Velocity-profile of torsional surface wave in a contorted watery-porous reservoir over a sand-dune deposit

Highlights

• Velocity-profile of Torsional wave in a porous layer over sand-dune deposits have been found.

• Single-step and double-step irregularity greatly influence phase velocity.

• Heterogeneity varies as sinusoidal function, linear function, and cosine function of depth.

• Biot's parameter, Initial stress in the layer, and the half-space affect the velocity marginally.

• Higher magnitude velocity has been observed in RT irregularity compared to the RP interface.

Abstract

The current study aims to study the velocity profile of torsional surface waves in a three-layer model consisting of sand-dunes deposit sandwiched between the watery-porous layer and the orthotropic half-space. The top and the middle layers exhibit irregular interfaces employing various geometrical shapes such as parabolic, rectangular, and triangular notch. The main objective is to analyze the effect of these irregular interfaces on torsional wave propagation. Furthermore, the three-layer model involves inhomogeneity through gravity, initial stress, rigidity, and medium density. These inhomogeneity parameters, coupled with irregularity, bears a remarkable impact on the velocity-profile. The generalized dispersion relation has been derived, and one particular case has been deduced to validate the current work. Detailed numerical simulation has been performed to characterize the computations, and graphical illustrations have been exhibited to support the mathematical investigation.

Introduction

The irregular surface of crustal layers affects the elastic wave propagation and plays a vital role in the energy distribution of reflected and refracted seismic waves. This is not uncommon that earthquake-generated elastic waves experience mountain basins and roots along with salt and ore bodies in their path. Such irregular surfaces affect the velocity-profile of the wave to a great extent. Therefore, it is of great concern to study the propagation of elastic waves at various types of irregular interfaces. Among all the elastic waves, the torsional wave is the most interesting one. Its amplitude decays exponentially with depth and gives a twist to the medium along with circumferential displacement. It was Kepceler [1] who studied torsional wave dispersion in a bi-material compounded cylinder with an imperfect interface. This imperfect interface may sometimes be called a corrugated interface along which reflection/refraction of waves have been studied by Ahmed and Dahab [2] and Kaur et al. [3]. Knowing that irregular undulation of the wave plays a vital role in the energy distribution, Ozturk and Akbbarov [4] investigated the torsional wave propagation in a pre-stressed circular cylinder embedded in the elastic medium.

The stress which is present in an elastic medium in the absence of external force is called initial stress, and the medium is said to be initially stressed. The Earth is an initially stressed medium possessing different layers where initial stresses exist because of dissimilarity in temperature, the overburden of the layers, slow process of creep, gravitation, etc. These stresses have a noteworthy effect on the propagation pattern of harmonic waves exhibited by earthquakes or explosions. Knowledge of the propagation behavior of different seismic waves is of great significance due to their practical importance in many engineering branches, including rock mechanics and Geophysical prospecting. Numerical simulation of elastic wave propagation forms an important ultrasonic inspection technique for non-destructive evaluation (NDE). It also gives a way to generate solution of the wave equation in the presence of arbitrary shaped defects and structural inhomogeneities and thus allows to study the complex wave-defect interactions. Several authors applied the theory given by Biot [5] for studying the propagation behavior of surface waves in an initially stressed medium. Dey and Sarkar [6] observed the effect of initial stress in a porous medium when a torsional wave propagates through it.

The propagation of the torsional surface wave in different types of elastic medium received the considered attention of many researchers and scientists for many decades. The phenomena of torsional surface wave in anisotropic and heterogeneous half-space subjected to initial compressive stress were studied by Chattopadhyay et al. [7]. Singh and Laxman [8] have investigated the propagation of the torsional wave in a corrugated doubly layered half-space. Recently, Paswan et al. [9] studied the propagation of the torsional surface wave in a heterogeneous fiber-reinforced layer lying over a heterogeneous half-space. Vishwakarma et al. [10] studied the torsional wave propagation in a dry sandy half-crust.

Pores and solid matrix together form the porous medium. The pores of these mediums are generally occupied with liquid or gas. Porous layers are naturally found beneath the Earth's surface. Almost every naturally occurring solid material is considered to be an example of porous media. Several authors dealt with the propagation of the wave in porous media, including Sharma and Gogna [11], and Shekhar and Parvez [12]. Gupta and Gupta [13] examined the phenomena of torsional surface wave propagation in anisotropic poroelastic half-space under the influence of gravity. The investigation of torsional surface waves in an anisotropic poroelastic layer lying over inhomogeneous half-space has been carried out by Chattaraj et al. [14]. The irregular boundaries at the interface may be of different geometrical shapes (like rectangular, triangular, parabolic, corrugated boundary, etc.). The problems with corrugated boundaries have much significance in seismology due to their closeness to the natural situation, as they lead to better comprehension and interpretation of seismic behavior at continental margins, mountain roots, etc. Propagation of seismic waves in the medium having uneven boundaries was discussed by Chattopadhyay and De [15], Tomar and Kaur [16], Vishwakarma and Kaur [17] and Kepceler [1].

Several authors have investigated torsional wave propagation in a two-layered geometry consisting of porous layers, heterogeneous-anisotropic substratum, reinforced pre-stressed medium and pre-stressed elastic layer [1,4,7,8,9]. Kepceler [1] although discussed about the imperfect interface but is only limited to a slight discussion. These studies are significant in explaining the impact of heterogeneous coefficients with free and rigid boundary plane. However, no one has ever studied the interactive phenomenon of torsional wave in three-layered geometry with single step and double step corrugated interface. Jeffreys & Bullen [18] and Bullen [19] have explained that the rigidities, densities, and initial stress along with other elastic parameters of the medium vary as a function of depth inside the Earth. It varies at different rates with different layers within the Earth. Bullen (1965) presented a detailed mathematical justification along with the heuristic proof and validation for the same. He further approximated the density law inside the Earth and confirmed that it varies quadratically (in-depth parameter) from 413 to 984 km in the depth direction. Ongoing further towards the central core (> 984 km), Bullen approximated that density varies as a linear function of the depth parameter and confirms that it would not be unrealistic to investigate the reaction of wave profile in various geo-media containing inhomogeneity in the form of a mathematical function of depth. Many authors have worked with inhomogeneity and considered the inhomogeneity as the linear, quadratic, and exponential function of depth parameter. There is a need to study for complicated functions. However, no one has ever considered these functions as the periodic functions of sine and cosine functions along with linear variation. In current study, we have considered,

Medium 1 (Watery-porous reservoir):

$$\begin{array}{c}{N}_1=N_{M1}\left(1-\sin \alpha z\right),L_1=L_{M1}\left(1-\sin \alpha z\right),\\ P_1=P_{M1}\left(1-\sin \alpha z\right),{\rho }_1={\rho }_{M1}\left(1-\sin \alpha z\right)\end{array}\},\alpha z\ne \frac{\pi }{2},$$

where N₁ and L₁ are directional rigidities along horizontal and longitudinal directions respectively. P₁ is the initial stress, and ρ₁ is the density of the medium 1. N_M1, L_M1, P_M1, and ρ_M1 are the elastic constants at $\alpha z=0$, α is an inhomogeneity parameter.

Medium 2 (Sand-dune deposits): μ = μ_M2(1 + az), ρ₂ = ρ_M2, P₂ = P_M2(1 + bz). where μ, ρ₂, and P₂ are the rigidity, density and initial stress of the medium 2. μ_M2, ρ_M2, and P_M2 are the elastic constants at az = −1. And, a is the inhomogeneity parameter.

Medium 3 (Orthotropic half-space):

$$\begin{array}{ccc}\hfill {\tilde{Q}}_1& =& Q_1(1+\cos {\delta }_{1}z),{\tilde{Q}}_3=Q_3(1+\cos {\delta }_{1}z),{\rho }_3={\rho }_{M_3}(1+\cos {\delta }_{1}z),\hfill \\ \hfill P_3& =& P_{M3}(1+\cos {\delta }_{1}z),{\delta }_{1}z\ne \left(2n+1\right)\pi ,n=0,1,2,\dots \hfill \end{array}$$

where ${\tilde{Q}}_1$ and ${\tilde{Q}}_3$ are the directional rigidities, ρ₃ and P₃ are the density and initial stress of the medium 3. δ₁ is the inhomogeneity parameter.

The gap in the existing literature have been understood and implemented in the current work. A three-layered model has been considered for the torsional surface wave in a watery-porous reservoir that lies above the sand-dune sediment followed by an orthotropic half-space. Different types of corrugations have been implemented at the interface of medium 1 and medium 2, whose mathematical representations are further used in the calculation of the dispersion equation. In order to support the findings (a closed-form of dispersion equation), numerical and graphical observations have been performed. Graphical representations have been executed to illustrates the various effects of the initial stress, Biot's gravity parameter, as well as inhomogeneity parameters on the torsional wave propagation. It has been observed that initial stresses associated with medium 1 and medium 2 affect the velocity to a significant level, while inhomogeneity parameters coupled with Biot's gravity parameters leave a marginal impact on the torsional wave velocity-profile. It presents a comprehensive study as to how complex corrugated interface with various geometrical shapes along with sinusoidal variations interacts with the wave-velocity.