Saturated Half-space Research Articles

Plane SV-waves scattering by seawater convex surface topography and underlying lined tunnel in a saturated half-space

Analytical solutions of boundary-valued problems for aboveground topography and underground buried structures, especially for sites with overlying seawater, have always been challenging. Based on the region decomposition technique (RDT) and Hankel integral transform method (HITM), SV-waves scattering by the convex circular topography with seawater and underlying tunnel in a saturated half-space is solved. The primary problem is tackling additional scattered waves introduced by the water–soil interface and convex topography, while ensuring boundary conditions are all met. To this end, scattering wave potential functions in cylindrical coordinates are first transformed into the Hankel integral form in Cartesian coordinates. This greatly facilitates the straightforward application of free boundary conditions at the soil–seawater flat interface, rather than making a large circular arc approximate assumption, which, in turn, avoids the resultant accumulating errors. Secondly, scattering problem to be solved is mathematically and analytically formulated as solving a 5 × 5 linear equation system comprised of wave functions inside the convex topography and lining tunnel while satisfying all physical continuous boundary conditions. This is the main innovation and resulting consequence from the theoretical derivation process. Finally, validation of numerical examples between this paper, existing studies and theoretical analysis further demonstrates the reliability and general adaptability of solutions. Parametric studies of the incident SV waves are conducted to investigate the spatially varying seismic response. The effects of SV-waves incident angles and frequencies, tunnel burial depth, tunnel lining thickness, saturated soil porosity, and seawater depth are investigated and analyzed in detail. This study provides a feasible scheme of investigating the dynamic response of complex submarine sites, which is of great significance for both theoretical value and engineering purposes.

A comparison of two numerical models for modeling a twin tunnel in a saturated half-space

In this work an analytical as well as a numerical model are established for computing vibrations induced by a twin tunnel buried in a saturated poroelastic half-space. In the analytical solution the mechanical response for the half-space ground is expressed as a superposition of planar waves from the ground surface and cylindrical waves from two tunnel-soil interfaces. While in the numerical solution, three-dimensional saturated poroelastic finite element and multi-transmitting formula are formulated. The comparisons are made between the two methodologies, and the advantages and disadvantages of the two models are highlighted. Using the analytical one, the influence of the fluid in the soils, the reflection effect of the ground surface and the existence of a neighboring tunnel with or without a source on the mechanical response is investigated. By using the three-dimensional numerical model, the influence of various irregular tunnels on the surface vibration is investigated. The results suggest that: for the displacement and pore pressure response, the difference between one source tunnel and twin source tunnel can be up to 20 dB; the rectangular tunnels lead to the largest response followed by horseshoes and circular ones.

3D elastodynamic analysis of transversely isotropic laminated water-saturated subsoils under harmonically moving loads

This article proposes an analytical layer element methodology for transversely isotropic (TI) laminated water-saturated subsoils to a harmonically moving load over the media surface. Biot’s elastodynamic equation is decoupled by the double Fourier transform combined with the Heaviside step function of the load source. Exact 3D dynamical stiffness matrices of the soil layer element and the TI saturated half-space are established based on the derived displacement and stress components in the transform domain. Assembling the stiffness matrices of the discrete layer elements and underlying half-space yields the total 3D dynamical stiffness matrices. Considering the element boundary conditions, the analytical layer-element solutions in the wavenumber domain are obtained. The space domain result is retrieved by the inverse Fourier transformation. The results have a good consistency with the current solutions. The effects of the soft soil layer thickness, the material anisotropy, stratification characteristics, the load frequency, and the load moving velocity are analyzed.

Seismogram Synthesis of Multi-Scale Layered Transversely Isotropic Saturated Half-Space Using a Revised Stiffness Matrix Method

Seismogram synthesis for a multi-scale transversely isotropic (TI) saturated layered half-space subjected to seismic dislocation source is studied by a revised stiffness matrix method in this paper. First, based on Biot’s theory of wave propagation in a fluid-saturated porous solid, the governing equations of motion in TI saturated medium are transformed by Fourier–Hankel integral transform. Next, the contribution of the seismic dislocation source can be imposed in the form of body forces. Then, the global stiffness matrix with well-conditioned properties is obtained by assembling each layer’s exact stiffness matrix, which can avoid accumulating errors in the matrix transmission. By extracting the divergence exponent terms of the stiffness matrix concerning thickness and wavenumber, the bottleneck of simulating the wave propagation process in multi-scale half-space, where the layer thickness and velocity vary sharply from the near-surface region to the crustal zone, could be overcome, and a fast and accurate seismogram synthesis could be achieved. The accuracy of the proposed method is verified by comparing the calculated results with those of published literature. Finally, the method is applied to the multi-scale system with realistic superficial fine layers. Numerical examples regarding thickness, TI properties and porosity are presented to illustrate the effects of superficial layers on ground motion. The numerical results show that the surface properties affect the wave propagation and energy distribution, leading to variable impacts on the surface response in different directions; this phenomenon is more pronounced in the case of high-frequency ground motions.

1-D Finite Element Artificial Boundary Method for Saturated Layered Half-Space Site Response under Obliquely Incident P-SV Waves

A novel finite element scheme is derived to evaluate the transient dynamic response of a multi-layered saturated porous half-space due to obliquely incident P or SV wave. The system consists of N − 1 layers with different thickness and properties that overlay on a saturated half-space from which the wave is input. Biot theory is utilized to describe the mechanism of saturated poroelastic media. In this method finite computational area is truncated from the infinite domain by a horizontal artifitial boundary introduced in the underlying half-space. Wave inputting effects and absorbing boundary condition at the bounday are derived. The spatial problem is then transformed approximatedly into a one-dimensional problem based on the Snell's law. The transient 1-D problem is descritized spacially by means of FEM and solved by the standard time intergration algorithm. Numerical examples show that the proposed 1-D finite element artificial boundary method can provide accurate results for seismic performance of saturated layerd half-space under obliquely incident waves with low values of permeability or frequency content. The method is applied in site response analysis of Hong Kong-Zhuhai-Macao immersed tunnel project. By comparison with the analytical solutions, it is demonstrated for geotechnic earthquake engineering problems, site response solutions with satisfactory accuracy can be achieved by the proposed method.

A semi-implicit material point method for coupled thermo-hydro-mechanical simulation of saturated porous media in large deformation

The coupled thermo-hydro-mechanical (THM) response of liquid-infiltrated porous media underpins the safe operation and maintenance of key engineering infrastructure. Challenges remain in modeling and understanding the complicated multiphysics processes of porous media subjected to THM loads and undergoing large deformation. In this study, we develop a stabilized material point method (MPM) for modeling the THM responses of large deformation problems in biphasic solid–fluid mixtures. A novel and efficient staggered solution scheme is proposed to solve the governing equations of the coupled system formulated in terms of four primary variables: solid displacement (u), liquid velocity (v), pore pressure (p), and temperature (T). The scheme solves the energy balance equation first and employs the resulting temperature to further advance the calculation of the momentum and mass balance equations using a semi-implicit fractional step method to facilitate equal-order interpolations. Both the incompressible and weakly compressible fluid are considered in the presented fractional step formulations. We also develop the axisymmetric form of the coupled MPM to increase the applicability and efficiency of the method in THM problems. The validity, stability, and robustness of the proposed method are demonstrated through three benchmark problems, including the heating of a saturated half-space, the non-isothermal consolidation of a soil column, and a three-dimensional axisymmetric problem pertaining to the thermoelastic response around a point heat source. The predictive capability of the proposed method for large deformation problems is further showcased by the simulation of the progressive failure process of a thermal-sensitive slope.

Surface motion of P/SV wave scattering by a semicircular canyon in saturated half-space by complex function and conformal mapping

Based on the complex function theory of elastic dynamics, a complex function solution of P/SV-wave scattering by a semicircular canyon in an elastic porous saturated half-space is obtained. Firstly, through Biot theory and Helmholtz decomposition, the wave potential functions with unknown coefficients in the model are obtained based on the wave function expansion method. Secondly, the complex function expressions of stress and displacement in the model are obtained by using the complex function method and conformal mapping. According to the boundary conditions of the horizontal ground surface and semicircular canyon surface, this boundary value problem can be converted into a series of algebraic equations and solved numerically by truncation. Then, the correctness of the complex function solution in this paper is verified by comparing it with the published results through degenerating the scattering problem in the saturated porous elastic half-space in this paper into that in the elastic half-space. Finally, the influence of the boundary drainage conditions, the medium porosity, and the incident angle and frequency of the incident wave on the ground surface displacement is investigated. Although the P/SV-wave scattering problem is a traditional problem, there is currently no accepted analytical solution that strictly satisfies the zero-stress condition on the horizontal ground surface. This paper is an attempt at this problem, which may provide insight into the search for the strictly analytical solution for P/SV-wave scattering.

Surface Motion for P-Wave Scattering by an Arbitrary-Shaped Canyon in Saturated Half-Space

This paper obtained a semianalytical solution for the P-wave scattering problem by an arbitrary-shaped canyon in a saturated half-space by using Biot’s theory, the wave function expansion method, and the moments method. Firstly, based on the Biot fluid-saturated porous media theory and the wave function expansion method, the wave potentials which automatically satisfy the zero-stress boundary condition on the surface of the half-space are obtained. Then, the boundary value problem is transformed into an algebraic problem by the method of moments according to the boundary conditions, and then solved numerically by truncation. By adjusting the parameters, the saturated medium in the original model approximately degenerates into a single-phase elastic medium, and the correctness of the proposed method is verified by comparing it with published results. Finally, the effects of the canyon shape, the porosity of the soil, and the incidence angle and frequency of the incident wave on the amplitude of ground surface motion are investigated. The results show that the incidence angle has a significant effect on the ground surface motion, while the porosity of the soil has little influence on the amplitude of surface motion. The influence of canyon shape on surface motion is mainly reflected in the shielding effect on the incident wave. Although P-wave scattering by a canyon is a traditional problem, most of the analytical solutions are limited to solving the scattering of seismic waves in geometric regular canyons. However, the actual shape of the canyon is not regular, which limits the application of closed analytical solutions in practical engineering. In this paper, the scattering of P-waves in arbitrary-shaped canyons is successfully solved by using a semianalytical method combined with a numerical method-moments method, which provides a possibility for engineering application.

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.

Semi-Analytical Solution for the Vertical Vibration of a Single Pile Embedded in a Frozen Poroelastic Half-Space

The theory of vertical pile vibration is the essential basis for pile integrity determination and dynamic analysis. The pile vibration characteristics are inevitably affected when the surrounding soil is frozen due to the low environmental temperature. Therefore, the investigation of pile vibration considering the surrounding soil as a saturated frozen porous medium is of great importance. In this paper, an analytical model for vertical pile vibration was established by employing the theory of composite saturated porous media, that is, by simplifying the upper frozen soil layer as a homogeneous isotropic saturated frozen porous medium and the foundation soil beneath the pile as an elastic half-space subjected to the motion of a rigid disk. By employing the integral transform and variable separation method, analytical solutions for the proposed model were derived under the three-dimensional axisymmetric condition. The analytical model and its solutions were verified by comparing them with the existing solutions for an end-bearing pile embedded in a homogeneously frozen soil layer as well as for a saturated half-space. A parametric study was conducted by utilizing the proposed solutions, and the results indicated that the pile bottom stiffness, the freezing temperature, the shear modulus of the unfrozen soil, etc., had a significant influence on the dynamic responses of the vertical pile vibration in both the frequency and the time domains.

Theoretical investigation on characteristics of field reactions of saturated ground subjected to vibrations of inclined pile groups

Inclined pile groups have been adopted in engineering practice (e.g. offshore wind turbines) to withstand strong horizontal environmental loads that are repetitive in nature. Field reactions including the additional stresses and excess pore water pressures are responsible for the cyclic degradation of foundation soil, which may induce the accumulative deformation and fundamental frequency shifting of the marine structure. In this study, a theoretical model incorporating an inclined pile group and a saturated half-space was established, wherein the pile and saturated soil were described by the E–B beam and Biot poroelasticity theories, respectively. The pile and soil were coupled through the receptance matrix, which was established at the pile–soil interaction points in the transform domain. The dynamic impedances of the inclined pile group are first obtained, and then the field responses, including stresses and excess pore water pressure in the saturated ground, are evaluated for different types of loads applied to the pile cap. The frequency response function and spatial distributions of the field responses were obtained for varying rake angle of pile and permeability of soil. It was found that the horizontal force induces the largest responses in the ground when compared to the vertical force and the overturning moment applied to the inclined pile group. It is beneficial to employ an inclined pile group to withstand the horizontal force and overturning moment because the foundation stiffness increases, and the additional stresses in the ground decrease with increasing rake angle.

Analytical approach to scattering of P, SV waves by twin tunnels in saturated half-space

Analytical approach to scattering of P, SV waves by twin tunnels in saturated half-space

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.

Scattering of plane SV-waves by a circular lined tunnel in an undersea saturated half-space

Considering the fluid-soil coupling of saturated soil and seawater-saturated soil-structure dynamic interaction, the Hankel function integral transformation method is employed herein to obtain an analytical solution to the scattering and diffraction problem posed by the plane SV-waves in an undersea lined tunnel. This study is based on the porous medium wave function theory of saturated soil proposed by Biot and the ideal fluid medium wave theory. The impact of factors, such as the incident angle, frequency, water depth, buried depth of tunnel, porosity, and Poisson's ratio of the saturated soil, on the horizontal and vertical displacement at the seawater–saturated soil interface, and on the pore water pressure and total circumferential stress at the saturated soil–tunnel interface, are computed and analyzed.

Seismic analysis of a lined tunnel in a multi-layered TI saturated half-space due to qP1-and qSV-waves

A special indirect boundary element method (IBEM) is presented to study the in-plane scattering by a lined tunnel embedded in a multi-layered TI saturated half-space. By integrating the full-space and half-space Green’s functions for the inclined distributed load and pore pressure, these singular-free kernels enables the method to treat an arbitrarily shaped tunnel with a superior flexibility and high efficiency in comparison to the traditional IBEM. For the total system, the substructure process is implemented to decompose the model into a layered TI saturated half-space region and a lining region filled with isotropic elastic medium. The dynamic stiffness matrix method developed by the authors is adopted to solve the free field of the exterior half-space, and the scattered fields existed in the exterior and interior lining regions are simulated by the half-space and full-space Green’s functions in a single-layer potential representation, respectively. Then the total wave-fields in the half-space region can be expressed as the superposition of free-field and scattered field, and the tunnel region merely contains the scattered field. The compatibility conditions on inner and outer interfaces enclosing the lining are satisfied to couple these sub-regions to the original one. The present method is calibrated by comparison with available limited cases. Displacement and dynamic hoop stress concentration at the inner surface of the lining are calculated in the frequency domain to portray the effects of anisotropy degree, drainage condition, incident angle, frequency and layering on the seismic response. Results show that the decrease of anisotropy degree generally leads to larger amplitudes of surface displacement and the stress. Prominent amplification of displacement and stress can be observed for the two-phase medium model in comparison to the one-phase model although the vertical displacement is hardly affected by the drainage condition at the lining-rock interface. When taking the layering into account, the vertical displacement maximum is increased by about 32%, while the DHSC amplitudes at different cases tend to become larger on the whole compared to those for the homogeneous half-space.

Scattering of plane waves by a 3D canyon in a transversely isotropic fluid-saturated layered half-space

Scattering of plane waves by a three-dimensional (3D) irregularity in a transversely isotropic (TI) fluid-saturated layered half-space is investigated for the first time. The derived exact stiffness matrix is applied to solve the dynamic response of free fields. The scattered fields, which arise from the presence of canyon irregularity, are described by applying dynamic Green's functions for uniformly distributed loads and pore pressure acting on an inclined plane, that is, the special indirect boundary element method (IBEM). The accuracy of the proposed method is verified via the comparison between the calculated results and those of published literature. By taking a semi-spherical canyon cut in a homogeneous, single-layered, and three-layered TI saturated half-space as examples, the influences of material anisotropy, boundary drainage condition, and sedimentary sequence on dynamic response are investigated. The results show that the material anisotropy and boundary drainage conditions all have a significant impact on the seismic responses, which is highly dependent on the waves type, incident angle, and frequency. For the layered medium, the displacement amplification is affected by the soil layer properties and sedimentary sequence.

HVSR analysis of a layered saturated half-space using diffuse-field theory

SUMMARYThe recently constructed diffuse field theory from isotropic energy equipartition has been well developed in elasticity for full-wave interpretation of horizontal-to-vertical ratio (HVSR), which links the signal autocorrelation with the imaginary part of Green's function. Here, the theory is extended to the saturated layered medium within the framework of Biot's theory to account for the offshore environment. The imaginary parts of Green's functions are obtained using direct stiffness method accompanied with Fourier–Hankel transform. In particular, the upgoing wave amplitudes are modified to tackle the overflow during wavenumber integral and allow for fast calculations. After validating the method from the perspectives of Green's function calculation, emphasis is laid on evaluating the inaccuracies of HVSR calculation induced by model misuses in the lack of prior geological and geotechnical information. The numerical results considering the effects of layer sequence, impedance ratio, porosity and drainage condition show that the predominant frequency of the one-phase medium is slightly less than the two-phase medium with the maximum shift no more than 0.1 Hz, while their amplitude differences can be prominent as impedance ratio and porosity increase, with the maximum difference up to 29 per cent. The shallowest soft layer has the dominant effects on HVSR amplitudes, whereas the buried low-velocity layer at depth over one-wavelength contributes little to the peak amplitude. Finally, the method is applied to a realistic case at Mirandola, Northorn Italy, which suffered extensive liquefaction-induced damages in 2012 Emilia earthquake. The well identified pattern of the experimental HVSR using the two-phase medium model illustrates the application potential of our method to further assist the subsurface geology retrieval.

Two-dimensional scattering of plane waves by irregularities in a multi-layered transversely isotropic saturated half-space

Wave scattering of plane waves by irregularities in a transversely isotropic (TI) saturated half-space has been rarely studied. Based on Biot's theory, this paper used an indirect boundary element method (IBEM) to solve the two-dimensional (2D) wave scattering of incident plane qP1- and qSV-waves by irregularities in a layered TI saturated half-space. 2D dynamic Green's functions for uniformly distributed loads as well as pore pressure acting on internal inclined line in a layered TI saturated half-space are derived. The method was verified by comparing with two published results for isotropic medium. The canyon, partially filled alluvial valley and fully filled alluvial valley with trapezoidal cross-section are covered in numerical calculations. Results show that the dynamic responses of the three types of irregularities are significantly different. The peak surface displacement amplitude of the fully filled alluvial valley case is up to approximately twice as much as that of the canyon case. The influences of TI property on the three irregularities are apparent, and the soft sedimentary material is more sensitive to TI property. Besides, the ground motion of the local irregularities strongly affected by the incident angle and frequency of plane wave.

Effects of water table on ground-borne vibration screening effectiveness by using open trenches

The seasonal variations in groundwater table would significantly change the dynamic responses of the ground and the screening performances of wave barriers. This paper develops a two-dimensional finite element model based on COMSOL to describe the propagation of waves in a layered ground with open trenches. This layered ground is modelled as a dry layer resting on a saturated half-space. The influence of the moving groundwater table on isolation efficiency of barriers is then investigated, considering the infiltrated water in trenches. The effects of soil hydraulic properties and geometrical parameters of the trench are also analyzed respectively. Results show that a low value of the permeability coefficient of saturated half-space may have a significantly detrimental effect on the screening performance due to the low frequency behaviors of the medium. Placing a trench in vibration-strengthening (weakening) zones results in a better (worse) isolation effect, and with a critical water table (Hw,c), increasing trench depth and width can hardly achieve a satisfactory screening efficiency (75% reduction) due to the occurrence of resonance in the dry layer. In addition, a multiple trench isolation system is simulated in the condition of high water levels and its shielding performance is compared to the single trench barrier. A triple trench barrier is found to be a preferred alternative because it can achieve a much better screening efficiency. Specially for Hw,c, a multiple trench barrier can effectively eliminate the adverse effect of the resonance and achieve more than 75% reduction of ground vibration.

Vibration of layered saturated ground with a tunnel subjected to an underground moving load

The transmission of vibrations within multi-layered saturated ground subjected to an underground moving load in a tunnel is investigated theoretically. A two-dimensional model of a layered saturated half-space with an embedded beam is established first. The beam simulating the tunnel is located between two horizontal soil layers with a moving point load acting upon it. The half-space soil is assumed as Biot’s poroelastic medium and the transmission and reflection matrices (TRM) method is applied to consider the wave propagation in the layered soil. Combined with the surface boundary conditions and the continuity conditions between the soil layers and the beam, the governing equations of the ground-beam coupled system are solved by the Fourier transform. Finally, the dynamic response of the layered saturated ground in the time-space domain is obtained by using the fast Fourier transform (FFT) method. Three different soil cases are selected to study the influences of the soft/hard interlayer on dynamic response. The displacement as well as the velocity and acceleration of the surface vibration are analyzed. The displacement amplitude spectra at different depths are presented to find the attenuation law of the ground vibration along depth. The numerical results show that both the stiffness and the embedded depth of the interlayer have impacts on the critical velocity of the coupled system and the surface vibration. When the vibration propagates from the tunnel to ground surface, the high-frequency components decrease and the low-frequency components increase.