Biot's Theory Research Articles

Theoretical, experimental, and numerical studies of flow field characteristics and incipient scouring erosion for slope with rigid vegetations

Under the action of continuous rainfall, the coupling linkage between runoff and seepage can cause erosion of non-cohesive soil particles on vegetated slopes. A nonlinear turbulent-seepage coupling model is established to study the flow field characteristics and incipient scouring erosion of the vegetated slope. The fluid in the turbulent region is governed by the Reynolds averaged Navier-Stokes. Biot's poroelasticity theory describes the flow in the vegetation region. The seepage in the soil region obeyed the Darcy-Brinkman equation. Analytical solutions of the vertical velocity profile and safety factor (sliding and rolling) of the particle incipient motion are derived based on the coupling between turbulence and seepage. The velocity expression is found to be in good agreement with the existing experimental data of both emerged and submerged vegetation conditions, respectively. The velocity distribution above the surface follows a logarithmic pattern, while below the surface, non-linear seepage occurs with a transition layer that deviates from Darcy seepage. In addition, the particle incipient motion is verified by the physical flume souring test under none, sparse, and dense vegetation, respectively. The absolute error between the theoretical calculation and the physical test is within 5%. Furthermore, the critical water depth is nonlinearly negatively correlated with the energy slope. Finally, a CFD-DEM coupling numerical model is employed to further observe the scouring erosion of the sparse and dense vegetated slopes with the same conditions as the physical test. The numerical results show that the scouring erosion exhibits discontinuous behavior and the soil particles between the two vegetation are more vulnerable to scouring than those in other locations. After scouring, the inertial porosity of the vegetated soil slope decreases while the porosity in the surface area increases.

Modelling of critical speed of railway tracks on a multi-layered transversely isotropic saturated ground

The responses of railway tracks are significantly amplified when the high-speed train travels around the speed of waves propagating in underlying soils. The dynamic amplification phenomenon corresponding to the issue of the critical speed may compromise the stability and safety of operation and increase track damage. This paper proposes an analytical model to analyze the effect of the transverse isotropy of ground on the critical speed of railway tracks on stratified saturated soils. By using the Biot's theory, the general solutions of the transversely isotropic saturated porous medium are derived by employing the potential decomposition and Fourier transform. The transmission and reflection matrix method is extended to derive the three-dimensional fundamental solution for moving point loads acting on a multi-layered transversely isotropic saturated half-space. The models for railway tracks and wheel-axle loads are subsequently coupled to the ground. The proposed model is validated through a comparison with two existing models. Numerical studies demonstrate that the material anisotropy changes the critical speed and the corresponding dynamic amplification coefficient by up to 20%. The anisotropy in the shear modulus has the most significant influence on the critical speed, followed by Young's modulus and the permeability coefficient. The material anisotropy and soil layering cannot be ignored for the assessment of the critical speed of a track-ground system.

Dynamic analysis of an infinite beam resting on layered transversely isotropic saturated media subjected to moving harmonic loads

AbstractThis paper investigates the dynamic behavior of an infinite beam on layered transversely isotropic (TI) poroelastic media due to moving harmonic loads. Based on the Euler‒Bernoulli beam theory and Biot theory, the governing equations of the beam and the layered TI poroelastic media are established. By introducing the compatibility conditions at the beam‐subgrade interface, the equivalent stiffness of the layered TI saturated media is derived via the extended precise integration method (PIM) in the wavenumber domain. By means of the inverse Fourier transform algorithm, the physical solutions for the beam‐saturated soil system are obtained after introducing the equivalent stiffness. Then, the accuracy of the proposed method is verified. Several numerical examples are presented to discuss the influences of the moving speed, beam bending rigidity, stratification, and soil anisotropy on the dynamic behavior of beams.

A hydro-mechanical phase field model for hydraulically induced fractures in poroelastic media

This paper presents a hydro-mechanical phase field model that integrates the unified phase field fracture theory and Biot's theory, emphasizing the correction of the volumetric strain term in the fluid mass conservation equation. After numerically implementing the multi-field finite element method, the proposed model is validated using examples with analytical solutions. The model accurately predicts fluid pressure, crack width, and crack length while capturing complex fracture behaviors, such as crack merging and non-planar propagation. The study reveals that the physical meaning of the volumetric strain in the fractured domain is the volume change of the space for fluid storage, and confirms the importance of considering the volumetric strain term in the fractured domain, as ignoring it can severely underestimate the injection volume required for fracturing. The proposed hydro-mechanical coupling method is also shown to be superior to previous methods, as it accurately simulates the relationship between crack width variation and pressure, and avoids the incorrect effective stress resulting from the assumption that the Biot coefficient equals the porosity. Overall, this study provides an effective tool for predicting the fracture process and essential insights into hydro-mechanical phase field modeling.

Vibration mitigation in porous soil using periodic rock-socketed pile barriers

This study aimed to reveal the vibration mitigation effect of periodic rock-socketed piles installed in porous soil. Biot's theory and the periodic theory from solid-state physics were applied to observe the ultralow and surface-wave bandgaps in rock-socketed pile barriers. The bandgaps of the pile barriers in dry and saturated soils were subsequently compared, indicating that the width of the ultralow bandgap decreased slightly when the porous soil was fully saturated. To overcome the limitations of the sound cone method for calculating the surface-wave bandgap of a periodic rock-socked pile barrier, a critical frequency for Rayleigh waves was defined. Next, the screening effectiveness of pile barriers for both the ultralow and surface-wave bandgaps was validated using three-dimensional numerical simulations. Finally, the influences of the geometric and material parameters of the piles and soil as well as the water table level on the ultralow bandgap were evaluated using a parametric study. The results of this investigation provide a reference for the application of periodic rock-socketed piles in geological conditions with groundwater.

Analytical simulation of the elastic moduli dispersion for an isotropic porous cylinder

There are extensive laboratory measurements of elastic moduli dispersion of porous materials, yet the mechanisms for the dispersion are not fully understood. Related analytical solutions are missing in the literature, potentially due to complex mathematical derivations. In this paper, we use Biot's theory of poroelastodynamics (PED) and present the first analytical solution for an isotropic fluid-saturated porous cylinder subject to a forced deformation test. This is done by introducing Helmholtz decomposition, two scalar potentials, and two vector potentials to decouple the original governing partial differential equations. The methods of matrix diagonalization and separation of variables are then used to solve the decoupled equations. The elastodynamics (ED) solution is also presented. We demonstrate the coupled responses of displacement, pore pressure, and stress and interpret systematically the mechanisms for the elastic moduli dispersion. Furthermore, we investigate comprehensively the effects of loading frequency, material's poromechanical characteristics, sample size, and boundary conditions on the elastic moduli dispersion. Finally, excellent matches are found between the analytical solution and published measurements of the dynamic Young's moduli and Poisson's ratios of a water-saturated clastic sediment rock and a glycerin-saturated limestone. The analytical solution captures the materials’ elastic moduli dispersion at both seismic and sonic frequencies. This work should be of interest to scientists and engineers who work on dynamics, vibration, and poromechanics.

Dynamic response of a multi-scale layered saturated porous half-space due to seismic dislocation source by using a revised dynamic stiffness matrix method

The study investigating the dynamic response of seismic dislocation source in a layered saturated porous half-space, particularly the multi-scale characteristic of half-space, from crustal to geotechnical scale, is still lacking in the literature. particularly in will lead to the computation bottlenecks for multi-scale seismic wave propagation. Based on Biot's theory of wave propagation in a fluid-saturated porous solid, a dynamic stiffness matrix method is proposed to investigate the dynamic response of a multi-scale layered saturated porous half-space due to the seismic dislocation source. Assembling each layer stiffness matrix in the frequency-wavenumber domain, the global dynamic stiffness matrix is constructed to avoid errors accumulative compared to the reflection and transmission matrix method. The divergent index terms of the stiffness matrix with respect to thickness and wavenumber are extracted to revise the dynamic stiffness matrix adapted for multi-scale half-space. The seismic dislocation source action is equal to the external forces imposed on the total saturated porous half-space, which can obtain the dynamic response by using revised dynamic stiffness matrix method. The accuracy of the proposed method is verified via the comparison between the calculated results and of three published literatures. The multi-scale model with realistic near-surface fine structure is selected to investigate the effects of superficial fine structures on the dynamic responses. The numerical simulations results show that the presence of low-velocity saturated porous layers has a prominent impact on ground motion. The dynamic response seems to be sensitive to the porosity conditions, especially for high frequency, which cause the difference in variation of peak displacement value.

A close-formed solution for the horizontal vibration of a pipe pile in saturated soils considering the radial heterogeneity effect

Based on the theory of Biot’s two-phase medium and Novak’s thin layer, a coupled mechanical model of pipe pile and radially inhomogeneous saturated soil is established by comprehensively taking into account the saturated surrounding soil’s construction disturbance effect and the influence of the pipe pile’s saturated inner soil. First, the displacements and horizontal impedances of the surrounding and inner soil are derived using the potential function, Laplace transformation and variable separation methods. Second, the analytical solution for the horizontal impedance of pile head is obtained by combining the conditions of the displacement continuity and stress equilibrium on the soil-pile interface. Finally, the obtained solution for the pile head’s horizontal impedance is reduced to validate the rationality of the proposed solution by making comparisons with existing solutions. On this basis, a detailed parametric analysis is given to explore the influence of the pile parameters and construction disturbance effect on the horizontal vibration characteristics of pipe pile.

Three-dimensional numerical simulation of wave-induced seabed response around a dumbbell-shaped cofferdam

The dumbbell-shaped cofferdam has a large-scale and complex geometric shape, which may result in a complicated fluid-structure-seabed coupling mechanism and sophisticated seabed response around the structure. To investigate the wave-induced oscillatory seabed response around a dumbbell-shaped cofferdam, this study simulates the wave-cofferdam-seabed interaction process using a numerical model within the OpenFOAM framework. In this study, the finite volume method is applied to both the wave model and seabed model, thus ensuring the computational accuracy of this study. Biot's poro-elastic theory (u-p approximation) is considered as the governing equation for the dynamic response of the porous seabed, and the fluid motions are simulated based on the Reynolds-averaged Navier–Stokes equation. First, the accuracy of the numerical model for simulating wave-induced soil response is validated against existing experiments. Next, the effects of the submerged depth, wave incidence angle, and wave height on the oscillatory pore pressure response are studied in detail. Finally, momentary liquefaction around the dumbbell-shaped cofferdam is examined. The numerical results indicate that the wave-induced oscillatory seabed response around the cofferdam is considerably affected by the wave height but less affected by the wave incident angle. Moreover, the distribution of the wave-induced pore pressure in the vicinity of the dumbbell-shaped cofferdam is complex, and the maximum liquefaction depth around the cofferdam occurs near the steel casing along the centerline of the cofferdam.

Finding wave equations for sediment acoustic theories

The two main theories for wave propagation in sediments, the Extended Biot theory and the Viscous Grain Shearing (VGS) theory, have been formulated in terms of dispersion equations only. Some have considered the lack of a wave equation to be a weakness that may indicate lack of a physical basis for these theories. In the Biot theory, it is the frequency-dependent viscodynamic operator which models viscous boundary effects in pores that poses the problem. It can be formulated as a ratio of Bessel functions for circular pores and as a ratio of tanh-functions for a 2D parallel plane duct. There is no simple equivalent time domain operator for use in the wave equation. In both geometries, the factor may, however, be approximated well with (1 + icω)1/2, where c is a constant. This factor also appears in the Cole-Davidson dielectric theory for complex media. The equivalent time domain operator, the fractional pseudo-differential operator, enables the formulation of wave equations for all three wave modes. The same operator turns out to be central for transforming the dispersion relations of the VGS theory to time-space wave equations. These wave equations enable a closer study of properties of the medium models than that obtained from dispersion relations alone.

Prediction of acoustical behavior of granular material stacks as measured in a standing wave tube by using a Biot theory-based model

The acoustical behavior of granular materials, such as activated carbon and silica gel, has drawn attention in recent studies, due to their favorable properties such as good low frequency sound absorption. Like other more traditional porous materials, granular materials can also be tested in a standing wave tube for a convenient assessment of their acoustical properties. However, the behavior of granular materials stacked in a standing tube is more complex than that of traditional materials. For example, the response of lightweight glass bubbles reveals a clear dependence on the sound pressure level of the input signal. Also, when tested in standing wave tubes of different diameters, the same type of granular materials displays differences in their behavior. The apparent stiffness of granule pack is also related to the depth of the stack. In the present work, a model based on Biot theory is proposed, together with a consideration of the effect of the change of boundary conditions and the granule stack stiffness in different test configurations. The model is realized by using a finite difference method, and the simulation results are compared with measurements of different types of granular materials.

An analytical solution for the horizontal vibration behavior of a cylindrical rigid foundation in poroelastic soil layer

AbstractThe large size embedded foundation is widely used in the engineering, but the finite thickness of soil layer underlying this foundation is usually neglected in design, which leads to the non‐negligible error of calculation. By virtue of Biot's elastodynamic theory, this paper proposes a simple method to discuss the horizontal dynamic response of the cylindrical rigid foundation partially embedded in a poroelastic soil layer. First, based on the Novak plane strain model, the shaft resistance from the surrounding soil is simulated and solved. Second, the foundation end soil is assumed as a continuous medium of finite thickness, whose initial mechanism is derived from the dynamic interaction between the rigid disk and soil. Finally, the horizontal dynamic response factor is calculated by adopting newton's second law. Several cases are set to verify the rationality of the presented solution and to develop the analysis of key parameters. The numerical results suggest that the soil layer thickness has a significant influence on the dynamic vibration of the embedded foundation, and its effect is consistent with that of poroelastic half‐space when the thickness exceeds certain value.

Influence factors on the nonlocal parameter and scale factor in strain gradient nonlocal Biot theory

In our previous work, the strain gradient nonlocal Biot poromechanics model was proposed, in which two key size parameters, i.e., the nonlocal parameter and scale factor, were introduced. However, an extensive discussion and the value range on these two parameters were absent. In this work, we present the method for determining these parameters and the influences of grain size, Porosity, and grain size distribution on these size parameters are also discussed for the first time. The results show that: Ⅰ) the nonlocal parameter increases while the scale factor decreases with increasing grain size. There exists a critical grain size, below which the scale factor has a more prominent effect on the wave dispersion. A unified exponential equation of li/dm=A+Bexp(−dm/C)(i=0,1) can be employed to describe the relationship between these two dimensionless parameters and the mean grain size. Ⅱ) Porosity significantly influences the wave velocity but has a negligible influence on the nonlocal parameter and scale factor. Ⅲ) For media with a mix of grain sizes, the nonlocal parameter increases, but the scale factor decreases when the proportion of the fine grain content decreases. Ⅳ) The values of the nonlocal parameter and scale factor are on the order of millimeters. This work determined the value range and influence factors of these two critical parameters introduced in strain gradient nonlocal Biot theory, which is of great significance for popularizing and applying this theory to engineering practices.

Brownian motion with radioactive decay to calculate the dynamic bulk modulus of gases saturating porous media according to Biot theory

We present a new stochastic simulation method for determining the long-wavelength effective dynamic bulk modulus of gases, such as ambient air, saturating porous media with relatively arbitrary microgeometries, i.e., simple enough to warrant Biot’s simplification that the fluid and solid motions are quasi-incompressible motions at the pore scale. The simulation method is based on the mathematical isomorphism between two different physical problems. One of them is the actual Fourier heat exchange problem between gas and solid in the context of Biot theory. The other is a diffusion-disintegration-controlled problem that considers Brownian motion of diffusing particles undergoing radioactive-type decay in the pore volume and instant decay at the pore walls. By appropriately choosing the decay time and the diffusion coefficient, the stochastic algorithm we develop to determine the average lifetime of the diffusing particles, directly gives the effective apparent modulus of the saturating fluid. We show how it leads to purely geometric stochastic constructions to determine a number of geometrical parameters. After validating the algorithm for cylindrical circular pores, its power is illustrated for the case of fibrous materials of the type used in noise control. The results agree well with a model of the effective modulus with three purely geometric parameters of the pore space: static thermal permeability divided by porosity, static thermal tortuosity, and thermal characteristic length.

Propagation of acoustic wave and analysis of borehole acoustic field in porous medium of heterogeneous viscous fluid

Sound field in fluid-saturated porous medium is closely related to the viscosity of fluid and the heterogeneity of porous medium. In order to improve the physical mechanism of wave propagation in porous medium and expand its application in borehole acoustic field, the shear stress of porous fluid and the heterogeneity of pore structure are considered. The wave theory in porous medium containing viscous fluid is deduced based on the Biot theory. The influence of porous medium parameters on slow shear wave is analyzed, and the dispersion and attenuation of elastic wave caused by shear stress balance in porous fluid under the influence of inhomogeneous pore structure are studied. The analytical solution of borehole acoustic field in porous medium containing viscous fluid is further derived. The phase velocity and attenuation of borehole mode waves in heterogeneous porous medium, and the waveform of borehole full wave are calculated. The influence of pore fluid viscosity on borehole full wave is analyzed. The results show that there are slow shear waves in the porous medium containing viscous fluid. The slow shear wave is characterized by low velocity and large attenuation. In heterogeneous porous medium, the balance process of shear stress related to slow shear wave not only leads to the dispersion and attenuation of fast P-wave, but also affects the propagation characteristics of borehole pseudo Rayleigh wave and Stoneley wave. In addition, the pore fluid viscosity has a great influence on the borehole Stoneley wave. The present work improves the physical mechanism of acoustic wave propagation in porous medium and provides theoretical guidance for the explanation and application of borehole acoustic waves in porous formations.

Modeling the Effect of Multiscale Heterogeneities on Wave Attenuation and Velocity Dispersion

Wave attenuation and velocity dispersion are of great significance to fluid evaluation in fluid-saturated rocks. Inspired by the double-porosity and cracked porous elastic wave theory, we develop a multi-scale heterogeneities elastic wave theory to better describe the attenuation characteristics caused by wave-induced fluid flow at various scales in saturated rocks. First, we introduce penny-shaped and narrow-shaped coupled cracks into double-porosity model. The potential energy function is given by the stress-strain relationship. The kinetic energy function and dissipation function are derived based on the generalized Biot theory. Next, we derive the multi-scale wave equations through the Lagrange equation and obtain three compressional waves and one shear wave. According to the novel wave equations, we work out the phase velocity and inverse quality factor in double-porosity and cracked media based on plane-wave analysis. The numerical simulation results show that there are four attenuation peaks in the whole frequency band, corresponding to mesoscopic fluid flow, two kinds of squirt flow and Biot flow respectively. Then, we investigate the effect of poroelastic parameters on wave propagation. It is found that porosity, inclusion size, crack aspect ratio and other parameters significantly affect the dispersion and attenuation. Finally, we compare experimental data and theoretical prediction. Moreover, the new elastic wave theory can degenerate into classical Biot’s theory, double porosity theory and pore-crack theory under certain conditions, which demonstrates the consistency of this method. The proposed theory can better predict and simulate the wave propagation by incorporating the effects of microscopic, mesoscopic, and macroscopic heterogeneities.

Dynamic analysis of beams vibrating on nonlinear poroelastic multi‐layered continuum

AbstractThe paper presents a framework for analysis of beams interacting with nonlinear‐poroelastic, layered continuums (e.g., clayey soils) when subjected to time‐dependent loads. The poroelastic layered continuum is characterized by a nonlinear‐elastic constitutive relationship that relates the secant shear modulus to the induced shear strain. The Biot's theory of consolidation is combined with a dynamic beam‐continuum interaction model to develop the analysis. The vertical consolidation settlement of the beam and the excess pore pressure in the porous continuum are assumed to be products of separable functions, and the extended Hamilton's principle of least action is applied to obtain the differential equations governing the inertial consolidation motion of the beam‐continuum system and the dissipation of excess pore pressure. An iterative numerical algorithm is used to solve these coupled differential equations following one‐dimensional finite element analysis in which the implicit Wilson‐Θ time integration scheme is used to obtain the time history of beam and continuum responses. The novelty of the framework is that it rigorously takes into account the nonlinear poroelastic soil‐structure interaction within a dynamic time‐integration framework with minimal computational resources. The characteristics of this newly developed model are illustrated through examples.

Effects of the fluid-solid interface conditions on the dynamic responses of the saturated seabed during earthquakes

In this paper, the effects of the fluid-solid interface conditions between seawater and seabed on the dynamic responses of the seabed during earthquakes are investigated. Based on Biot's porous media theory, a semi-analytical solution of the 1D transient response and an analytical solution of the 1D steady-state response of the saturated seabed under different fluid-solid interface conditions are given, respectively. Based on the transient response solution, it is found that the fluid-solid interface conditions have a significant effect on slow waves and a small effect on fast waves. Based on the steady-state response solution, it is found that the effects of the fluid-solid interface conditions on the displacement amplification factor and pore-water pressure vary under different incident wave frequencies. In the low-frequency region, the lower the interface permeability, the smaller the displacement amplification factor. In the high-frequency region, the lower the interface permeability, the larger the displacement amplification factor. Moreover, the fluid-solid interface conditions affect the dynamic response of the seabed with different permeability coefficients to different degrees. As the seabed permeability coefficient decreases, the effects of the fluid-solid interface conditions on the dynamic responses of the seabed gradually decrease.

Study on Sound Velocity and Attenuation of Underwater Cobalt-Rich Crust Based on Biot and BISQ Theories

A prediction model of the sound velocity and sound attenuation of underwater cobalt-rich crusts (CRCs) was established to solve the problem that it is difficult to predict the sound velocity in thickness measurements of cobalt-rich crusts. Based on Biot theory and BISQ theory, a simplified Biot and BISQ model was proposed for the prediction of the sound velocity and sound attenuation of CRCs by using the Kozeny–Carman (KC) equation. The models could calculate the sound velocity and attenuation by the porosity and detection frequency. Based on the physical and mechanical properties of CRCs, a similarity model of the sound velocity and sound attenuation of CRCs was made by using the similarity theory to solve the problem that it is difficult to measure the acoustic propagation characteristics of CRCs. The sound velocity and sound attenuation of CRC similarity models with different porosities were measured by an underwater transmission experiment and the results of the simplified model calculation and experimental measurements were compared. The results showed that the simplified Biot model was suitable for the CRC sound velocity prediction and the simplified BISQ model was suitable for the CRC sound attenuation prediction, which had a high prediction accuracy.

Excess pore water pressure and seepage in slopes induced by breaking internal solitary waves

Internal solitary waves (ISWs) can induce the dynamic response of the seabed when they propagate over the seabed. In particular, the interaction between ISWs and the seabed is complex on slopes. In our study, the breaking of ISWs was simulated by Large Eddy Simulation (LES), and the pressure induced by ISWs was analyzed. In addition, the excess pore water pressure in the seabed was calculated based on Biot's consolidation theory. We found that the ISW-induced pressure changed the polarity when they propagated over the slope. The change rate of polarity increased with an increase in slope angle. However, the change of polarity was not obviously related to the breaking mechanisms of ISWs. Two distinct seepage fields existed in the whole seabed with sloping topography. In the flat region and near the foot of the slope, pore water flowed upward under the action of the ISW-induced negative pressure. The pore water near the top of the slope flowed downward because the ISW-induced pressure turned positive. Then, the two different parts of the seabed faced different failure risks. This study provides a basis for evaluating the seabed stability influenced by ISWs and contributes to researching the transformation of seabed topography induced by ISWs.