Biot Waves Research Articles

A novel analytical solution for horizontal vibration of partially embedded offshore piles considering the distribution effect of wave loads

According to the Novak's plane strain theory and Biot's wave equation, this paper proposes an analytical model for horizontal vibration of partially embedded offshore piles under distribution effect of wave loads (DEWL) incorporating the diffraction effects of waves. Firstly, for the pile above the mudline, the wave force acting on the pile is obtained through the method of separation of variables. Secondly, for the embedded section of offshore piles, the governing equations of the seabed are decoupled by introducing potential functions, and then the boundary condition of the seabed and the continuity condition at the soil-pile contact surface are combined to obtain the frequency domain analysis of the soil reaction forces. Furthermore, by applying the continuity condition of displacement at mudline and using the matrix transfer method, the analytical solution for the horizontal vibration characteristics (HVC) is derived. Finally, a comparison analysis with existing theoretical solutions is conducted to validate the rationality and accuracy of the analytical solution proposed in this paper. Based on this, a parametric analysis is conducted to discuss the effects of the parameters of the soil, the pile and the wave on the HVC of partially embedded offshore piles.

A numerical extension of White's theory of P-wave attenuation to non-isothermal poroelastic media.

Mesoscopic P-wave attenuation in layered, partially saturated thermo-poroelastic media is analyzed by combining the theories of Biot poroelasticity and Lord-Shulman thermoelasticity (BLS). The attenuation is quantified by estimating the quality factor Q. The mesoscopic attenuation effect, commonly referred to as wave-induced fluid flow (WIFF), is the process that converts fast compressional and shear waves into slow diffusive Biot waves at mesoscopic heterogeneities larger than the pore scale, but much smaller than the dominant wavelengths. This effect was first modeled in White's isothermal theory by quantifying the seismic response of a periodic sequence of planar porous layers that are alternately saturated with gas or water. This work presents a numerical extension of White's theory for the non-isothermal case in this type of sequence. For this purpose, an initial-boundary-value problem (IBVP) for the BLS wave propagation equations is solved using the finite element method, where the particle velocity field is recorded at uniformly distributed receivers. The quality factor is estimated using spectral-ratio and frequency-shift methods. The Q-estimates show that thermal effects influence the attenuation of the P-wave and the velocity dispersion compared to the isothermal case.

Ground motion around a semi-cylindrical canyon in saturated soil: P wave incidence

Based on Biot wave theory in saturated two-phase medium, the analytical solution of plane P-wave scattering by semi-circular canyon in half space in saturated two-phase medium is given by using complex function method and Hankel function integral transformation method. Firstly, a semi-circular depression model in saturated half space is constructed, the wave field potential function in the medium is solved, and its general solution form is obtained. Secondly, the unknown coefficient value in the potential function is obtained by substituting the corresponding stress boundary conditions. With the help of Hankel function integral transformation method, the wave potential function in complex coordinates is transformed to meet the horizontal boundary conditions of half space in rectangular coordinates. Finally, the surface displacement amplitude at the semi-circular depression and the surrounding horizontal boundary is obtained through the total wave field potential function, and the variation law of surface displacement amplitude under the influence of different parameters is analyzed.

Quantifying Tortuosity in Porous Lithium-Ion Battery Materials Using Ultrasound

Active electrode materials in lithium-ion batteries are porous. These materials are composed of a solid frame containing interconnected pores/channels, in which mass transport follows a tortuous pathway. Tortuosity is defined as the ratio of the average tortuous pathway to the projected straight path, and is a particularly important parameter indicating the diffusivity and conductivity properties of porous battery materials; so quantifying tortuosity is highly desirable to safeguard battery performance. Existing quantification methods are mostly based on impedance and polarisation measurements from specially-fabricated cells and numerical diffusion simulations using the microstructural tomographic images of porous materials. These methods, however, are generally cumbersome and barely applicable to in-production measurement. Here we report a novel alternative quantification method based on ultrasonic Biot waves in porous materials. This method utilises the fundamental fact that in a porous material with a rigid frame (which the electrode sheets are when immersed in the air), the slow Biot wave only travels in the liquid/gaseous phase, through the same tortuous pathway that the diffusion of ions and electrons happens in a finished battery. To achieve the quantification, we first develop a physical model relating the propagation of ultrasonic Biot waves to the tortuosity of porous battery materials. Based on this physical model, we then propose an inversion method to infer tortuosity from ultrasonic readings. The readings are acquired using air-coupled ultrasonic transducers in a non-destructive fashion, thus enabling real-time tortuosity quantification. This ultrasonic method measures the average tortuosity in a range of ~20 mm across the testing material, and can easily scan over a sheet to provide spatially resolved information. With these exciting features, the proposed method could offer a next-generation tortuosity quantification tool for quality control of porous battery material productions and a better understanding of battery performance.

Propagation of Pore Pressure and Stress in Saturated Porous Media Based on a Darcy-Brinkman Formulation

Propagation of pore pressure and stress in water-saturated elastic porous media is theoretically investigated when considering the Darcy-Brinkman law. The wave mode, phase velocity, phase lag, damping factor, and characteristic frequency are found from the updated mathematic model. The Brinkman term describes the fluid viscous shear effects and importantly contributes to the dispersion relation and wave damping. The coincidence of the properties of Biot waves of the first and second kinds occurs at a characteristic frequency, which is remarkably influenced by the Brinkman term. A key finding is that, compared to the Darcy-Brinkman law, Darcy’s law overestimates the phase velocity, damping, and phase lag of the first wave, while underestimates the phase velocity, damping, and phase difference of the second wave. The introduction of the Darcy-Brinkman law yields an improved description of the damping of the compressional wave modes in saturated porous media.

Compressional wave propagation in saturated porous media and its numerical analysis using a space-time conservation element and solution element method.

Compressional waves in saturated porous media are relevant to many fields from oil exploration to diagnostic of human cancellous bone and can be used to interpret physical behaviors of materials. In this work, based on Biot's theory in the low frequency range, a key finding is that there exists a critical frequency of Biot's theory in the low frequency range, which determines the coincidence of the properties of Biot waves of the first and second kinds. Furthermore, we have investigated the dispersion and attenuation of the coalescence of the first and second compressional waves in the low frequency range. The coalescence of the first and second waves is strongly attenuated with a moderate phase velocity and shows the in-phase feature. In addition, acoustic wave propagation has been calculated numerically using the space-time conservation element and solution element (CESE) method. The CESE-simulated results are compared to the experimental data and to those of the classical transfer function approach. We show that the CESEscheme preserves the local and global flux conservations in the solution procedure of Biot's theory. It is found that the CESEmethod provides more accurate predictions of high dispersion and strong attenuation of compressional waves in the low frequency range and is well suitable for predicting compressional wave fields in saturated porous media.

Nonlocal thermo-hydro-mechanical (THM) coupling dynamic response of saturated porous thermoelastic media with temperature-dependent physical properties

The coupled thermal-hydro-mechanical (THM) analysis of saturated porous media is of great significance and has been widely used in various engineering fields. However, the classical Biot theory assumes that the physical parameters of porous media are constant and ignores the influence of pore size on wave propagation characteristics, which cannot reflect the actual situation. Based on the nonlocal theory and Biot wave equation, a nonlocal coupled Biot THM dynamic model for porous media with variable physical properties is established by introducing the temperature-dependent factor function. Utilizing the nonlocal coupled Biot THM theory, the dynamic response of forced convection in a half-space saturated porous media under surface harmonic thermal action is investigated in this paper. The analytical solutions of temperature increment, displacement, stress, and pore water pressure of saturated porous media are obtained by means of differential operator decomposition. In addition, the effects of the temperature-dependent factor and porosity on the nonlocal model at different frequencies, and the effects of the nonlocal parameter on the distribution rules of various physical fields at different frequencies are explored in depth.

Implementation of hybridizable discontinuous Galerkin method for time‐harmonic anisotropic poroelasticity in two dimensions

AbstractWe apply a hybridizable discontinuous Galerkin (HDG) method to numerically solve two‐dimensional anisotropic poroelastic wave equations in the frequency domain given by Biot theory. The motivation for choosing HDG method comes from the complexity of the considered equations and the high number of unknowns. The HDG method possesses all the advantages of discontinuous Galerkin method (hp‐adaptivity, accuracy, ability to model complex tectonics, etc.) without a drastic increase in the number of degrees of freedom. After a description of its implementation, we illustrate the accuracy of the proposed method by comparisons with analytical solutions. We then perform a sensitivity analysis of the method as a function of stabilization parameters and frequency, and show in particular that there exists optimal values of these parameters in order to obtain a very good level of accuracy. We also show the ability of the method to reproduce the different types of poroelastic waves including the slow Biot wave.

An analytical solution for self-weight consolidation based on one-dimensional small-strain consolidation wave theory

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 (Cw) 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 Cwand 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 Cw.

DYNAMIC COUPLED THERMO-HYDRO-MECHANICAL PROBLEM FOR SATURATED POROUS VISCOELASTIC FOUNDATION

Natural soil often has the characteristics of rheology due to different depositional conditions and stress states. The present paper focuses on investigated the effects of different porosity and permeability coefficient in saturated porous foundation which considered the viscoelastic relaxation times with coupled thermo-hydro-mechanical fields under external load. A two-dimensional coupled thermo-hydro-mechanical dynamics problem for a half-space on an isotropic, uniform, fully saturated, and poroviscoelastic soil (THMVD) whose surface is subjected to either mechanical force or thermal load based on the Biot's wave theory of porous media, Darcy's law, and Lord-Shulman (L-S) generalized thermoelastic theory with Kelvin-Voigt viscoelastic model is investigated. The general relationships among the non-dimensional vertical displacement, excess pore water pressure, vertical stress, and temperature distribution are then deduced via normal mode analysis and depicted graphically. Normal mode analysis is a method using weighted residuals to derive analytical solutions. Via this method, the equation can be divided into two parts without integral transformation and inverse transformation, thereby increasing the speed of decoupling and eliminating the limitation of numerical inverse transformation. The effects of the porosity and the permeability coefficient on the four different physical variables have been investigated. It can be shown that: whatever load is being considered, the variation of load frequencies have obvious effect on all the considered physical variables; the porosity and permeability have the most obvious influence on non-dimension excess pore water pressure. When thermal loads were considered only, the variation of porosity and permeability coefficient had barly effect on non-dimension temperature. This proposed derivation method can be widely applied in the geotechnical engineering field, especially with regard to the mechanical and thermal behaviors of commercial buildings, high-speed railways, and highway energy foundations. The research results of this problem can lay a certain theoretical foundation for engineering construction and have a certain guiding significance.

SeismicQof inhomogeneous plane waves in porous media

Seismic waves in an attenuative porous medium are generally inhomogeneous waves, which have different directions of propagation and attenuation. The dissipation factors (1/ Q) of inhomogeneous waves are strongly dependent on the degree of wave inhomogeneity and cannot be expressed correctly with the usual 1/ Q expressions valid only for homogeneous waves. We have used the differing definitions of 1/ Q for inhomogeneous waves (i.e., the ratio of the time-averaged dissipated energy density to the time-averaged strain energy density or time-averaged total energy density) and the complex form of the energy balance equations of poroviscoelastic media to derive concise and explicit expressions for the dissipation factors. They are given as simple functions of the material parameters and the wave inhomogeneity parameter for inhomogeneous SV-waves and fast and slow P-waves. The isotropic, poroviscoelastic medium under consideration is upscaled from effective Biot theory for a double-porosity solid, which is the most general theory to describe wave propagation in a reservoir. We find that, if the inhomogeneity parameter is infinite (i.e., the inhomogeneity angle is 90°) for all three Biot waves, then the dissipation factors only depend on the ratio of the imaginary to the real part of the complex shear modulus. Our explicit expressions for the dissipation factors of poroviscoelastic materials also are reduced to obtain their counterparts for viscoelastic media as a special case. The inhomogeneous waves in an example poroviscoelastic material are used to demonstrate that the 1/ Q values of the three Biot waves strongly depend on the inhomogeneity parameters and furthermore the different definitions may cause significant differences of 1/ Q values. We find that the dissipation factor of fast P-waves may decrease with the increasing degree of inhomogeneity, which contradicts previously published results.

Frequency-domain FD modeling with an adaptable nearly perfectly matched layer boundary condition for poroviscoelastic waves upscaled from the effective Biot theory

To approximate seismic wave propagation in double-porosity media, one can use the effective Biot theory, which can explain the high level of attenuation observed at seismic frequencies, but which is unaccounted for with classic Biot theory. The governing equations of the effective Biot theory for homogeneous media with complex frequency dispersion characteristics need to be upscaled and extended to S-waves using a poroviscoelastic approach when simulating heterogeneous porous media. We have applied a frequency-space domain mixed grid finite-difference modeling method for this purpose and computed the wavefields for solid particle velocity, fluid flux, and pore pressure. A homogeneous full space model with a single Cole-Cole relaxation function (fractional Zener model) representing the attenuation mechanism is used to indicate that the numerical solutions match fairly well the analytical waveform solutions of the effective Biot theory for source center frequencies ranging from 25 to 10,000 Hz. The computed pore pressure wavefield for a two-layer example porous model is used to further support our numerical method. This model and a laterally heterogeneous three-layer porous medium model further illustrate the applicability of the procedure and even indicate the presence of the slow Biot wave near the interfaces as a result of mode conversion on reflection and transmission. The nearly perfectly matched layer (NPML) absorbing boundary condition is chosen to truncate the computational grid and avoid reflections from the model edges. We found and proved that the NPML performs identically to the PML for either elastic wave modeling or Biot porous medium wave modeling for 3D and 2D scenarios. Two methods to adjust the maximum damping factor (one for the source center frequency and the other for each frequency component) are suggested to adjust the damping factor function in the NPML condition to ensure its effectiveness for various frequency ranges.

The role of rheology in modelling elastic waves with gas bubbles in granular fluid-saturated media

Elastic waves in fluid-saturated granular media depend on the rheology which includes elements representing the fluid and, if necessary, gas bubbles. We investigated the effect of different rheological schemes, including and excluding the bubbles, on the linear Frenkel–Biot waves of P1 type. For the wave with the bubbles the scheme consists of three segments representing the solid continuum, fluid continuum, and a bubble surrounded by the fluid. We derived the Nikolaevskiy-type equations describing the velocity of the solid matrix in the moving reference system. The equations are linearized to yield the decay rate lambda as a function of the wave number k. We compared the lambda(k)-dependence for the cases with and without the bubbles, using typical values of the input mechanical parameters. For the both cases, the lambda(k)-curve lies entirely below zero, which is in line with the notion of the elastic wave being an essentially passive system. We found that the increase of the radius of the bubbles leads to faster decay, while the increase in the number of the bubbles leads to slower decay of the elastic wave.

Observation of Biot compressional waves of the second kind in granular soils

The compressional wave of the second kind predicted theoretically by Biot was observed in granular soils using triaxial testing system. This theory models both the individual and coupled behavior of soil skeleton and pore fluid using the properties of pore space and elastic constants of the solid, pore fluid and solid skeleton. The wave is characterized by its large damping and hence detection of the wave in the laboratory is difficult. This difficulty was overcome by adopting highly sensitive sensor, signal stacking and appropriate filtering. The propagation of the P- and S-waves in granular materials was measured under different confining pressures and in both dry and saturated conditions. The phase of the Biot wave was the opposite to that of the P- and S- waves in all observed. According to the theory, the Biot wave results from the out-of-phase movement of the soil skeleton and pore fluid. The measured Biot wave velocities agree well with the theoretical values computed using the properties and parameters which were obtained primarily from laboratory tests. The good agreement demonstrates that the theory is valid for granular soils.

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

In this paper, it is proposed a model for deformable porous media saturated by compressible nematic liquid crystal subjected to slowly varying electric fields. from a mechanical point of view, we assume that such a system can be described by means of a Biot-type model and that the mechanical action of the NLC on the solid matrix can be modeled by means of a suitable modification of Biot constitutive equations for pore pressure only. The nonlinear nature of NLCs and the presence of bifurcations make the analysis particularly challenging. We prove that suitable electrical stimulus applied on the NLC specimen may induce both type of Biot waves, fast and slow, along with shear waves in the porous matrix. This effect may be of use when one may wish to damp mechanically induced pressure waves using Darcy dissipation.

Rock anelasticity due to patchy saturation and fabric heterogeneity: A double double‐porosity model of wave propagation

AbstractHeterogeneity of rock's fabric can induce heterogeneous distribution of immiscible fluids in natural reservoirs, since the lithological variations (mainly permeability) may affect fluid migration in geological time scales, resulting in patchy saturation of fluids. Therefore, fabric and saturation inhomogeneities both affect wave propagation. To model the wave effects (attenuation and velocity dispersion), we introduce a double double‐porosity model, where pores saturated with two different fluids overlap with pores having dissimilar compressibilities. The governing equations are derived by using Hamilton's principle based on the potential energy, kinetic energy, and dissipation functions, and the stiffness coefficients are determined by gedanken experiments, yielding one fast P wave and four slow Biot waves. Three examples are given, namely, muddy siltstones, clean dolomites, and tight sandstones, where fabric heterogeneities at three different spatial scales are analyzed in comparison with experimental data. In muddy siltstones, where intrapore clay and intergranular pores constitute a submicroscopic double‐porosity structure, wave anelasticity mainly occurs in the frequency range (104–107 Hz), while in pure dolomites with microscopic heterogeneity of grain contacts and tight sandstones with mesoscopic heterogeneity of less consolidated sands, it occurs at 103–107 Hz and 101–103 Hz (seismic band), respectively. The predicted maximum quality factor of the fast compressional wave for the sandstone is the lowest (approximately 8), and that of the dolomite is the highest. The results of the diffusive slow waves are affected by the strong friction effects between solids and fluids. The model describes wave propagation in patchy‐saturated rocks with fabric heterogeneity at different scales, and the relevant theoretical predictions agree well with the experimental data in fully and partially saturated rocks.

Does Biot’s theory have predictive power?

Biot’s theory of elastic waves in fluid-saturated porous solids has two free parameters: the tortuosity α, characterizing the dynamic coupling between the solid and the fluid, and the structural factor δ, representing the geometric properties of the porous space. The meaning and significance of these parameters have not been sufficiently understood. The tortuosity has the physical meaning of the normalized mean square of the velocity of the pore fluid relative to the solid wall; it has a low-frequency but no high-frequency limits. The analytical calculation of the tortuosity for Biot’s slit-like pore provides its range of variability from approximately 1–100 in the frequency range of practical interest. The tortuosity has a significant effect on the properties of the Biot waves of the second kind in the high-frequency range. On the other hand, in realistically complex pore geometries, the values of the tortuosity are virtually unpredictable. This limits the usefulness of the Biot theory in predicting the wave propagation at high frequencies. At all frequencies, the effect of the structural factor is insignificant relative to the effect of the tortuosity. The conventional compressional wave (the wave of the first kind) is insensitive to both parameters at all frequencies. The frequencies of interest to seismic exploration are also free of the uncertainty imposed by the lack of constraints on the tortuosity as the only free parameter in Biot’s theory.

Finite-element harmonic experiments to model fractured induced anisotropy in poroelastic media

Fractures in a fluid-saturated poroelastic–Biot–medium can be modeled as very thin highly permeable and compliant layers within a porous background. A Biot medium containing a dense set of aligned fractures behaves as an effective transversely isotropic and viscoelastic (TIV) medium at the macroscale when the predominant wavelength is much larger than the average distance between fractures. One important mechanism in Biot media at seismic frequencies is wave-induced fluid flow generated by fast compressional waves at mesoscopic-scale heterogeneities, generating slow diffusion-type Biot waves. In this work, we present and analyze a collection of time-harmonic finite element experiments that take into account the effects of the presence of aligned fractures and interlayer fluid flow occurring at the mesoscale, allowing us to determine the complex and frequency dependent stiffnesses of the effective TIV medium at the macroscale.These numerical upscaling experiments are defined as boundary value problems on representative samples of the fractured material, with boundary conditions associated with compressibility and shear tests, which are solved using the finite element (FE) method. The FE space chosen to discretize each component of the solid displacement vector is that of globally continuous piecewise bilinear functions, while for the fluid phase the vector part of the Raviart–Thomas–Nedelec space of zero order is employed. We present results on the uniqueness of the solution of the continuous and discrete problems, and derive optimal a priori energy error estimates. First, the numerical results are validated with those of a theory valid for fluid flow perpendicular to the fracture layering and independent of the loading direction, so that the attenuation mechanism can be represented by a single relaxation function. Then, the methodology is applied to cases for which no analytical solutions are available, such as a fractured Biot medium saturated with brine and patches of CO2 and a brine saturated sample of uniform background and fractures with fractal variations in their petrophysical properties.

Seismic velocity and Q anisotropy in fractured poroelastic media

A fluid-saturated poroelastic isotropic medium with aligned fractures behaves as a transversely isotropic and viscoelastic (TIV) medium when the predominant wavelength is much larger than the average distance between fractures. A planar fracture embedded in a fluid saturated poroelastic background medium can be modeled as a extremely thin and compliant porous layer. P-waves traveling in this type of medium induce fluid-pressure gradients at fractures and mesoscopic-scale heterogeneities, generating fluid flow and slow (diffusion) Biot waves, causing attenuation and dispersion of the fast modes (mesoscopic loss). A poroelastic medium with embedded aligned fractures exhibits significant attenuation and dispersion effects due to this mechanism, which can properly be represented at the macroscale with an equivalent TIV medium. In this work, we apply a set of compressibility and shear harmonic finite-element (FE) experiments on fractured highly heterogeneous poroelastic samples to determine the five complex and frequency dependent stiffnesses characterizing the equivalent medium. The experiments consider brine or patchy brine-CO2 saturated samples and a brine saturated sample with a heterogeneous (fractal) skeleton with fractures. We show that fractures induce strong seismic velocity and Q anisotropy, both for qP and qSV waves, enhanced either by patchy saturation or frame heterogeneity.

The Partition of Unity Finite Element Method for the simulation of waves in air and poroelastic media.

Recently Chazot et al. [J. Sound Vib. 332, 1918-1929 (2013)] applied the Partition of Unity Finite Element Method for the analysis of interior sound fields with absorbing materials. The method was shown to allow a substantial reduction of the number of degrees of freedom compared to the standard Finite Element Method. The work is however restricted to a certain class of absorbing materials that react like an equivalent fluid. This paper presents an extension of the method to the numerical simulation of Biot's waves in poroelastic materials. The technique relies mainly on expanding the elastic displacement as well as the fluid phase pressure using sets of plane waves which are solutions to the governing partial differential equations. To show the interest of the method for tackling problems of practical interests, poroelastic-acoustic coupling conditions as well as fixed or sliding edge conditions are presented and numerically tested. It is shown that the technique is a good candidate for solving noise control problems at medium and high frequency.