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.     

The effect of inertial coupling on seismic reflection amplitudes

A problem of reflection and transmission of elastic waves at a plane interface between a uniform elastic solid half-space and a porous elastic half-space containing two immiscible fluids is investigated. The theory developed by Lo, Sposito and Majer for porous media containing two immiscible fluids is employed to find out the reflection and transmission coefficients. The incident wave is assumed to propagate through the uniform elastic half-space and two cases are considered. In the first case, a beam of plane longitudinal wave is assumed to be incident and in the second case, a beam of transverse wave is assumed to be incident at the interface. By taking granite as impervious elastic medium and columbia fine sandy loam containing air-water mixture as porous medium, reflection and transmission coefficients are obtained. By neglecting the inertial coupling coefficients, these coefficients are reduced to those obtained by Tomar and Arora using the theory of Tuncay and Corapcioglu. It is found that the inertial coupling parameters significantly affect the phase speeds and the amplitude ratios of the transmitted waves.     

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.     

含两项不混合/单项黏性流体孔隙介质分界面上反透射系数特征研究

弹性孔隙介质分界面上的反透射系数特征,在岩性划分、流体识别、储层边界判识等方面有重要的应用.本文研究上层为含两项不混合黏性流体孔隙介质、下层为含单项黏性流体孔隙介质分界面上的反透射理论.首先根据两种孔隙介质分界面上的能量守恒得到边界条件,再将波函数、位移、应力与应变关系代入边界条件,推导出完全连通孔隙情况下,第一类纵波入射到孔隙介质分界面上的反透射系数方程.通过建立砂岩孔隙介质模型,分别分析不同孔隙流体类型、不同含油饱和度及不同入射角情况下,各类波的反透射系数特征.研究表明,第二、三类纵波反透射系数数值比第一类纵波小多个数量级,且两者对入射角的变化不敏感,但对孔隙流体性质、含油饱和度的变化较敏感,而横波反透射系数特征恰好与此相反;第一类纵波反透射系数特征比较复杂,入射角、孔隙流体的性质及含油饱和度的变化都对其产生影响.不同孔隙流体弹性物性的差异、孔隙介质中含油饱和度的变化及不同入射角引起垂向和切向应力分量的变化都会影响各类波的反透射系数特征,分析这些特征可以为研究储层含油气性提供理论基础.     

Indirect Boundary Element Method applied to fluid–solid interfaces

In this paper scattering of elastic waves in fluid–solid interfaces is investigated. We use the Indirect Boundary Element Method to study this wave propagation phenomenon in 2D models. Three models are analyzed: a first one with an interface between two half-spaces, one fluid on the top part and the other solid in the bottom; a second model including a fluid half-space above a layered solid; and finally, a third model with a fluid layer bounded by two solid half-spaces. The source, represented by Hankel's function of the second kind, is always applied in the fluid. This indirect formulation can give to the analyst a deep physical insight on the generated diffracted waves because it is closer to the physical reality and can be regarded as a realization of Huygens' principle. In any event, mathematically it is fully equivalent to the classical Somigliana's representation theorem. In order to gauge accuracy we test our method by comparing with an analytical solution known as Discrete Wave Number. A near interface pulse generates scattered waves that can be registered by receivers located in the fluid and it is possible to infer wave velocities of solids. Results are presented in both time and frequency domain, where several aspects related to the different wave types that emerge from this kind of problems are pointed out.     

Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium

An analytical model for describing the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids is developed in the present study based on the poroelastic equations formulated by Lo et al. [Lo WC, Sposito G, Majer E. Wave propagation through elastic porous media containing two immiscible fluids. Water Resour Res 2005;41:W02025]. The dispersion equation obtained is complex-valued due to viscous dissipation resulting from the relative motion of the solid to the pore fluids. As an excitation frequency is stipulated, the dispersion equation that is a cubic polynomial is numerically solved to determine the phase speed and attenuation coefficient of Rayleigh waves in Columbia fine sandy loam permeated by an air–water mixture. Our numerical results show that, corresponding to three dilatational waves, there is also the existence of three different modes of Rayleigh wave in an unsaturated porous medium, which are designated as the R1, R2, and R3 waves in descending order of phase speed, respectively. The phase speed of the R1 wave is non-dispersive (frequency-independent) in the frequency range we examined (10 Hz–10 kHz) and decreases as water saturation increases, whose magnitude ranges from 20% to 49% of that of the first dilatational wave with respect to water content. However, it is revealed numerically that the R2 and R3 waves are functions of excitation frequency. Given the same water saturation and excitation frequency, the phase speeds of the R2 and R3 waves are found to be approximately 90% of those of the second and third dilatational waves, respectively. The R1 wave has the lowest attenuation coefficient whereas the R3 wave attenuates highest.     

基于横向各向同性BISQ方程的弹性波传播数值模拟

Biot流动和喷射流动是含流体多孔隙介质中流体流动的两种重要力学机制. 近年来，利用同时处理这两种力学机制的BISQ(Biot-Squirt)模型，弹性波衰减和频散的问题已被广泛研究；然而基于BISQ方程的波场数值模拟尚未见到公开的报道．本文从BISQ方程出发，利用交错网格方法对横向各向同性孔隙介质中不同频率和相界情况，以及双层介质中的弹性波传播进行数值模拟，研究了在同时考虑两种流动机制作用情况下地震波和声波的传播特性及传播过程中出现的各种波动现象．      

Pseudo Rayleigh wave in a partially saturated non-dissipative porous solid

Propagation of surface waves is studied at the pervious boundary of a porous solid saturated with a mixture of two immiscible fluids. An approach, based on continuum mixture theory, is used to derive a secular equation for the propagation of harmonic waves at the stress-free plane surface of this non-dissipative medium. Numerical analysis shows that this secular equation may not represent the propagation of true surface wave in the porous aggregate. Then, this equation is solved numerically for the propagation of pseudo Rayleigh wave or the leaky surface waves. To ensure the existence of pseudo Rayleigh wave, capillary effect between two (wetting and non-wetting) pore-fluids is related to the partial saturation. Effects of porosity and partial saturation coupled with capillary effect are observed on the phase velocity of pseudo Rayleigh waves in sandstone saturated with water-CO₂ mixture.     

Numerical simulation of elastic wave propagation based on the transversely isotropic BISQ equation

Introduction More real models are being developed by the modern seismology. As we all know, the earth is not a simple elastic body. Oil and gas reservoir, ground surface, seashore zone, sea bottom layer, etc, are porous solid media with fluids. It has been confirmed that there are two main fluid flow mechanisms in these media (Dvorkin, Nur, 1993), i.e., the Biot flow mechanism (Biot, 1956, 1962) based on the macroscopic property and the Squirt-flow mechanism (Mavko, Nur, 1979) based on the …     

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.     

Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks

Saturation of porous rocks with a mixture of two fluids (known as partial saturation) has a substantial effect on the seismic waves propagating through these rocks. In particular, partial saturation causes significant attenuation and dispersion of the propagating waves, due to wave-induced fluid flow. Such flow arises when a passing wave induces different fluid pressures in regions of rock saturated by different fluids. As partial fluid saturation can occur on different length scales, attenuation due to wave-induced fluid flow is ubiquitous. In particular, mesoscopic fluid flow due to heterogeneities occurring on a scale greater than porescale, but less than wavelength scale, is responsible for significant attenuation in the frequency range from 10 to 1000 Hz.Most models of attenuation and dispersion due to mesoscopic heterogeneities imply that fluid heterogeneities are distributed in a periodic/regular way. In 1D this corresponds to periodically alternating layering, in 3D as periodically distributed inclusions of a given shape (usually spheres). All these models yield very similar estimates of attenuation and dispersion.Experimental studies show that mesoscopic heterogeneities have less idealized distributions and that the distribution itself affects attenuation and dispersion. Therefore, theoretical models are required which would simulate the effect of more general and realistic fluid distributions.We have developed two theoretical models which simulate the effect of random distributions of mesoscopic fluid heterogeneities. The first model assumes that one fluid forms a random ensemble of spherical inclusions in a porous medium saturated by the other fluid. The attenuation and dispersion predicted by this model are very similar to those predicted for 3D periodic distribution. Attenuation (inverse quality factor) is proportional to ω at low frequencies for both distributions. This is in contrast to the 1D case, where random and periodically alternating layering shows different attenuation behaviour at low frequencies. The second model, which assumes a 3D continuous distribution of fluid heterogeneities, also predicts the same low-frequency asymptote of attenuation. However, the shapes of the frequency dependencies of attenuation are different. As the 3D continuous random approach assumes that there will be a distribution of different patch sizes, it is expected to be better suited to modelling experimental results. Further research is required in order to uncover how to relate the random functions to experimentally significant parameters.     

Calculation of reflection and transmission coefficients for qP waves incident on a planar interface between isotropic and triclinic media

In the paper by Chattopadhyay and Rajneesh (2006, “Reflection and refraction of waves at the interface of an isotropic medium over a highly anisotropic medium’, Acta Geophysica, vol. 54, no. 3, pp. 239–249), the authors proposed a process to calculate R/T (reflection and transmission) coefficients at the interface between isotropic and triclinic half-spaces, with incident qP waves in triclinic media. Unfortunately, besides several misprints, the authors made a fatal assumption that there is no transmitted SH wave generated in isotropic media, which led the successive analytical derivations and numerical calculations thoroughly wrong. In this paper, the errors are analyzed at length and corrections are given. Then an alternative approach to solve the problem is proposed and numerical results are shown and discussed.     

Stochastic analysis of immiscible displacement in porous media

The interface of two immiscible fluids flowing in porous media may behave in an unstable fashion. This instability is governed by the pore distribution, differential viscosity and interface tension between the two immiscible fluids. This study investigates the factors that control the interface instability at the wetting front. The development of the flow equation is based on the mass balance principle, with boundary conditions such as the velocity continuity and capillary pressure balance at the interface. By assuming that the two-phase fluids in porous media are saturated, a covariance function of the wetting front position is derived by stochastic theory. According to those results, the unstable interface between two immiscible fluids is governed by the fluid velocity and properties such as viscosity and density. The fluid properties that affect the interface instability are expressed as dimensionless parameters, mobility ratio, capillary number and Bond number. If the fluid flow is driven by gravitational force, whether the interface undergoes upward displacement or downward displacement, the variance of the unstable interface decreases with an increasing mobility ratio, increases with increasing capillary number, and decreases with increasing Bond number. For a circumstance in which fluid flow is horizontal, our results demonstrate that the capillary number does not influence the generation of the unstable interface.     

Seismic attenuation in porous rocks with random patchy saturation

Saturation of porous rocks with a mixture of two fluids has a substantial effect on seismic‐wave propagation. In particular, partial saturation causes significant attenuation and dispersion of the propagating waves due to the mechanism of wave‐induced fluid‐flow. Such flow arises when a passing wave induces different fluid pressures in regions of rock saturated by different fluids. Most models of attenuation and dispersion due to mesoscopic heterogeneities imply that fluid heterogeneities are distributed in a regular way. However, recent experimental studies show that mesoscopic heterogeneities have less idealized distributions and that the distribution itself affects attenuation and dispersion. Based on an approximation for the coherent wavefield in random porous media, we develop a model which assumes a continuous distribution of fluid heterogeneities. As this continuous random media approach assumes that there will be a distribution of different patch sizes, it is expected to be better suited to modelling experimental data. We also show how to relate the random functions to experimentally measurable parameters.     

基于三相孔隙弹性理论的火山岩气层岩石物理响应特征分析

Unlike previous theories with velocity and/or elastic modulus averaging, we use a three-phase porous rock physics model developed by Santos for analyzing the seismic response of two immiscible fluids in saturated porous media. Considering reservoir reference pressure and coupling drag of two fluids in pores, the effects of frequency, porosity, and gas saturation on the phase velocities of the P-and S-waves are discussed in detail under field conditions. The effects of porosity and gas saturation on Vp/Vs are also provided. The data for our numerical experiments are from a sample of deep volcanic rock from Daqing. The numerical results show that the frequency dispersion effect can be ignored for deep volcanic rocks with low porosity and low permeability. It is concluded that for deep volcanic rocks the effect of gas content in pores on Vp/Vs is negligible but the effect of porosity is significant when there is a certain amount of water contained in the pores. The accurate estimate of lithology and porosity in this case is relatively more important.     

Effects of pore fluids on quasi-static shear modulus caused by pore-scale interfacial phenomena

It is evident from the laboratory experiments that shear moduli of different porous isotropic rocks may show softening behaviour upon saturation. The shear softening means that the shear modulus of dry samples is higher than of saturated samples. Shear softening was observed both at low (seismic) frequencies and high (ultrasonic) frequencies. Shear softening is stronger at seismic frequencies than at ultrasonic frequencies, where the softening is compensated by hardening due to unrelaxed squirt flow. It contradicts to Gassmann's theory suggesting that the relaxed shear modulus of isotropic rock should not depend upon fluid saturation, provided that no chemical reaction between the solid frame and the pore fluid. Several researchers demonstrated that the shear softening effect is reversible during re-saturation of rock samples, suggesting no permanent chemical reaction between the solid frame and the pore fluid. Therefore, it is extremely difficult to explain this fluid–rock interaction mechanism theoretically, because it does not contradict to the assumptions of Gassmann's theory, but contradicts to its conclusions. We argue that the observed shear softening of partially saturated rocks by different pore fluids is related to pore-scale interfacial phenomena effects, typically neglected by the rock physics models. These interface phenomena effects are dependent on surface tension between immiscible fluids, rock wettability, aperture distribution of microcracks, compressibility of microcracks, porosity of microcracks, elastic properties of rock mineral, fluid saturation, effective stress and wave amplitude. Derived equations allow to estimate effects of pore fluids and saturation on the shear modulus and mechanical strength of rocks.     

Reflection and refraction at an imperfectly bonded interface between poroelastic solid and cracked elastic solid

Porous solid is in contact with a cracked elastic solid at a plane interface between them. For the presence of vertically aligned microcracks, the elastic solid behaves transversely isotropic to wave propagation. The coefficients of elastic anisotropy depend on the crack density and crack porosity in the medium. A loose bonding is considered between the two solids so that a limiting case could be the welded contact. At the plane interface, the imperfection in welded bonding is represented by tangential slipping and, hence, results in the dissipation of a part of strain energy. Three types of waves propagate in an isotropic fluid-saturated porous medium, which are considered for incidence at the interface. Incidence of a wave results in three reflected waves and two refracted waves. Partition of incident energy among the reflected and refracted waves is studied for each incidence, varying from normal to grazing directions. Numerical example calculates the energy shares of reflected and refracted waves at the plane interface between water-saturated sandstone and basalt. These energy shares are computed and analyzed for different values of crack parameters as well as loose bonding parameter.     

On the shape of the non-steady interface intersecting discontinuities in permeability

The simplifying assumption is often made, that when two fluids (whether miscible or immiscible) occupy the void space of a porous medium, they are separated by a sharp interface. Examples are the phreatic surface (between air and water) and the interface between fresh and salt water in a coastal aquifer. The orientation of such a sharp interface as it crosses a surface of discontinuity between media of different permeabilities, and as it intersects an impervious boundary, is shown to depend not only on the fluid and porous media properties, but also on the direction and rate of motion of the interface. Thus, advancing and retreating interfaces intersect boundaries of discontinuity in permeability at different angles.     

Propagation of seismic waves in patchy-saturated porous media: double-porosity representation

Propagation of harmonic plane waves is studied in a patchy-saturated porous medium. Patchy distribution of the two immiscible fluids is considered in a porous frame with uniform skeletal properties. A composition of two types of patches, connected through continuous paths, constitutes a double-porosity medium. Different compressibilities of pore-fluids in two porous phases facilitate the wave-induced fluid-flow in this composite material. Constitutive relations are considered with frequency-dependent complex elastic coefficients, which define the dissipative behaviour of porous aggregate due to the flow of viscous fluid in connected patches. Relevant equations of motion are solved to explain the propagation of three compressional waves and one shear wave in patchy-saturated porous solids. A numerical example is solved to illustrate dispersion in phase velocity and quality factor of attenuated waves in patchy-saturated porous materials. Role of fluid–solid inertial coupling in Darcy's law is emphasized to keep a check on the dispersion of wave velocities in the porous composite. Effects of patchy saturation on phase velocities and quality factors of attenuation are analysed using the double-porosity formulation as well as the reduced single-porosity equivalents.     

Seismoelectric and electroseismic responses to a point source in a marine stratified model

We investigate the seismoelectric/electroseismic wavefields excited by a point source in an air/seawater/three-layered porous medium configuration containing a hydrocarbon layer. The results show that if an explosive source for excitation is used, receivers at seafloor can record the coseismic electromagnetic fields accompanying the P, S, fluid acoustic waves and the interface responses converted from the acoustic waves at seafloor interface and from the seismic waves at the interfaces beneath the seafloor. Employing a vertical electric dipole source shows that, with the exception of the interface responses converted from electromagnetic waves at seafloor, the interface responses converted from transmitted electromagnetic waves at the interfaces beneath the seafloor can also be identified. Given that the strength of the explosive source is within excitation capability of industry air guns, the generated interface responses from the hydrocarbon layer can be detected by current electromagnetic sensors considering the low ambient noise at the seafloor. Our results demonstrate the feasibility of the seismoelectric method applied to marine hydrocarbon exploration. Electroseismic modelling results suggest that it is not practical to employ this method to prospect marine hydrocarbon layer due to the weak interface response signal, unless a much larger current is injected into seafloor.