Wave propagation analysis in a bonded transversely isotropic structure: an axisymmetric potential-based approach

In the present time, self-reinforced materials are the basic requirement for civil engineering construction. These constructions encounter surface wave propagation during earthquakes and similar disturbances. Therefore, the study of surface wave propagation in such material is of great significance. The present paper, in the light of practical situation like dams, canals etc. aims to study the effect of gravity on the propagation of Rayleigh-type surface wave in a self-reinforced semi-infinite medium bounded and loaded by an inviscid liquid layer. Secular equation for the propagation of Rayleigh-type wave has been derived in closed form. The effect of gravity, reinforcement and liquid loading on phase velocity of Rayleigh-type wave has been distinctly observed. Numerical computation has been carried out and graphical illustration is provided for analysis of the presented study. Moreover, the effect of reinforced semi-infinite medium is compared to the effect of reinforced free semi-infinite medium on the phase velocity of Rayleigh-type surface wave to unravel the reinforcement effect. The effect of presence and absence of liquid loading on self-reinforced semi-infinite medium for Rayleigh-type wave propagation is also analysed and depicted by means of graphs.

Purpose The purpose of this paper is to investigate the propagation of Rayleigh-type surface wave in a porothermoelastic half-space. This study addresses the impact of surface pores characteristics, specific heat, temperature, porosity, wave frequency, types of rock frame and types of pore fluids on the propagation characteristics of Rayleigh-type wave. Design/methodology/approach A secular equation is derived, based on the potential functions for both sealed and open surface pores boundary conditions at the stress-free insulated surface of the porothermoelastic medium. Findings Propagation characteristics (velocity, attenuation and particle motions) of Rayleigh wave are significantly influenced by boundary conditions (opened or sealed surface pores) and thermal characteristics of materials. Furthermore, the path of particles throughout the propagation of Rayleigh-type waves is identified as elliptical. Originality/value A numerical example is considered to examine the effect of thermal characteristics of materials on the existing Rayleigh wave’s propagation characteristics. Graphical analysis is used to evaluate the behavior of particle motion (such as elliptical) at both open and sealed surface of the porothermoelastic medium.

Transversely isotropic (TI) medium is a widely studied anisotropic solid medium in seismology. The numerical simulation of seismic wave propagation in TI media is an effective tool to analyze the mechanism of seismic waves in complex anisotropic media. In this study, we introduce a double-weighted Runge-Kutta discontinuous Galerkin (RKDG) method for numerically solving wave propagation problems in 3D TI media with surface topography. This method incorporates the discontinuous Galerkin spatial discretization with an explicit double-weighted two-step iteration time discretization. The local Lax-Friedrichs flux is used as the numerical flux in the formulations. This method can solve the first-order velocity-stress seismic wave equations including TI media as well as more general anisotropic media. Due to the large scale of 3D problems, the parallel technology is adopted. Regions with irregular boundaries are discretized into unstructured tetrahedral meshes. Numerical experiments for various anisotropic media including the vertical and tilted TI media are presented. The results demonstrate the effectiveness of the double-weighted RKDG method in wavefield simulations in 3D complicated anisotropic media.

Rayleigh-type wave propagation in incompressible visco-elastic media under initial stress

Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer

In this paper, we study the propagation of Rayleigh-type wave in a heterogeneous isotropic elastic layer with initial stress resting on a rigid foundation. Frequency equation is obtained in closed form. The frequency equation being a function of phase velocity, wave number, initial stress and heterogeneous parameter associated with the rigidity and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced by above stated parameters. In Numerical and graphical computation, the significant effects of distortional velocity have been carried out. Moreover, the obtained dispersion relation is found in well–agreement to the classical case in homogeneous isotropic layer resting on a rigid foundation.

Influence of Heterogeneity and Initial Stress on the Propagation of Rayleigh-type Wave in a Transversely Isotropic Layer

This study uses ultrasonic physical modelling to test the accuracies of numerical calculations of traveltimes and conversion-point (CP) positions for P-SV wave propagation in a horizontal transversely isotropic (TI) medium. Study results show that the traveltimes and CP positions for P-SV wave propagation on the isotropic plane of a TI medium computed using Fermat’s minimum-time principle are the same as those of using the isotropic non-hyperbolic moveout equation and the isotropic CP equation. However, for P-SV wave propagation on the symmetry-axis plane of a TI medium, the arrival times and CP positions of SV-waves are difficult to determine by any ray methods when the propagation directions of SV-waves are within the cuspoidal SV-wave group velocities zone. But the first arrival times and the propagation of the dominant energy of P-SV waves can still be analysed by ray methods. Based on the calculation of Fermat’s minimum-time principle, if the source-receiver offset is greater than a critical distance, the reflection angles of the converted SV-waves are fixed at a specific angle with a local maximum SV-wave group velocity of the neighbourhood area. This is because the converted SV-waves prefer to propagate along the cuspoidal directions with larger amplitude and higher velocity. Verified by the physical modelling, the Fermat’s minimum-time principle used to calculate traveltimes of P-SV waves is better than the anisotropic non-hyperbolic moveout equation. The physical modelling for the CP position experiment can give a clearer visualisation of the variations of CP positions in the profile, and the feasibility of using Fermat’s minimum-time principle to determine CP positions is also better than that of the anisotropic CP equations. Therefore, in the seismic data processing, Fermat’s minimum-time method is recommended to accurately determine the arrival times and CP positions of P-SV wave propagation in TI media.

Based on a particle lattice model, a dynamic lattice method is proposed to simulate seismic wave propagation in transversely isotropic (TI) media with tilted symmetry axis (TTI media) in the presence of free surface topography. Different from other wave equation based numerical methods, the dynamic lattice approach calculates the micro-mechanical interactions between particles in the lattice instead of solving the wave equation. In the case of TI media, it is a challenge to find the correct particle lattice model which can reflect the anisotropic nature of TI media. Our study reveals the theoretical connection between TI medium and the particle lattice model, allowing us to model elastic seismic waves in TI media. On the other hand, the particle feature of this method makes it convenient to incorporate free surface topography. To achieve this, we only need to remove the particles above the surface topography from the lattice. The free surface boundary condition is automatically implemented through the interactions between particles near the free surface. Numerical results demonstrate the effect of our method in simulating elastic waves in TI media with free surface topography.

The effect of frictional boundary on the propagation of Rayleigh-type wave in an initially stressed inhomogeneous fiber-reinforced layer overlying an initially stressed homogeneous semi-infinite medium has been analyzed by an approximate analytical method. A realistic model has been considered for sliding boundary friction at the interface. The frequency equation has been obtained in closed form. The substantial effects of various affecting parameters, viz. reinforcement, inhomogeneity, bonding parameter, spectral decay parameter, and horizontal initial stress on phase and damped velocity have been discussed graphically in detail. The remarkable observation has been obtained through the comparative study in the presence and the absence of reinforcement in the layer.

The present article undertakes the study of propagation of SH-wave and Rayleigh-type wave in a layered structure with a layer overlying a semi-infinite medium composed of distinct initially stressed exponentially graded fiber-reinforced viscoelastic materials in two separate cases. Adopting analytical treatment closed form of complex frequency equation is obtained for both the surface waves. The real part of complex frequency equation imparts dispersion equation whereas imaginary part accounts for the damped velocity equation. Dispersion and damped velocity equations for both the surface waves are in well-agreement with the classical results. Also as a special case of the problem, the deduced results are validated with the pre-established results. Numerical computation of the analytical finding has been accomplished and manifested through dispersion curves and damped velocity curves for both SH-wave and Rayleigh-type wave in order to analyze the influence of various affecting parameters, viz., initial stress, reinforcement, exponential gradient, volume and shear viscosity on their phase and damped velocities. Moreover, comparative study has also been carried out for various distinct cases associated with the considered layered structure for both the waves, which serves as a salient feature of the present study.

The objective of this study is to develop a theory to study the propagation of Rayleigh-type waves in an inhomogeneous layer having yielding surface. A detailed study of a Rayleigh-type wave propagating in an exponentially graded incompressible layer resting on yielding surface is considered. The frequency equation being a function of phase velocity, wave number and heterogeneity parameter associated with the yielding parameter and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced by the above-stated parameters. In particular cases, the dispersion relation has been discussed for stress-free foundation by taking yielding parameter tending to zero. In numerical and graphical computation, the significant effects of distortional velocity have been carried out. Moreover, the obtained dispersion relation is found in well agreement to the classical case in homogeneous isotropic layer resting on a yielding foundation.

Rayleigh-type wave propagation through liquid layer over corrugated substrate

ABSTRACTThe present study investigates the propagation of Rayleigh-type wave in a transversely isotropic viscoelastic layer in effect of yielding foundation and rigid foundation in two different cases for two considered models. Numerical computation and graphical demonstration have been carried out for the case when layer is comprised of transversely isotropic viscoelastic Mesaverde clay shale material (Model I) and simply isotropic viscoelastic material (Model II). Closed-form expression of phase velocity and damped velocity for both the cases are deduced analytically. Obtained result is found in well agreement to the established standard results existing in the literature. Significant effect of dilatational viscoelasticity, volume viscoelasticity and yielding parameter on phase and damped velocities for both the considered models has been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the context of the present problem. Moreover, comparison of phase and damped velocities for the case of layer with stress-free foundation, layer with rigid foundation and layer with yielding foundation serve as a major highlight of the present work.

Shale has been usually recognized as a transverse isotropic (TI) medium in conventional geomechanical log interpretation due to its laminated nature. However, when natural fractures (NFs) exist in the rock body, additional elastic anisotropy can be introduced, converting Shale to an orthorhombic (OB) medium. Previous study illustrates that treating the naturally fractured shale rock as a TI medium by ignoring the NF-induced anisotropy can cause the erroneous estimation of the geomechanical properties (i.e. Young’s modulus, Poisson’s ratio, brittleness index, and etc.) and in-situ stress. In this paper, the study is extended to quantify the impact of NF-induced elastic anisotropy on completion and frac designs in different actual shale reservoirs in U.S. Published acoustic log data from five different shale formations (Bakken, Marcellus, Haynesville, Eagle Ford, and Niobrara) are collected and examined to determine their availability to generate the stiffness tensor of the representative TI background rock of each Shale reservoir. Natural fractures with different intensity values from 0 to 10 per foot, with shear wave splitting ranging from 0-5%, are introduced in the TI background rock to create the corresponding OB rock stiffness tensor. The OB stiffness tensors of different shale cases are finally converted back to the compressional and shear acoustic signals, which can be interpreted based on the TI or OB assumptions. The final output elastic moduli and in-situ stress results interpreted from different assumptions are compared, and the impact of NF-induced elastic anisotropy on completion and fracturing designs is quantified and fully understood for different shales. The results show that the higher the natural fracture intensity within the shale rock body, the outcomes interpreted from TI and OB models are more deviated from each other. In addition to that, the impact of natural fracture induced anisotropy on geomechanical log interpretation is different in different shale reservoirs. Specifically, the magnitudes of Young’s modulus are overestimated for all five shale when ignoring natural fracture induced anisotropy in log interpretation. The overestimation is different for different layers of a single shale formation as well as different shales. Similarly, the magnitudes of the minimum horizontal stress are also overestimated by different extents for different shales. Moreover, ignoring natural fracture induced anisotropy leads to incorrect interpretation of stress contrast. The stress depth profiles of all five shales are identified for upper, middle, and lower zones. The stress difference between upper and middle zones (upper stress contrast) and between middle and lower zones