Seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid

A general solution is deduced of the differential equations describing the propagation of elastic waves in a dissipative liquid-filled viscoelastic porous solid. The velocities of three existing waves have been expressed in convenient form using the moduli of the solid phase and by introducing the frequency-dependent equivalent mass densities. The solution is then used to examine some of the phenomena which arise when each of the three-body waves, in turn, are incident on a traction-free plane boundary. Analytic expressions for the reflection coefficients are obtained. Numerical calculations have been made, for a particular model, in case of incidentP _I wave. Effect of viscoelasticity and viscosity on the reflection coefficients has also been exhibited.     

Reflection and transmission of seismic waves at an interface betwen two saturated soils

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Reflection of elastic waves at an interfaces between two porous half-spaces filled with different fluids

The problem about the refraction of elastic waves at the interface of two half-spaces (rocks) filled with immiscible fluids is solved with the use of modified boundary conditions of the dynamics of saturated porous media [Nagy and Nayfeh, 1995]. The solution is obtained within the framework of the Frenkel’-Biot theory with allowance for the surface tension at the interface of fluids for a porous medium formed by two half-spaces, which differ only in the properties of the fluids filling them. Practically important cases of the reflection from the fluid-gas (water-air) and fluid-fluid (oil-water) interfaces are considered in detail. The calculations are performed for both harmonic waves and pulses. The possibility to determine in principle the structural factor characterizing the pore-space geometry of rocks from measured dynamic parameters of reflected waves is shown.     

含黏滞流体各向异性孔隙介质中弹性波的频散和衰减

基于Biot理论,考虑液相的黏弹性变形和固液相接触面上的相对扭转,提出了含黏滞流体VTI孔隙介质模型.从理论上推导出,在该模型中除存在快P波、慢P波、SV波、SH波以外,还将存在两种新横波－慢SV波和慢SH波.数值模拟分析了6种弹性波的相速度、衰减、液固相振幅比随孔隙度、频率的变化规律以及快P波、快SV波的衰减随流体性质、渗透率、入射角的变化规律.结果表明慢SV波和慢SH波主要在液相中传播,高频高孔隙度时,速度较高;大角度入射时,快P波衰减表现出明显的各向异性,而快SV波的衰减则基本不变;储层纵向和横向渗透率存在差异时,快SV波衰减大的方向渗透率高.     

Shear waves at a corrugated interface between anisotropic elastic and visco-elastic solid half-spaces

Rayleigh’s method of approximation is employed to find out the reflection and transmission coefficients due to an incident plane SH wave at a corrugated interface between a laterally and vertically inhomogeneous anisotropic elastic solid half-space and a laterally and vertically inhomogeneous isotropic visco-elastic solid half-space. The lateral and vertical inhomogeneities are described by the exponential variations of elastic parameters. The formulae of reflection and transmission coefficients are derived in closed form for the first-order approximation of the corrugation. The effects of the corrugation of the interface, the inhomogeneity, the anisotropy, the visco-elasticity and the frequency of the incident wave on these coefficients are studied analytically and numerically for a specific model containing a periodic interface. The results of earlier workers have been reduced as particular cases from the present formulation.     

Surface wave propagation in a transversely isotropic elastic layer overlying a liquid saturated porous solid half-space and lying under the uniform layer of liquid

Dispersion of Rayleigh-type surface wave is studied in a homogeneous transversely isotropic elastic layer overlying a nondissipative liquid-saturated porous solid half-space and lying under a uniform layer of homogeneous liquid. The frequency equation in the form of ninth-order determinant is obtained.Special cases have been deduced by reducing the depth of the layers to zero and by changing the transverse isotropic layer to an isotropic layer. Dispersion curves for the phase velocity have been plotted for a particular model.     

Reflection and transmission of inhomogeneous waves in a composite porous solid saturated by two immiscible fluids

This paper is concerned with reflection and transmission of a plane, elastic, and inhomogeneous wave striking obliquely at some discontinuity inside a porous medium composed of two distinct solids and saturated by two immiscible fluids. It is found that four P‐ and two SV‐waves are reflected, whereas four P‐ and two SV‐waves are transmitted at the interface. All reflected and transmitted waves are inhomogeneous in nature and specified with different directions of propagation and attenuation vectors. An expression for the Umov–Poynting energy flux vector is derived for the system. Continuity of energy flux along normal to the interface gives 12 required boundary conditions. Expressions of amplitude and energy ratios of various reflected and transmitted waves are derived. Variations in amplitude and energy coefficients of reflected and transmitted waves with angle of incidence are numerically studied for a porous matrix composed of shaley sandstone and clay, saturated with water and oil. The effects of change in oil saturation and volume fraction of clay are also observed on amplitude ratios. Numerical simulation reveals that the change in sign in the difference of capillary pressure across the interface causes jump in the values of amplitude ratios of all waves.     

The reflection and transmission of elastic waves at a interface between a transversely isotropic elastic soils and saturated soils

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Reflection and refraction of waves at the interface of an isotropic medium over a highly anisotropic medium

In this paper, we have considered the reflection and refraction of a plane wave at an interface between two half-spaces. The lower half-spaces is composed of highly anisotropic triclinic crystalline material and the upper half-space is homogeneous and isotropic. It has been assumed that due to incidence of a plane quasi-P (qP) wave, three types of waves, namely, quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH), will be generated in the lower half space whereas P and S waves will be generated in the upper half space. The phase velocities of all the quasi waves have been calculated. It has been assumed that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. Some specific relations have been established between directions of motion and propagation, respectively. The expressions for reflection coefficients of qP, qSV, qSH and refracted coefficients of P and SV waves are obtained. Results of reflection and refraction coefficients are presented.     

Reflection and transmission of attenuated waves at the boundary between two dissimilar poroelastic solids saturated with two immiscible viscous fluids

The phenomenon of reflection and transmission of plane harmonic waves at the plane interface between two dissimilar poroelastic solids saturated with two immiscible viscous fluids is investigated. Both porous media are considered dissipative due to the presence of viscosity in pore‐fluids. Four attenuated (three dilatational and one shear) waves propagate in such a dissipative porous medium. A finite non‐dimensional parameter is used to define the effective connections between the surface‐pores of two media at their common interface. Another finite parameter represents the gas‐share in the saturation of pores. An attenuated wave in a dissipative medium is described through the specification of directions of propagation and maximum attenuation. A general representation of an attenuated wave is defined through its inhomogeneous propagation, i.e., different directions for propagation and attenuation. Incidence of an inhomogeneous wave is considered at the interface between two dissipative porous solids. This results in four reflected and four transmitted inhomogeneous waves. Expressions are derived for the partition of incident energy among the reflected and transmitted waves. Numerical examples are studied to determine the effects of saturating pore fluid, frequency, surface‐pore connections and wave inhomogeneity on the strengths of reflected and transmitted waves. Interaction energy due to the interference of different (inhomogeneous) waves is calculated in both the dissipative porous media to verify the conservation of incident energy.     

Reflection and transmission ofSH waves in an initially stressed medium consisting of a sandy layer lying over a fluid-saturated porous solid

Biot's theory is employed to study the reflection and transmission ofSH waves in a sandy layer lying over a fluid-saturated porous solid half-space. The entire medium is considered under constant initial stress. Effects of sandiness, initial stress, anelasticity and viscosity of the interstitial fluid on the partitioning of energy are studied. In the presence of initial stress the incident wave starts attenuating when incider beyond a certain angle (depending upon the amount of initial stress), even if the medium is perfectly clastic. Anelasticity of the solid layer results in the dissipation of energy during transmission. The direction of attenuation vector of incident wave affects the dissipation energy to a large extent. Effect on partitioning of energy reverse at incidence after the critical angle. A complete account of energy returmed back to the underlying half-space and that which is dissipated in the overlying layer has been discussed analytically as well as numerically.     

Seismic reflection and transmission coefficients at an air-water interface of saturated porous soil

Based on the modified Biot＇s theory of two-phase porous media, a study was presented on seismic reflection and transmission coefficients at an air-water interface of saturated porous soil media. The major differences between air-saturated soils and water-saturated soils were theoretically discussed, and the theoretical formulas of reflection and transmission coefficients at an air-water interface were derived. The characteristics of propagation and attenuation of elastic waves in air-saturated soils were given and the relations among the frequency, the angle of incidence and the reflection, transmission coefficients were analyzed by using numerical methods. Numerical results show that the propagation characteristic of the wave in air-saturated soils is great different from that in water-saturated soils. The frequency and the angle of incidence can have great influences on the reflection and transmission coefficients at interface. Some new cognition about the wave propagation is obtained and the study suggests that we may carefully pay attention to the influence of air on the dynamic analysis of seismic wave.     

含黏性流体孔隙介质分界面上反透射系数特征研究

地下岩石由岩石骨架和孔隙流体组成,通常流体含黏性.地震波在地下介质中传播时受岩石骨架和黏性流体的影响会呈现出复杂的变化.本文将流、固体位移和应力连续作为边界条件,推导出含黏性流体孔隙介质分界面上反透射系数方程;通过建立上层为饱油、下层为饱盐水的砂岩孔隙介质模型,开展反透射系数特征研究,分别分析不同频率、不同黏滞系数条件下,含黏性流体孔隙介质分界面上反透射系数随入射角的变化.研究表明,孔隙介质分界面上和等效介质分界面上的反透射系数分别随入射角的变化趋势基本一致,说明方程推导和数值计算的正确性;快纵波反透射系数受频率、流体黏性的影响较小,而快横波反透射系数在一定入射角范围内受频率、流体黏性的影响比较大;由于黏性孔隙流体的作用,慢纵波和慢横波的反透射系数受入射角、频率及流体黏性的影响都很大.