A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space

https://doi.org/10.1016/j.enganabound.2021.11.012Get rights and content

Abstract

A new indirect boundary integral equation method (IBIEM) is developed to solve the seismic wave scattering problem in a three-dimensional fluid-saturated layered half-space. Based on Biot's theory, the Green's functions of inclined circular loads in porous elastic layered half-space are firstly deduced. According to the single-layer potential theory, when establishing the scattered wave field, the uniform surface loads and fluid sources are located directly on the boundary surfaces of the scatterers, so as to avoid determining the optimal location of fictitious wave sources surface. At the same time, the radiation condition of wave in semi-infinite layered media can be accurately realized by using the dynamic Green's function of concentrated load, and the computational memory can be greatly reduced. The scattering of seismic waves by a canyon topography in saturated layered half-space is examined in detail. The numerical results indicate that with the increase of the porosity of the overlying soil, the amplitude of surface displacement can be amplified by 1∼6 times, and the amplification effect is more significant near the corner of the canyon, which can be attributed to the superposition of wave diffraction effect and resonance amplification effect of saturated soil layer.

Introduction

The fluid saturated porous media, as a material with solid skeleton and porous fluid, exist widely in nature and engineering. According to Biot's theory [1], duo to the dynamic interaction between the solid skeleton and porous fluid, the analysis based on saturated porous media is more scientific and reasonable than that based on single solid medium. It is of high theoretical and practical value to study the seismic wave scattering caused by heterogeneities such as surficial cavity, crack or sedimentary valley in saturated two-phase media. Thus, in recent decades, the related problems have received considerable attention in geotechnical engineering, seismic engineering, geophysical exploration and other fields. In specially, the seismic amplification effect by local sites can be explained and quantified by the seismic wave scattering theory.

For the wave scattering in a poroelastic media, at present, the research methods mainly include analytical methods and numerical methods. The analytical methods mainly refer to wave function expansion methods, has been used by several scholars to study the scattering of seismic waves in a saturated media [2], [3], [4], [5], [6], [7], [8], [9]. However, the analytical methods are only suitable for solving relatively simple operations, and it is difficult to solve complex partial differential operators and boundary conditions in reality.

Compared with analytical solutions, numerical methods can more easily deal with complex media problems. Methodologies can be divided into domain-type methods, boundary-type methods, and hybrid methods. The finite element method [10], [11], [12], [13], [14], [15] and the finite difference method [16,17] are two typical domain-type methods, which are capable of addressing wave motion in complex geometry and heterogeneous material more conveniently. However, two key challenges of domain-type methods need to be addressed: (1) the high-frequency numerical dispersion [18]; and (2) For infinite layered poroelastic half-plane, the exact satisfaction of no-reflection boundary conditions is a significant challenge as of yet [19].

The boundary-type methods have several advantages, such as dimension reduction, automatically fulfilling the Somerfield radiation conditions, no high-frequency dispersion, and the same order precision for the stress and displacement, etc. Thus the boundary-type methods have been extensively used in wave motion simulation in an infinite domain. This method can be divided into boundary element method [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], boundary integral equation method [29], [30], [31], [32], [33], [34], [35], Aki-Larner method [36,37], and discrete-wave number method [38], etc.

By using the boundary method, previous scholars have made many important achievements in the study of the effect of local site on ground motion in saturated soil, but most of these studies assume that these local site are located in saturated full or half-space. This assumption brings convenience to theoretical analysis, but neglects that due to different sedimentary ages, near-surface soil layers are stratified. After the 1995 Osaka-Kobe earthquake in Japan, researchers found that the saturation of shallow soil layers is an important factor affecting site amplification [39]. In recent years, more and more construction projects have been built in saturated soil regions. It is of great theoretical significance and application value to study the local site effect and seismic wave scattering in a fluid saturated layered half-space.

However, the studies on seismic response in saturated layered sites are very limited. To the authors’ best knowledge, by using the boundary element method, Liang, et al. solved the scattering of plane elastic waves by a 2-D topography in a poroelastic layered half-space [40,41]. Furthermore, Ba, et al. used 3-D indirect IBEM to investigate the scattering of incident plane SV-waves by an alluvial valley embedded in a saturated layered half-space [42]. Liu et al. studied the 2-D elastic wave scattering in a fluid-saturated poroelastic layered half-plane by the method of fundamental solution [43].

But it's important to note that, to avoid treating the singularity of Green's function, in the 2-D case of Liu et al. [43], the fictitious wave surfaces were located at some distance from the boundary surfaces. However, the location and density of the fictitious wave source, which affected by the incident frequency, the incident wave type, as well as the shape of scatterers, need to be carefully determined, which limits its broad application in actual engineering problems with complex geometry.

To overcome above-mentioned disadvantages, Liu et al. further developed the IBIEM by applying fictitious surface loads directly on the boundary of a scatterer to construct the scattered wave field, but they only studied the elastic wave scattering in a single phase layered half-space [44]. On the basis of paper [44], this paper further develops this method to solve 3-D seismic wave scattering in a fluid-saturated poroelastic layered half-space. The uniform loads and fluid source are loaded on a circular surface to construct the 3-D scattered wave field in a multi-layered fluid-saturated half-space. In addition, the Green's function for a poroelastic layered half-space is used to omit the element discretization on the layer interfaces and the free surface, which greatly reduce the memory requirement and makes the calculation simpler and more convenient.

In the next section, the calculation procedure of IBIEM for the 3-D seismic wave scattering in a poroelastic layered half-space is illustrated in detail. And the corresponding Green's function and Biot's theory are also introduced. In Section 3 the accuracy is verified by comparing with available solutions and the numerical convergence of this method is also confirmed. In Section 4, the scattering of plane P1-, SV-waves by a 3-D canyon in a poroelastic layered half-space is solved using this methodology, with several important conclusions presented in the last Section.

Section snippets

IBIEM for 3-D wave scattering problem in a poroelastic layered half-space

As shown in Fig. 1, several irregular local sites, such as canyon, cavity, inclusion, etc., exist in a poroelastic layered half-space. Taking this model as an example, the applicability of IBIEM for solving the 3-D poroelastic layered irregular sites is calculated. The total wave filed is firstly divided into the free field and the scattered wave filed. Herein, the free field is considered as the incidence of plane waves from the bottom of the half-space. Then, to construct the scattered waves

Verification of accuracy and stability of the new IBIEM

The scattering of elastic waves by a semi-spherical canyon in a poroelastic layered half-space are tested by the proposed IBIEM based on the poroelastic layered model. Available results are utilized to verify the accuracy and convergence of present IBIEM.

Numerical results and discussion

In this part, detailed parameters analysis will be presented for the scattering of P1-, SV-waves by a 3-D canyon in a poroelastic layered half-space (Fig. 5). The properties of the poroelastic medium are set as follows: Poisson's ratio ν=0.25; critical porosity ncr=0.36; critical bulk modulus of the solid skeleton kcr =200 MPa; bulk modulus of the solid grain kg =36,000 MPa; bulk modulus of fluid kf =2000 MPa; mass density of the solid grain ρg=2650 kg/m3; mass density of pore fluid ρf=1000 kg/m

Conclusions

In this paper, a new efficient indirect boundary integral equation method (IBIEM) is proposed, to solve the scattering of seismic wave by 3-D local site in a fluid-saturated poroelastic layered half-space. The scattering wave fields are constructed by applying uniform surface load and fluid sources on the virtual wave source surface. This new method has several key advantages in dealing with the problem of saturated layered ground motion: (1) The fictitious loads can be located directly on the

Declaration of Competing Interest

The authors declare no conflict of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grants (51878434, 51878108, 52108466), Key projects of Tianjin science and technology support program (17YFZCSF01140).

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