in previously published laboratory experiments with glycerol-saturated Berea sandstone. Several additional macroscopic mechanical properties of the rock are obtained: average porosity of the microcracks, effective high-pressure bulk modulus of the drained frame, internal stiffness defect within the rock frame, solid viscosities associated with bulk and shear deformations, and an exponent of nonlinearity for viscosity. These parameters constitute a rigorous, Biot-consistent mechanical model of the rock which can be used to explain its behavior in arbitrary experimental environments. Consequently, this model should be useful for developing detailed and accurate models for various types of laboratory experiments, reservoir characterization, geothermal exploration, thermal-enhanced oil recovery, and exploration for deep oil and gas resources in high-temperature environments.
... Other models exist that also acknowledge that acoustic velocity varies despite constant porosity, mineralogy and fluid properties (Berge et al., 1992;Berryman, 1980a;Dvorkin and Nur, 1993;Korringa et al., 1979;Kuster and Toksoz, 1974b;Mavko and Mukerji, 1995). One such model presented by Mavko and Mukerji (1995) incorporates the concept of "pore space stiffness." ...
... The modulus of the matrix is calculated by the Voigt-Reuss-Hill average equation (Hill, 1952). We figure out the moduli of the dry skeleton and the saturated background medium without the aligned fractures using the self-consistent approximation (SCA) (Berryman, 1980) and the Gassmann equation (Mavko et al., 2020), respectively. The isometric pores and aligned fractures are filled with gas-water and oil-water mixtures, respectively. ...
... 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%.
... where F s is the structure factor, which can be expressed as [36] F s = 0. 5 ...
Carbon dioxide geological utilization and storage (CGUS) is an effective way to mitigate climate warming. In this paper, we resorted to Lo’s model to analyze the dispersion and attenuation characteristics of unsaturated porous media. Based on this, we analyzed the sensitivity of the first compressional wave (P1) and the shear wave (S) to various physical parameters. In addition, the modified models of live oil’s velocity and density were proposed, which were verified by experimental data under the consideration of CO2 dissolution. It is shown that the velocities and attenuations of P1 and S waves are influenced by various parameters, especially CO2 saturation and pore fluid parameters, such as density and velocity. In particular, with increasing CO2 saturation, the sensitivity of P1 velocity decreases, while that of the S velocity increases. Better monitoring results can be achieved by combining P1 and S waves. Finally, the acoustic response was analyzed under the modified model. With the increase in CO2 saturation, the P1 velocity decreases, while the S velocity becomes almost constant and then linearly increases, with the trend changing at the critical saturation. The study provides a more precise basis for monitoring the security of CO2 injection in CGUS.
... where d f is the average diameter of the PPFF. For the tortuosity (a 1 ), it is conditioned using the Berryman's formula [39] as follows: ...
Composite materials made from palmyra palm fruit fiber (PPFF) were formed using urea–formaldehyde (UF) and polyvinyl acetate (PVAc). We prepared two sets of five different cylindrical samples with varying PPFF contents. The PPFF composites’ normal-incident sound absorption coefficient (SAC) and transmission loss (TL) were measured by using the impedance tube method. The sample with higher PPFF content shows a lower SAC spectrum. It is the opposite for the TL, where a sample with high PPFF content demonstrates a higher TL spectrum. We conducted the least-square fitting method on the experimental SAC and TL spectra utilizing the Johnson-Champoux-Allard (JCA) equivalent fluid model. Non-acoustic parameters were acquired from the fitting. The optimized porosity (ϕ), viscous characteristic length (Λ), and thermal characteristic length (Λ′) are inversely proportional to the PPFF content. The airflow resistivity (σ) and tortuosity (α∞), on the other hand, demonstrate a direct correlation with the PPFF content. Even though the UF samples have an average density of 14.7 % higher than the PVAc samples, their σ,Λ′, and α∞ are just 7.7 %, 4.5 %, and 0.39 % higher than PVAc samples. On the other hand, PVAc samples show higher average Λ and ϕ of 1.4 % and 0.73 %, respectively. The optimized porosity values obtained from the JCA model (ϕJCA) are coherent with ones from the direct estimation method assuming adhesive-coated fiber (ϕfa). It can be concluded that the adhesive’s quantity and density contribute to the composites’ porosity value, ultimately affecting material acoustic properties. Researchers can control and predict how the SAC and TL of fibrous sound absorbers would behave by varying the quantity and density of an adhesive.
... The approach was extended to the case of N different type of inclusions by Berryman (1980Berryman ( , 2013. However, the prediction of this model for crack density >0.1 is non-physical: the predicted moduli decreases rapidly with the increasing the crack density (Figure 1.4) and reaches a value of 0 for a crack density ~9/16. ...
The dispersion and attenuation of the elastic velocities related to fluid-flow in porous rock are of interest in the context of oil-gas exploration, production or CO2 storage. Two stress-strain apparatus with the ability of capturing the seismic dispersion and attenuation were firstly introduced. Secondly, a three-dimensional model was proposed to predict the influence of the global-drainage flow. Thirdly, we investigate experimentally the influence of micro-heterogeneity on the global-drainage flow and squirt flow. Fourthly, the effect of mechanical compaction on the seismic dispersion and attenuation was investigated using a porous sandstone. Finally, series of experiments were conducted to investigate the influence of a biphasic saturation on the dispersion and attenuation.
... partially saturated medium). For the propagation of elastic waves in saturated medium, many researchers including Biot (1956a;1956b), Jones (1961), Plona (1980), Berryman (1980), Hajra and Mukhopadhyay (1982), Sharma and Gogna (1991), Carcione (1996), Wang et al. (2016), Zhou and Ma (2016), Ciarletta et al. (2018), Xiong et al. (2021), Tung (2021), Zhang et al. (2021), Rohan et al. (2021), and Ding et al. (2022) have investigated the propagation behaviors of compression, shear and Rayleigh waves through theoretical and experimental means. These works drew and verified some common conclusions that there are two compression waves (fast compression wave typically denoted as P 1 wave and slow compression wave typically denoted as P 2 wave) and one shear wave (typically denoted as S wave) in saturated medium; Rayleigh wave in saturated medium is the superposition of P 1 , P 2 and S waves at the stress-free surface; the propagation behaviors of the compression, shear and Rayleigh waves not only depend on the frequency but also the medium parameters. ...
Based on the mixture theory, considering the flow-independent viscosity related to solid skeleton, the present work investigates the propagation characteristics of Rayleigh wave in partially saturated viscoelastic soil. Firstly, the complex history-dependent viscoelastic behavior of solid skeleton is characterized by the fractional standard linear solid constitutive model. The generalized governing equations of motion for the partially saturated viscoelastic soil are established theoretically. Secondly, the analytical solution of Rayleigh wave fields in the partially saturated viscoelastic soil is obtained using the wave function expansion method. Finally, the influences of flow-independent viscosity characterized by fractional viscoelastic parameter on the propagation behaviors of Rayleigh wave are implemented analytically and then discussed in detail. The results show that the fractional index, stress relaxation time and strain relaxation time in the fractional standard linear solid constitutive model have significant influence on the phase velocity and attenuation coefficient of Rayleigh wave. The fractional index, stress relaxation time and strain relaxation time have similar effects on the P1 and S waves included in body waves. Additionally, the influences of the fluid hydraulic conductivity and liquid saturation on the wave propagation cannot be ignored.
... Biot's poroelasticity theory (BT) only predicts two P-waves (fast and slow P-wave) and one S-wave (fast S-wave) [12,13]. Although the existence of these three waves has been confirmed by various experiments [9,15,34,44,55], in 2008 Sahay [49] showed that the constitutive equations of Biot's theory are incomplete. To eliminate the inconsistency, it is necessary to add a fluid strain rate term to the constitutive relations. ...
Using Biot viscosity-extended theory, this work gives a methodology to obtain analytical formulas for the torsional phase velocities and torsional attenuations in fully saturated, homogeneous, isotropic, axially symmetric hollow poroelastic cylinders with stress-free boundary conditions on both surfaces. These phase velocities and attenuations formulas are given in terms of the poroelastic medium, such as porosity, permeability, solid density, fluid shear viscosity, and solid-frame shear modulus, among others. We use computational simulations to compare how torsional waves propagate in axial symmetric hollow cylinders in Biot viscosity-extended theory and Biot’s theory. A comparison of torsional wave propagation in hollow and solid cylinders is also provided.
... Following Williams [39] and Hosokawa and Otani [2], tortuosity is related to porosity and can be defined as = 1 − (1 − 1∕ ), where = 0.25 is a variable that can be calculated from a microscopic model of a porous frame moving in the fluid, [41]. ...
The paper discusses the fundamental mechanisms underlying the interaction between ultrasound and trabecular bone, which is considered a two-phase material. When fluid-saturated cancellous bone is interrogated by ultrasound, in some cases, one or two wave modes are observed. Many authors claim that these waves correspond to the fast and slow waves predicted by Biot’s theory of elastic wave propagation in fluid-saturated porous media. Within our analysis of the physical conditions, predictions of the existing two-phase models of the propagation of ultrasonic waves in the material as well as numerical simulations for fluid-saturated trabecular bone were performed. On the basis of the theoretical results (from numerical studies) and arguments presented in this paper, we aimed to answer the question of whether two waves observed in ultrasonic wave transmission studies can be interpreted as the fast and slow waves predicted by Biot’s theory.
... Poroelastic materials consist of a solid matrix with pores that are completely fluid-filled. Biot's theory of poroelasticity [2,3,4,5] describes the interaction between the fluid and the solid phase and is widely accepted and validated [6,7,8]. The resulting system of partial differential equations (PDEs) describes seismic wave propagation in porous media, extending the elastic model often used in computational seismology by additional quantities (e.g., fluid velocities) and, in particular, by a stiff reactive source term that is required to model viscosity effects of the fluid-solid interaction. ...
Many applications from the fields of seismology and geoengineering require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff reactive source term (Darcy's Law) into the elastodynamic wave equations, thereby increasing computational cost of respective numerical solvers and motivating efficient methods utilising High-Performance Computing.
We present a novel realisation of the discontinuous Galerkin scheme with Arbitrary High-Order DERivative time stepping (ADER-DG) that copes with stiff source terms. To integrate this source term with a reasonable time step size, we utilise an element-local space-time predictor, which needs to solve medium-sized linear systems – each with 1,000 to 10,000 unknowns – in each element update (i.e., billions of times). We present a novel block-wise back-substitution algorithm for solving these systems efficiently, thus enabling large-scale 3D simulations. In comparison to LU decomposition, we reduce the number of floating-point operations by a factor of up to 25, when using polynomials of degree 6. The block-wise back-substitution is mapped to a sequence of small matrix-matrix multiplications, for which code generators are available to generate highly optimised code.
We verify the new solver thoroughly against analytical and semi-analytical reference solutions in problems of increasing complexity. We demonstrate high-order convergence of the scheme for 3D problems. We verify the correct treatment of point sources and boundary conditions, including homogeneous and heterogeneous full space problems as well as problems with traction-free boundary conditions. In addition, we compare against a finite difference solution for a newly defined 3D layer over half-space problem containing an internal material interface and free surface. We find that extremely high accuracy is required to accurately resolve the slow, diffusive P-wave at a or near a free surface, while we also demonstrate that solid particle velocities are not affected by coarser resolutions. We demonstrate that by using a clustered local time stepping scheme, time to solution is reduced by a factor of 6 to 10 compared to global time stepping. We conclude our study with a scaling and performance analysis on the SuperMUC-NG supercomputer, demonstrating our implementation's high computational efficiency and its potential for extreme-scale simulations.
... According to Biot, the speed and modes of the propagating elastoacoustic waves, which are the products of the coupling between elastic waves in solid and pressure waves in fluid, can be accurately identified. It was analytically predicted [12] and experimentally confirmed [24,25] that the coupled compressional waves split into fast and slow pressure modes in the long wavelength regime. The balance between the boundary layer of Poiseuille flow and the characteristic size of the porous media plays a key role in determining the mode of wave propagation. ...
Metamaterials with microscale architectures, e.g., microlattices, can exhibit extreme quasi-static mechanical response and tailorable acoustic properties. When coupled with pressure waves in surrounding fluid, the dynamic behavior of microlattices in the long wavelength limit can be explained in the context of Biot’s theory of poroelasticity. In this work, we exploit the elastoacoustic wave propagation within 3D-printed polymeric microlattices to incorporate a gradient of refractive index for underwater ultrasonic lensing. Experimentally and numerically derived dispersion curves allow the characterization of acoustic properties of a fluid-saturated elastic lattice. A modified Luneburg lens index profile adapted for underwater wave focusing is demonstrated via the finite element method and immersion testing, showcasing a computationally efficient poroelasticity-based design approach that enables accelerated design of acoustic wave manipulation devices. Our approach can be applied to the design of acoustic metamaterials for biomedical applications featuring focused ultrasound.
... (We may note the relation between and T found by Berryman (1980) for the solid matrix: ...
We present a new methodology of the finite-difference modelling of seismic wave propagation in a strongly heterogeneous medium composed of poroelastic (P) and (strictly) elastic (E) parts. The medium can include P/P, P/E and E/E material interfaces of arbitrary shapes. The poroelastic part can be with a) zero resistive friction, b) non-zero constant resistive friction or c) JKD model of the frequency-dependent permeability and resistive friction. Our finite-difference scheme is capable of sub-cell resolution: a material interface can have an arbitrary position in the spatial grid. The scheme keeps computational efficiency of the scheme for a smoothly and weakly heterogeneous medium (medium without material interfaces). Numerical tests against independent analytical, semi-analytical and spectral-element methods prove the efficiency and accuracy of our finite-difference modelling. In numerical examples we indicate effect of the P/E interfaces for the poroelastic medium with a constant resistive friction and medium with the JKD model of the frequency-dependent permeability and resistive friction. We address the 2D P-SV problem. The approach can be readily extended to the 3D problem.
... They observed the slow compressional wave, P2 wave, in saturated soils as well. Berryman [16] tested the velocities of the three waves in saturated porous media through experiments. The velocities were in a good agreement with the theory velocities of the Biot theory. ...
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.
... Poroelastic materials consist of a solid matrix with pores that are completely fluid-filled. Biot's theory of poroelasticity [2,3,4,5] describes the interaction between the fluid and the solid phase and is widely accepted and validated [6,7,8]. The resulting system of partial differential equations (PDEs) describes seismic wave propagation in porous media, extending the elastic model often used in computational seismology, by additional quantities (e.g., fluid velocities) and, in particular, by a stiff reactive source term that is required to model viscosity effects of the fluid-solid interaction. ...
Many applications from geosciences require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff source term, thereby increasing computational cost and motivating efficient methods utilising High-Performance Computing. We present a novel realisation of the discontinuous Galerkin scheme with Arbitrary DERivative time stepping (ADER-DG) that copes with stiff source terms. To integrate this source term with a reasonable time step size, we use an element-local space-time predictor, which needs to solve medium-sized linear systems - with 1000 to 10000 unknowns - in each element update (i.e., billions of times). We present a novel block-wise back-substitution algorithm for solving these systems efficiently. In comparison to LU decomposition, we reduce the number of floating-point operations by a factor of up to 25. The block-wise back-substitution is mapped to a sequence of small matrix-matrix multiplications, for which code generators are available to generate highly optimised code. We verify the new solver thoroughly in problems of increasing complexity. We demonstrate high-order convergence for 3D problems. We verify the correct treatment of point sources, material interfaces and traction-free boundary conditions. In addition, we compare against a finite difference code for a newly defined layer over half-space problem. We find that extremely high accuracy is required to resolve the slow P-wave at a free surface, while solid particle velocities are not affected by coarser resolutions. By using a clustered local time stepping scheme, we reduce time to solution by a factor of 6 to 10 compared to global time stepping. We conclude our study with a scaling and performance analysis, demonstrating our implementation's efficiency and its potential for extreme-scale simulations.
... Biot's theories (Biot 1956(Biot , 1962) provide the mathematical model for wave dynamics in porous solids, which accommodates the anisotropy and anelasticity of solid matrix as well as the viscosity of pore-fluid. For waves in real materials, Biot's theories are found to be in agreement with experimental observations (Berryman 1980;Lakes et al. 1983;Plona and Johnson 1984). ...
An attenuating wave-field in an anisotropic poroviscoelastic medium is represented through the three-dimensional inhomogeneous propagation of four bulk waves. Propagation of each wave is governed by a complex slowness vector, which specifies its phase direction, phase velocity and coefficients for homogeneous/inhomogeneous attenuation. Partially-opened surface pores restrict the seepage of pore-fluid at the boundary. A generalised reflection phenomenon is illustrated for incidence of inhomogeneous waves at this boundary. Horizontal slowness of this incidence derives the slowness vectors of four inhomogeneous waves reflected into the medium. Slowness vectors and polarisations of incident and reflected waves are used to calculate the amplitudes, phase shifts and energy fluxes of reflected waves in comparison to the incident wave. A particular example illustrates the numerical implementation of the derived mathematical model for anisotropic poroviscoelastic reflection.
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... An additional parameter required is a geometrical quantity τ > 1 that increases as the tortuosity of the pore space increases. Sidler (2015) used the expression derived by Berryman (1980a) for the case of isolated spherical solid particles in the fluid: ...
Measurements of elastic wave velocities enable non-destructive estimation of the mechanical properties, elastic moduli and density of snow and firn. The variation of elastic moduli with porosity in dry snow and firn is modeled using a differential effective medium scheme modified to account for the critical porosity above which the bulk and shear moduli of the ice frame vanish. A comparison of predicted and measured elastic moduli indicates that the shear modulus of ice in snow is lower than that computed from single crystal elastic stiffnesses of ice. This may indicate that the bonds between snow particles are more deformable under shear than under compression. A partial alignment of ice crystals also may contribute. Good agreement between elastic stiffnesses of the ice frame obtained from elastic wave velocity measurements and the predictions of the theory is observed. The approach is simple and compact, and does not require the use of empirical fits to the data. Owing to its simplicity, this model may prove useful in a variety of potential applications such as construction on snow, interpretation of seismic measurements to monitor and locate avalanches and estimation of density within compacting snow deposited on glaciers and ice sheets.
... where r 12 defines the geometrical parameters of the pores (r 12 = 1/2 for spheres), a 12 represents the tortuosity parameter, which was given by Berryman 27 : ...
A complete and formal theoretical derivation of the equations of motion and the stress–strain relations in Carcione–Leclaire three-phase theory is presented. The Lagrangian formulation is obtained on the basis of the potential and kinetic energies. We provided a new and more clear method to express the kinetic and potential energy density in an elastic solid. In particular, the deduction of the kinetic energy density in a fluid-saturated porous medium is established from a physical point of view, for which Biot’s theory did not give a detailed description. Moreover, to make the establishment of the equations more clearly expressed, the potential, kinetic and dissipation energy densities are described in great detail to obtain the equations of motion by using the Lagrangian formulation. In order to show the self-consistency of the three-phase model, the deduction of the degradation of a three-phase medium into a simplified form of a two-phase medium is proved in two physically distinct situations. One case is that the medium degenerates into a saturated porous medium. The second case is that the medium degenerates into a porous medium composed of two solids. The results show that, if the pore solid is replaced by the pore fluid, Carcione–Leclaire three-phase theory is completely consistent with Biot’s theory. Besides, if the pore fluid is substituted by the pore solid, two compressional waves and two shear waves can be generated when the reference value of the friction coefficient between two solids [Formula: see text] is zero. When [Formula: see text] is not neglected, there are only one compressional wave and one shear wave, which is equivalent to the single-phase elastic medium. Finally, we compute the phase velocities and attenuations versus frequency with three pore solid saturations to analyze the characteristics of the five modes.
A better understanding of the temperature effects on the propagation characteristics of elastic waves in frozen soils and rocks is imperative for accurately quantifying their freezing degrees. While existing rock-physics models based on the three-phase Biot (TPB) theory adeptly interpret observed velocity versus temperature (VVT) curves, they often lack a comprehensive understanding of the mechanisms underlying attenuation versus temperature (AVT) curves. In this study, we first extend the TPB theory to incorporate the temperature-dependent properties of ice, including changes in volumetric fraction, morphology, and viscoelasticity, by integrating relevant thermodynamic laws. Model parameters related to ice properties and interactions, such as rigidity, shear moduli, density, and friction, are redefined. Then, using a numerical rock-physics modeling approach, we examine influential factors and modes of wave VVT and AVT responses. Our results show that both P- and S-wave velocities increase with source frequency, consolidation degree, and frame-supporting ice content, while decreasing with temperature and pore-floating ice content. Both P- and S-wave attenuation factors increase with frame-supporting ice content and decrease with consolidation degree. Rising temperatures tend to amplify the peak magnitude of P-wave attenuation factors and shift the central frequency of S-wave attenuation factors. Finally, within a temperature-controlled laboratory environment, we conduct ultrasonic wave transmission testing on brine-saturated sediment and rock specimens. Results demonstrate that as the temperature increases from 15 to 3 °C, both the P- and S-wave velocities decrease, while the P-wave attenuation factors decrease and the S-wave attenuation factors initially rise before declining. Our viscoelastic TPB theory outperforms existing ones in interpreting S-wave AVT observations. This temperature-dependent rock-physics model holds promise for interpreting sonic logging data in time-lapse monitoring of permafrost, glaciers, and Antarctica.
هدف از اين تحقيق، تحليل خاك بر اساس فرمولبندي تنش مؤثر با بكارگيري مكانيك محيط متخلخل اشباع براي بررسي اثر جريان غير دارسي و توپوگرافي نامنظم لايه بندي بر روي روانگرائي مي باشد. اين که قانون دارسي براي عدد رينولدز بسيار پايين صادق است، موضوعي کاملاً شناخته شده است. قانون جريان در محيط متخلخل براي سرعت تراوش بالا غيرخطي است. با اين وجود، جريان غيرخطي در معادلات اندرکنش خاک- سيال حفرهاي و مکانيک خاک مرسوم استفاده نشده است. در اين تحقيق روش اجزاء محدود ديناميکي جهت تحليل محيط متخلخل اشباع بر پايه فرمولاسيون اوليه ارائه شده توسط بيوت (1962-1941) براي جريان غيردارسي توسعه داده شده است. رفتار ارتجاعي- خميري سيکلي خاک تحت بارگذاري لرزهاي با استفاده از نظريه پلاستيسيته عمومي منتج شده از مفهوم سطح تسليم همراه با قانون جريان غير همراه شبيه سازي شده است. براي بررسي اثر حرکت غيردارسي سيال، نرم افزاري به زبان برنامه نويسي فورترن نوشته شده است. شبيه سازي هاي عددي محيط متخلخل با نفوذپديري بالا و زلزلههاي ورودي فرکانس بالا نشان ميدهد که اختلاف قابل ملاحظه اي در نتايج تحليل روانگرائي بين محيط دارسي و غير دارسي وجود دارد. در ادامه تحقيق اثر توپوگرافي نامنظم لايه بندي بر روانگرائي براي دو لايه از خاک نيز بررسي شده است. در کليهي روشهاي ساده شده براي شناسايي پتانسيل روانگرائي، خاک به صورت لايه هاي همگن مسطح فرض شده است، اما در واقع لايه هاي خاک داراي نامنظمي در توپوگرافي است. نتايج بدست آمده حاکي از آن است که لايه بنديهاي نامنظم در بسياری از موارد بر پاسخ ديناميکي خاک اثر گذار بوده که اين امر مي تواند در ارزيابي روانگرائي تاثير داشته باشد.
Fractures, often existing across various scales, control the mechanical and fluid flow properties of upper crustal rocks. Inferring fracture properties at different scales from multiband geophysical measurements is essential for many fields of Earth and energy sciences Under the framework of multiscale homogenization, we develop a rock physics model to characterize the elastic and anisotropic properties of multiscale fractured rocks following a certain statistical law. The isotropic differential effective medium theory is employed to model the elasticity of randomly oriented microcracks, and linear-slip theory is used to model the elasticity of oriented or randomly oriented macroscale fractures. For a multiscale fractured rock covered by 6 orders of fracture length (10 ⁻⁴ to 10 ² m), it is found that velocity exhibits a decreasing trend with the increment of scale (wavelength). Nevertheless, the different statistical distribution of multiscale fractures significantly affects the velocity and anisotropy variation pattern with scale. The velocity for the fractal distribution decreases significantly at the seismic scale; while for the log-Gaussian distribution the dramatic change in velocity occurs more at the ultrasonic and logging scales, depending on the length of the dominant fractures. We apply the proposed methodology to interpret a set of multiscale geophysical data of P-wave velocity in a fractured carbonate formation, and estimate that the multiscale fracture is possibly distributed in a log-Gaussian manner. The proposed elastic and anisotropic modeling strategy has the potential to predict the distribution pattern of fractures, especially by reconciling the multi-frequency geophysical measurement (e.g., ultrasonic, logging, and seismic).
To study the effects of underwater terrain on plane seismic waves (P-SV wave) propagation, the theoretical solution of ground displacement, strain, rocking, and energy for the underwater terrain is derived based on Biot’s theory. Numerical results in terms of incident angle and Poisson’s ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the response of a saturated-fluid porous soil under overlying water is controlled by the solid frame stiffness and Poisson’s ratio. Considering the effect of complex site amplification, the scattering theoretical solution of plane SV wave by a circular-arc canyon with overlying water is first derived based on the free-wave field above obtained. The effects of the incident frequencies, the incident angles, and the circular arc canyon on the ground motion of the soil-water interface are investigated parametrically. In particular, the effect of overlying water on ground motion is studied, and the results indicate that the existence of an overlying water layer absorbs the energy of a seismic wave to a certain extent. Meanwhile, the accuracy of the results is also verified by reducing the complex site to that without an overlying water layer and no canyon. Finally, based on the theoretical solution derived above, the coherence functions and the underground variable seismic motion of the site also are simulated, respectively. Compared with ground motions, the numerical results show that underground motions are smaller, which indicates that the canyon has a certain amplification effect on the seismic motions.
The rock permeated by tilted aligned fractures, which is common in the earth, can be considered as the tilted transversely isotropic (TTI) medium under the assumption of the long wavelength. We developed a feasible method to predict fracture parameters (namely weakness parameters and dip angle) and fluid type in the TTI rock using azimuthal seismic data. Based on an approximate stiffness matrix, we first deduced a linear reflection coefficient of the gas-bearing TTI medium in terms of the new fluid indicator and fracture parameters. The reflection coefficient was then rewritten in the form of the Fourier series to decouple the fracture and skeleton-fluid information. Whereupon the sequential Bayesian inversion method was proposed, which consists of three steps. The first two steps conquer inversion instability owing to coupling of the fracture parameters by constructing the linear relationship between the second-order and fourth-order Fourier coefficients. The last step aims at estimating the skeleton-fluid parameters. The sequential Bayesian inversion method alleviates the ill posedness of the multi-parameter simultaneous inversion caused by large differences between contributions of the fracture and skeleton-fluid parameters to the reflectivity. Synthetic and field cases proved the proposed method stable and rational in fracture and fluid detections. Finally, we draw some conclusions from numerical experiments that the approximate stiffness coefficients and derived reflection coefficient are of satisfactory accuracy for the gas-bearing reservoir with low fracture density; The new fluid indicator is sensitive to the fluid type but very weakly dependent on the mineral composition and porosity; Deconvolution processing can improve accuracies of different seismic components calculated using the discrete Fourier transformation.
In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biots theory and Lagrangian formulas, we obtain 3D Biots equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single-porosity or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs (LLF) flux and an implicit weighted RK time discretization scheme are used for this computation. We use two types of mesh elements quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P-waves, P1- and P2-waves, in double-porosity media, whereas three kinds of slow P-waves, P1-, P2-, and P3-waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation utilizing the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multi-porosity media computed by DGM. The slow P-wave in multi-porosity media with dissipation decay rapidly, which will also lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and show that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.
Near‐field wave propagation of the fluid‐saturated porous media (FSPM) is an important issue in soil dynamics and geotechnical earthquake engineering. In the current study, a fully explicit staggered algorithm for the near‐field wave propagation analysis of the FSPM in the time domain is developed based on the u‐p dynamic formulation. The decoupling of dynamic formulation is implemented by the diagonalization of the mass matrix and pore fluid compressibility matrix. The central difference method and Newmark constant average acceleration method are used for the solution of the displacement and velocity of solid skeleton, and the backward difference method is used for the solution of pore fluid pressure in the time domain. The effectiveness and feasibility of the matrix diagonalization are verified by the comparison with the classical numerical algorithm. The proposed algorithm is also validated by the comparison of the numerical results with the corresponding analytical solution results. Combing with the transmitting artificial boundary condition, the proposed algorithm is then applied to investigate the seismic response of a FSPM free field, and its applicability to the typical near‐field wave propagation problems of the FSPM is indicated. The stability analysis of the proposed algorithm is finally conducted, and the corresponding stability criterion is presented. For the proposed algorithm, all the dynamic variables of the FSPM are solved in an iterative way and the coupling dynamic equations need not to be solved at each time step. The computational effort can be reduced considerably, and the computational efficiency can be improved remarkably.
Biot theory is the basis to describe wave propagation in porous media, starting with Terzaghi law, Gassmann equation, and the static approach leading to the concept of effective stress, much used in soil mechanics. The coefficients of the strain energy are obtained by the so-called jacketed and unjacketed experiments. The theory includes anisotropy and dissipation due to viscodynamic and viscoelastic effects. The interface boundary conditions and the Green functions are treated in detail since they provide the basis to obtain the reflection coefficients and the solution of wave propagation in inhomogeneous media. Biot theory is extended from first principles to composite media. The mesoscopic-loss mechanism is described by means of the White theory for plane-layered and fractured media. Moreover, the Biot-Gardner effect, bounds, and averages for composite media and partially molten media are considered. Finally, the theories of anisotropic poro-viscoelasticity and thermo-poroelasticity are developed in detail.
p>The theoretical modelling of ultrasonic propagation in cancellous bone is pertinent to the improvement of ultrasonic techniques for diagnosing the bone disease osteoporosis. For such techniques to be confidently used in the clinical management of osteoporosis, fundamental research is required to establish an understanding of how ultrasonic waves travel in porous, or cancellous bone. This thesis concerns investigation into various theoretical models of propagation in porous media. These studies are supported by in vitro experiments on bovine cancellous bone around 1 MHz.
Previous applications to bone of established theories, such as Biot's theory for fluid-saturated porous media, have enjoyed limited success. This thesis begins by considering Biot's theory in more detail than previously reported in the literature. Biot's theory predicts that two longitudinal waves travel in cancellous bone in response to insonation with a single wave. The existence of two waves, known as fast and slow waves, is confirmed, which had not been reported in the literature prior to the start of this work. The importance of the presence of marrow in the pores on these waves is investigated.
The phase velocities of fast and slow waves are observed to be strongly dependent on direction, relative to the internal cancellous structure. However, the isotropic form of Biot is not appropriate for modelling this response. Therefore, a second approach is proposed, which uses Schoenberg's theory to model propagation in a parallel-plate model of cancellous bone. Direction dependent measured velocities are observed to give qualitative agreement with the predictions of the Schoenberg model. The two theoretical approaches are compared when anisotropic mechanical and fluid motion effects are introduced into Biot's theory.</p
A systematic study of wave theory in thermoviscoelastic soil is essential for engineering applications such as geophysical exploration. In the present work, the influences of flow-independent viscosity of the soil skeleton and the thermal effect on elastic waves are considered, and the propagation behaviors of body waves in thermoviscoelastic saturated soil are investigated. Firstly, the thermoviscoelastic dynamic coupling model of saturated soil were established by employing the Biot model, the generalized thermoelastic theory, and the Kelvin–Voigt linear viscoelastic model. Secondly, the dispersion equations of body waves in thermoviscoelastic saturated soil were theoretically derived with structural symmetry considered. Finally, the variations of wave velocity and the attenuation coefficient of the body waves with the thermophysical parameters are discussed. The results revealed that the enhancement of the relaxation time of soil caused an increase of wave velocity and the attenuation coefficient of P1, P2, and S waves, and a decrease of the wave velocity and attenuation coefficient of the thermal wave. Different ranges of the permeability coefficient and frequency have different effects on the P1, P2, and S waves. The variation of thermal conductivity and the phase-lags of heat flux and temperature gradient only affect the thermal wave.
Seismic exploration of complex geological structure model has been paid much attention as the increasing complexity of geological structure in seismic exploration, and especially, the research on saturated porous medium saturated with fluid has become a hot topic. In our study, we propose a new Padé approximation (PAM) method of solving the elastic wave equation in the porous medium of low-frequency case. This method uses an implicit scheme derived from the rational function of time difference operator for the time discretization, which has the characters of low-dispersion and high-efficiency for time advancing. An explicit iteration for this implicit algorithm is obtained for avoiding solving a large linear system with a block tridiagonal coefficient matrix at each time step. Then, we employ the stereo-modelling method with the eighth-order accuracy for space discretization, which uses linear combination of wave field displacements and their gradients to discretize spatial derivatives and obtains a high-order approximation. Compared with the traditional method, this discretization operator has shorter operator radius and better compactness, which is beneficial for increasing the precision and imaging quality of seismic inversion and seismic migration. Theoretical analysis and numerical experiments verify that the PAM method is an accurate forwarding modelling tool. Waveforms obtained by the PAM method can well match the analytical solutions. Moreover, the seismic wave fields including fast P wave, S wave, and slow P wave for the saturated fluid porous medium can be observed clearly on coarser grids. This is in contrast with the FD method, which suffers from serious numerical dispersion.
The complex heterogeneities of underground rocks will cause wave-induced fluid flows at different scales, which consequently leads to the wave velocity dispersion and energy loss. Mastering and modeling the frequency-dependent elastic and attenuation behaviors are of great significance to characterize underground rocks using multi-scale geophysical data. Following Biot's approach, the constitutive relationship, kinetic energy and dissipation functions in regard to wave induced global fluid flow, interlayer local fluid flow and squirt flow in the annular and penny-shaped cracks are established. From the Lagrange equations, the wave equations considering multi-scale wave induced fluid flow are further derived, which yields three P waves and one S wave. The frequency-dependent velocity and attenuation of fast P wave calculated by the multi-scale wave equations present nice match with that of the single-scale or dual-scale theories in the corresponding frequency bands. Besides, the multi-scale wave theory, under certain circumstance, can be degenerated to the widely known theories including Tang's pore-crack theory, the layered double-porosity theory and Biot's theory, which theoretically illustrates the rationality of the novel wave equation. In order to adjust the low-frequency velocity of the multi-scale wave equations to Gassmann velocity, the dynamic fluid modulus (DFM) is introduced into the multi-scale wave theory. However, the original multi-scale wave theory behaves better fit with the experimental data in comparison with the DFM multi-scale wave theory. The effect of micro-parameters on the dispersion and attenuation calculated by the multi-scale wave theory indicates that the annular crack deforms more with weaker stiffness than the penny-shaped cracks under the same aspect ratio.
Please find the published article (open access) at the publisher's site:
https://www.frontiersin.org/articles/10.3389/fphy.2021.697990/full
In shear wave elastography, rotational wave speeds are converted to elasticity measures using elastodynamic theory. The method has a wide range of applications and is the gold standard for non-invasive liver fibrosis detection. However, the observed shear wave dispersion of in vivo human liver shows a mismatch with purely elastic and visco-elastic wave propagation theory. In a laboratory phantom experiment we demonstrate that porosity and fluid viscosity need to be considered to properly convert shear wave speeds to elasticity in soft porous materials. We extend this conclusion to the clinical application of liver stiffness characterization by revisiting in vivo studies of liver elastography. To that end we compare Biot’s theory of poro-visco-elastic wave propagation to Voigt’s visco-elastic model. Our results suggest that accounting for dispersion due to fluid viscosity could improve shear wave imaging in the liver and other highly vascularized organs.
Acoustic waves in a poroelastic medium with periodic structure are studied with respect to permanent seepage flow which modifies the wave propagation. The effective medium model is obtained using the homogenization of the linearized fluid–structure interaction problem while respecting the advection phenomenon in the Navier–Stokes equations. For linearization of the micromodel, an acoustic approximation is introduced which yields a problem for the acoustic fluctuations of the solid displacements, the fluid velocity and pressure. An extended Darcy law of the macromodel involves the permeability and advection tensors which both depend on an assumed stationary perfusion of the porous structure. The monochromatic plane wave propagation is described in terms of two quasi-compressional and two quasi-shear modes. Two alternative problem formulations in the frequency domain are discussed. The one defined in terms of displacement and velocity fields leads to generalized eigenvalue problems involving non-Hermitean matrices whose entries are constituted by the homogenized coefficients depending on the incident wave frequencies, whereby degenerate permeabilities can be accounted for. The homogenization procedure and the wave dispersion analysis have been implemented to explore the influence of the advection flow and the microstructure geometry on the wave propagation properties, namely the phase velocity and attenuation. Numerical examples are reported.
Porous materials contain a network structure composed of distributed pores. Due to the distribution of pores, the volume of the material will be less than the actual dense volume. The material coefficients such as density and elastic modulus can change due to the influence of porosity. Therefore, the characteristics of porosity can be used as a measurement standard in the properties detection. This study explores the dispersion relationship of the wave propagation behaviour of the Lamb wave of porous material plates such as 316L stainless steel and permeable alumina ceramic with different porosities in water fluid environment. We used laser to excite the ultrasonic wave and used probe receiving as a method to measure the guided wave dispersion relationship. The results of the study show that the effect of porosity of the porous plate on the density and elastic modulus can be shown through the dispersion relationship. The theoretical model and the experimentally measured dispersion relationship results were in good agreement. In addition, by altering the density, Poisson's ratio, shear modulus and independent porosity of the porous plate, the dispersion curve can be shifted, which can be used for the development of ultrasonic measurement in the future.
Terzaghi's consolidation theory neglects inertial effects on the consolidation of saturated soils. To quantify the inertial effects, in this paper an original one-dimensional small-strain consolidation wave (C-wave) theory is developed, based upon a proposed modified Darcy's law with relaxation time and the equation of motion for soil ensemble. The one-dimensional governing equations were first formulated for self-weight consolidation, followed by a closed-form solution employing the method of separation of variables. The proposed model was then validated against wave velocity measurements and verified against finite-difference analysis. The half-closed self-weight consolidation behaviour was subsequently investigated, compared with Terzaghi's theory, Fillunger–Heinrich's dynamic theory and the u–p form of Biot's wave theory. This research indicates that: (a) superior to conventional models under comparison, the C-wave model enhances the predictability of the C-wave velocity; (b) the dimensionless C-wave coefficient (C w ) dominates the fundamental consolidation behaviour; (c) a wave-diffusion duality underlying the consolidation mechanism contributes qualitatively to the spatial bottom-up pattern and temporal response delay in consolidation observations; and (d) Terzaghi's theory can afford a practically accurate solution provided the C w and time factor are below and above approximately 0·01, respectively. The C-wave theory may enrich the understanding of consolidation-related phenomena involving an appreciable C w .
Published in Petroleum Transactions, AIME, Volume 210, 1957, pages 331–340.
Abstract
In order to obtain a better insight into the pressure-volume relationship of reservoir rocks a theory of pore and rock bulk volume variations is presented. The theory is independent of the shape of the pores but is restricted to isotropic porous media built up of continuous homogeneous matrix material. The main conclusion obtained from this theory is that only three elastic constants and three viscous constants are required for describing pore and rock bulk volume variations if the porosity is explicitly introduced into the treatment.
In addition, reasonable approximations are introduced for various types of reservoir rock, e.g., sandstones, limestones, and shales, which lead to further simplifications of the basic formulas. In consequence there is then a further reduction in the number of deformation constants which have to be determined experimentally. It is shown how measurements of these remaining deformation constants can be performed most conveniently.
Finally the application of the theory to reservoir studies is discussed and the translation of experimental results obtained in the laboratory into reservoir behavior is considered.
Introduction
The decline of fluid pressure in connection with the withdrawal of fluid from an underground reservoir gives rise to a change in volume of both reservoir fluids and reservoir rock. The volume variation of the reservoir rock results in a decrease of both the pore volume and the total volume of the fluid-filled formation. Whereas the variation in volume of the reservoir fluids with pressure is usually known from PVT analysis, that in the volume of the porous medium is rarely measured, as it is considered of minor importance in reservoir engineering. Nevertheless, certain experimental results suggest that in a number of cases the neglection of the variation in pore volume may introduce errors into material balance calculations of reservoirs producing above the bubble point.
Recent theoretical work suggests that attenuation of acoustic waves in ocean sediments depends on two distinct types of energy loss; these losses result from inelasticity of the skeletal frame and motion of the pore water relative to the frame. In this paper, experimental evidence is presented which verifies the type of response predicted by the theoretical model. Torsional resonance and logarithmic decrement are measured over a wide frequency range in both water-saturated and dry sediments. A changeover from dominance by one type of loss to the other is clearly demonstrated by a change of response that is observed as the frequency is varied. Specially designed equipment is used which allows the intergranular stress to be controlled to simulate various depths of embedment in the sea floor. Both the theory and the experiments suggest that for some sediments the attenuation will vary in a manner quite different from the usual dependency on the first power of frequency that is often assumed.-Author
This paper considers a mathematical model to describe the propagation of low‐amplitude waves in saturated sediments. Losses due to inelasticity of the skeletal frame and to motion of the pore fluid relative to the frame are both accounted for, and each is found to be significant in a different frequency range. The theory shows favorable agreement with experimental results where available for both sands and finer‐grained sediments over a wide range of frequencies.
A self‐consistent method of estimating effective macroscopic elastic constants for inhomogeneous materials with ellipsoidal inclusions has been formulated based on elastic‐wave scattering theory [J. G. Berryman, Appl. Phys. Lett. 35, 856 (1979)]. The self‐consistent effective medium is determined by requiring the scattered, long‐wavelength displacement field to vanish on the average. The resulting formulas are simpler to apply than previous self‐consistent scattering theories due to the reduction from tensor to vector equations. The results are compared to the rigorous Hashin‐Shtrikman bounds, Miller bounds, and Kuster‐Toksö estimates for the elastic module. For spherical inclusions, our formulas agree with statically derived self‐consistent moduli of Hill and Budiansky. For general ellipsoidal inclusions, our results always satisfy both the Hashin‐Shtrikman bounds and the more stringent Miller bounds. Furthermore, our theory reduces correctly to all known exact results in the appropriate limits. The theory is used to calculate velocity and attenuation of elastic waves in fluid‐saturated media.
Biot's equations for the propagation of dilational waves in fluid-saturated porous solids in the low-frequency range are analyzed for the purpose of application in geophysical research. The deformation constants of the system are unraveled in terms of compressibilities and porosity, and suitable approximate solutions for wave velocity and attenuation of the waves of both the first and the second kind are obtained. Additional results of the investigation are the following: a rather simple formula for the speed of sound in sedimentary rocks (the wave of the first kind) is obtained, which has to replace the so-called 'time-average relation' now sometimes used. A comparison between the results obtained here and published results on wave propagation in simpler fluid-solid systems, such as, for instance, suspensions, showed some weak points in the older theories. Suggestions for possible improvements are given.-from Authors
The theory of the deformation of a porous elastic solid containing a compressible fluid has been established by Biot. In this paper, methods of measurement are described for the determination of the elastic coefficients of the theory. The physical interpretation of the coefficients in various alternate forms is also discussed. Any combination of measurements which is sufficient to fix the properties of the system may be used to determine the coefficients. For an isotropic system, in which there are four coefficients, the four measurements of shear modulus, jacketed and unjacketed compressibility, and coefficient of fluid content, together with a measurement of porosity appear to be the most convenient. The porosity is not required if the variables and coefficients are expressed in the proper way. The coefficient of fluid content is a measure of the volume of fluid entering the pores of a solid sample during an unjacketed compressibility test. The stress-strain relations may be expressed in terms of the stresses and strains produced during the various measurements, to give four expressions relating the measured coefficients to the original coefficients of the consolidation theory. The same method is easily extended to cases of anisotropy. The theory is directly applicable to linear systems but also may be applied to incremental variations in nonlinear systems provided the stresses are defined properly.
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 new method of estimating effective macroscopic elastic constants for microscopically inhomogeneous materials is formulated using elastic‐wave scattering theory. The self‐consistent medium is determined by the condition that the scattered long‐wavelength displacement field must vanish on the average. The resulting formulas are simpler to apply than previous methods due to the reduction from tensor to vector equations. Our formulas are automatically symmetric under interchange of constituent labels whereas some other ’’self‐consistent’’ formulations for needle and disk inclusions do not possess this property. For spherical inclusions, the standard self‐consistent elastic constants are reproduced.
Themacroscopic elastic moduli of two-phase composites are estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kröner theory of crystalline aggregates. The phases may be arbitrarily aeolotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results arc given for an isotropic dispersion of spheres.
A heuristic analysis is given for the determination of the elastic moduli of a composite material, the several constituents of which are each isotropic and elastic. The results are intended to apply to heterogeneous materials composed of contiguous, more-or-less spherical grains of each of the phases.
The theory of propagation of stress waves in a porous elastic solid developed in Part I for the low‐frequency range is extended to higher frequencies. The breakdown of Poiseuille flow beyond the critical frequency is discussed for pores of flat and circular shapes. As in Part I the emphasis of the treatment is on cases where fluid and solids are of comparable densities. Dispersion curves for phase and group velocities along with attenuation factors are plotted versus frequency for the rotational and the two dilational waves and for six numerical combinations of the characteristic parameters of the porous systems. Asymptotic behavior at high frequency is also discussed.
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.