Full frequency-range transient solution for compressional waves in a fluid-saturated viscoacoustic porous medium1

An analytical transient solution is obtained for propagation of compressional waves in a homogeneous porous dissipative medium. The solution, based on a generalization of Biot's poroelastic equations, holds for the low- and high-frequency ranges, and includes viscoelastic phenomena of a very general nature, besides the Biot relaxation mechanism. The viscodynamic operator is used to model the dynamic behaviour associated with the relative motion of the fluid in the pores at all frequency ranges. Viscoelasticity is introduced through the standard linear solid which allows the modelling of a general relaxation spectrum. The solution is used to study the influence of the material properties, such as bulk moduli, porosity, viscosity, permeability and intrinsic attenuation, on the kinematic and dynamic characteristics of the two compressional waves supported by the medium. We also obtain snapshots of the static mode arising from the diffusive behaviour of the slow wave at low frequencies.     

Analysis of attenuation and dispersion of propagating wave due to the coexistence of three fluid phases in the pore volume

In exploration geophysics, the efforts to extract subsurface information from wave characteristics exceedingly depend on the construction of suitable rock physics model. Analysis of different rock physics models reveals that the strength and magnitude of attenuation and dispersion of propagating wave exceedingly depend on wave-induced fluid flow at multiple scales. In current work, a comprehensive analysis of wave attenuation and velocity dispersion is carried out at broad frequency range. Our methodology is based on Biot's poroelastic relations, by which variations in wave characteristics associated with wave-induced fluid flow due to the coexistence of three fluid phases in the pore volume is estimated. In contrast to the results of previous research, our results indicate the occurrence of two-time pore pressure relaxation phenomenon at the interface between fluids of disparate nature, that is, different bulk modulus, viscosity and density. Also, the obtained results are compatible with numerical results for the same 1D model which are accounted using Biot's poroelastic and quasi-static equation in frequency domain. Moreover, the effects of change in saturation of three-phase fluids were also computed which is the key task for geophysicist. The outcomes of our research reveal that pore pressure relaxation phenomenon significantly depends on the saturation of distinct fluids and the order of saturating fluids. It is also concluded that the change in the saturation of three-phase fluid significantly influences the characteristics of the seismic wave. The analysis of obtained results indicates that our proposed approach is a useful tool for quantification, identification and discrimination of different fluid phases. Moreover, our proposed approach improves the accuracy to predict dispersive behaviour of propagating wave at sub-seismic and seismic frequencies.     

Dynamic response of a multi-layered poroelastic medium

An exact stiffness matrix method is presented to evaluate the dynamic response of a multi-layered poroelastic medium due to time-harmonic loads and fluid sources applied in the interior of the layered medium. The system under consideration consists of N layers of different properties and thickness overlying a homogeneous half-plane or a rigid base. Fourier integral transform is used with respect to the x-co-ordinate and the formulation is presented in the frequency domain. Fourier transforms of average displacements of the solid matrix and pore pressure at layer interfaces are considered as the basic unknowns. Exact stiffness (impedance) matrices describing the relationship between generalized displacement and force vectors of a layer of finite thickness and a half-plane are derived explicitly in the Fourier-frequency space by using rigorous analytical solutions for Biot's elastodynamic theory for porous media. The global stiffness matrix and the force vector of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces. The numerical solution of the global equation system for discrete values of Fourier transform parameter together with the application of numerical quadrature to evaluate inverse Fourier transform integrals yield the solutions for poroelastic fields. Numerical results for displacements and stresses of a few layered systems and vertical impedance of a rigid strip bonded to layered poroelastic media are presented. The advantages of the present method when compared to existing approximate stiffness methods and other methods based on the determination of layer arbitrary coefficients are discussed.     

Viscoelastic representation of surface waves in patchy saturated poroelastic media

Wave-induced flow is observed as the dominated factor for P wave propagation at seismic frequencies. This mechanism has a mesoscopic scale nature. The inhomogeneous unsaturated patches are regarded larger than the pore size, but smaller than the wavelength. Surface wave, e.g., Rayleigh wave, which propagates along the free surface, generated by the interfering of body waves is also affected by the mesoscopic loss mechanisms. Recent studies have reported that the effect of the wave-induced flow in wave propagation shows a relaxation behavior. Viscoelastic equivalent relaxation function associated with the wave mode can describe the kinetic nature of the attenuation. In this paper, the equivalent viscoelastic relaxation functions are extended to take into account the free surface for the Rayleigh surface wave propagation in patchy saturated poroelastic media. Numerical results for the frequency-dependent velocity and attenuation and the time-dependent dynamical responses for the equivalent Rayleigh surface wave propagation along an interface between vacuum and patchy saturated porous media are reported in the low-frequency range (0.1–1,000 Hz). The results show that the dispersion and attenuation and kinetic characteristics of the mesoscopic loss effect for the surface wave can be effectively represented in the equivalent viscoelastic media. The simulation of surface wave propagation within mesoscopic patches requires solving Biot’s differential equations in very small grid spaces, involving the conversion of the fast P wave energy diffusion into the Biot slow wave. This procedure requires a very large amount of computer consumption. An efficient equivalent approach for this patchy saturated poroelastic media shows a more convenient way to solve the single phase viscoelastic differential equations.     

Explicit use of the Biot coefficient in predicting shear-wave velocity of water-saturated sediments

Predicting the shear‐wave (S‐wave) velocity is important in seismic modelling, amplitude analysis with offset, and other exploration and engineering applications. Under the low‐frequency approximation, the classical Biot–Gassmann theory relates the Biot coefficient to the bulk modulus of water‐saturated sediments. If the Biot coefficient under in situ conditions can be estimated, the shear modulus or the S‐wave velocity can be calculated. The Biot coefficient derived from the compressional‐wave (P‐wave) velocity of water‐saturated sediments often differs from and is less than that estimated from the S‐wave velocity, owing to the interactions between the pore fluid and the grain contacts. By correcting the Biot coefficients derived from P‐wave velocities of water‐saturated sediments measured at various differential pressures, an accurate method of predicting S‐wave velocities is proposed. Numerical results indicate that the predicted S‐wave velocities for consolidated and unconsolidated sediments agree well with measured velocities.     

Steady-state harmonic response of a rigid plate bearing on a liquid-saturated poroelastic halfspace

Soil–structure interaction problems are typically modelled by assuming subgrade behaviour to be either elastic or viscoelastic. Herein, compliance functions that may be used to solve soil–structure interaction problems are evaluated by treating the subgrade as a liquid-saturated poroelastic material whose behaviour is governed by Biot's theory. The compliances are evaluated for the harmonic rocking and vertical motions of rigid permeable and impermeable plates bearing on a poroelastic halfspace. Comparisons are made with elastic solutions which assume the subgrade to be either completely drained or undrained. Also, solid and fluid contact stresses are reported for the poroelastic case and compared to the solid contact stresses for the elastic cases.     

ANISOTROPIC Q AND VELOCITY DISPERSION OF FINELY LAYERED MEDIA1

When a seismic signal propagates through a finely layered medium, there is anisotropy if the wavelengths are long enough compared to the layer thicknesses. It is well known that in this situation, the medium is equivalent to a transversely isotropic material. In addition to anisotropy, the layers may show intrinsic anelastic behaviour. Under these circumstances, the layered medium exhibits Q anisotropy and anisotropic velocity dispersion. The present work investigates the anelastic effect in the long-wavelength approximation. Backus's theory and the standard linear solid rheology are used as models to obtain the directional properties of anelasticity corresponding to the quasi-compressional mode qP, the quasi-shear mode qSV, and the pure shear mode SH, respectively. The medium is described by a complex and frequency-dependent stiffness matrix. The complex and phase velocities for homogeneous viscoelastic waves are calculated from the Christoffel equation, while the wave-fronts (energy velocities) and quality factor surfaces are obtained from energy considerations by invoking Poynting's theorem. We consider two-constituent stationary layered media, and study the wave characteristics for different material compositions and proportions. Analyses on sequences of sandstone-limestone and shale-limestone with different degrees of anisotropy indicate that the quality factors of the shear modes are more anisotropic than the corresponding phase velocities, cusps of the qSV mode are more pronounced for low frequencies and midrange proportions, and in general, attenuation is higher in the direction perpendicular to layering or close to it, provided that the material with lower velocity is the more dissipative. A numerical simulation experiment verifies the attenuation properties of finely layered media through comparison of elastic and anelastic snapshots.     

Algorithms for staggered-grid computations for poroelastic,elastic, acoustic,and scalar wave equations

Heterogeneous wave equations are more complicated numerically than homogeneous wave equations, but are necessary for physical validity. A wide variety of numerical solutions of seismic wave equations is available, but most produce strong numerical artefacts and local instabilities where model parameters change rapidly. Accuracy and stability of heterogeneous equations is achieved through staggered-grid formulations. A new pseudospectral staggered-grid algorithm is developed for the poroelastic (Biot) equations. The algorithm may be reduced to handle the elastic and acoustic limits of the Biot equations. Comparisons of results from poroelastic, elastic, acoustic and scalar computations for a 2D model show that porous medium parameters may affect amplitudes significantly. The use of homogeneous wave equations for modelling of a heterogeneous medium, or of a centred rather than a staggered grid, or of simplified (e.g. acoustic) wave equations when elastic or poroelastic media are synthesized, may produce erroneous or ambiguous interpretations.     

Numerically quantifying energy loss caused by squirt flow

In interconnected microcracks, or in microcracks connected to spherical pores, the deformation associated with the passage of mechanical waves can induce fluid flow parallel to the crack walls, which is known as squirt flow. This phenomenon can also occur at larger scales in hydraulically interconnected mesoscopic cracks or fractures. The associated viscous friction causes the waves to experience attenuation and velocity dispersion. We present a simple hydromechanical numerical scheme, based on the interface-coupled Lamé–Navier and Navier–Stokes equations, to simulate squirt flow in the frequency domain. The linearized, quasi-static Navier–Stokes equations describe the laminar flow of a compressible viscous fluid in conduits embedded in a linear elastic solid background described by the quasi-static Lamé–Navier equations. Assuming that the heterogeneous model behaves effectively like a homogeneous viscoelastic medium at a larger spatial scale, the resulting attenuation and stiffness modulus dispersion are computed from spatial averages of the complex-valued, frequency-dependent stress and strain fields. An energy-based approach is implemented to calculate the local contributions to attenuation that, when integrated over the entire model, yield results that are identical to those based on the viscoelastic assumption. In addition to thus validating this assumption, the energy-based approach allows for analyses of the spatial dissipation patterns in squirt flow models. We perform simulations for a series of numerical models to illustrate the viability and versatility of the proposed method. For a 3D model consisting of a spherical crack embedded in a solid background, the characteristic frequency of the resulting P-wave attenuation agrees with that of a corresponding analytical solution, indicating that the dissipative viscous flow problem is appropriately handled in our numerical solution of the linearized, quasi-static Navier–Stokes equations. For 2D models containing either interconnected cracks or cracks connected to a circular pore, the results are compared with those based on Biot's poroelastic equations of consolidation, which are solved through an equivalent approach. Overall, our numerical simulations and the associated analyses demonstrate the suitability of the coupled Lamé–Navier and Navier–Stokes equations and of Biot's equations for quantifying attenuation and dispersion for a range of squirt flow scenarios. These analyses also allow for delineating numerical and physical limitations associated with each set of equations.     

Wave propagation analysis of porous rocks with the thermal activated relaxation mechanism

Wave attenuation and phase velocity dispersion in the temperature domain are more complicated than those in the frequency domain. To describe wave propagation properties in the temperature domain, a so-called thermal activation mechanism model is built on the experimental result that increasing the temperature or decreasing frequency could obtain similar results on the attenuation. A rheological model (the Zener model) is employed to describe viscoelastic attenuation in saturated porous rocks. The Arrhenius relation is introduced to describe the thermal activation mechanism. The wave propagation model with thermal effects in porous media is then obtained, and 1-D P-wave and S-wave propagation characteristics are analyzed in numeric process, respectively.Two attenuation mechanisms are found in this model, the Biot loss and the thermal activation relaxation. The thermal relaxation attenuation peak and the Biot attenuation peak are observed in both frequency and temperature spectra. These two peaks move towards each other when the temperature increases on frequency spectra. The thermal relaxation peak shifts towards higher frequencies while the Biot peak shifts towards lower frequencies. At some temperature, these two peaks will superpose. The combination of the thermal relaxation and the Biot loss leads to the complexity of wave velocity curves. Similar phenomena could be observed on temperature spectra. The thermal relaxation features may relate to a so-called “local heat transfer” mechanism. These two peaks in the temperature domain have been observed in the experiments by other investigators. The characteristics of velocity and attenuation are more remarkable for high porosity rock samples. The model is helpful for the understanding of wave propagation in the temperature domain.     

Dynamic mechanisms of the post-seismic deformation following large events: Case study of the 1999 Chi-Chi earthquake in Taiwan of China

The mechanism of postseismic deformation related to strong earthquakes is important in geodynamics, and presumably afterslip or viscoelastic relaxation is responsible for the postsesimic deformation. The 1999 Chi-Chi, Taiwan of China, earthquake occurred in the region where GPS observation station is most densely deployed in the world. The unprecedented GPS data provides a unique opportunity to study the physical processes of postseismic deformation. Here we assume that the interactions of viscoelastic relaxation, afterslip, fault zone collapse, poroelastic rebound, flow of underground fluids, and all these combined contribute to the surface displacements following the main shock. In order to know the essence of the postseismic deformation after the strong event, fault zone collapse, poroelastic rebound, flow of underground fluids, and so on, are represented equivalently by the variations of the focal medium properties. Therefore, the viscoelastic relaxation, afterslip, and the variations of the equivalent focal medium properties are inverted by applying the GPS temporal series measurement data with viscoelastic finite element method. Both the afterslip rate distribution along the fault and the afterslip evolution with time are obtained by means of inversion. Also, the preliminary result suggests that viscosities of the lower crust and the upper mantle in Taiwan region is 2.7×10¹⁸ and 4.2×10²⁰ Pa·s, respectively. Moreover, the inversion results indicate that the afterslip contributing to postseismic deformation of 44.6% in 450 days after the Chi-Chi earthquake, with 34.7% caused by the viscous relaxation and 20.7% by other factors such as fault zone collapse, poroelastic rebound, and the flow of liquids.     

THE INFLUENCE OF PERMEABILITY ON COMPRESSIONAL WAVE VELOCITY IN MARINE SEDIMENTS*

To investigate the effect of permeability on the propagation of seismo-acoustic waves through marine sediments, a theoretical model based on Biot's equations is established which relates the compressional wave velocity measured at a fixed frequency to computed velocities at zero and infinite frequencies in terms of sediment porosity and permeability. The model is examined experimentally in a standard soil mechanics consolidation test (itself dependent, among other things, on sediment porosity and permeability) which has been modified to include measurements of compressional wave velocity at 1 MHz and shear-wave velocity at 5 kHz. This test allows the elastic modulus of the sediment frame to be assessed under different load conditions simultaneous with the velocity determinations. From a number of tests on different samples, five samples are chosen to typify the range of sediment sizes. The results show that the difference between the measured velocity at 1 MHz and the model-derived velocity at zero frequency increases with increasing particle size (from clays to fine sand), with decreasing porosity, and with increasing permeability. For sediments coarser than fine sand the simple model breaks down, possibly because of the dominance of scattering/diffraction effects at the high frequency of the experiment. Within this limitation the model seems satisfactory to offer a capability of predicting the permeability of a sea floor sediment to an order of magnitude by the in situ measurement of seismic velocities over a wide range of frequencies; the prediction process requires a good in situ determination of sediment porosity such as that offered by electrical formation factor measurements.     

Drained-to-undrained transition of bulk modulus in fluid-saturated porous rock induced by dead volume variation

We examine the effect of poroelastic boundary conditions when determining elastic properties of fluid-saturated porous rocks from forced-oscillation laboratory experiments. One undesired yet often unavoidable complication in the estimation of the undrained bulk modulus is due to the presence of the so-called dead volume. It implies that some fluid mass can escape the rock sample under applying a confining pressure perturbation. Thus, the dead volume compromises the undrained state required to unambiguously determine the undrained bulk modulus. In this paper, we model data of recently performed low-frequency (0.1 Hz) measurements. Therein, the dead volume has been systematically varied from 10% to 1000% of the pore volume. For the smallest dead volume, the inferred bulk modulus is close to the Biot–Gassmann undrained bulk modulus. With increasing dead volume, the experimentally inferred bulk modulus approaches the drained bulk modulus. We show that the transition from undrained to drained state as a function of dead volume can be modelled with a 1D poroelastic model for the rock sample-dead volume system with a boundary condition that honours the continuity of the fluid volume flux. We discuss the limitations of the 1D model when applied to data recorded at higher frequencies (up to 100 Hz).     

Dynamic effects of moving loads on road pavements: A review

This review paper deals with the dynamic response of road pavements to moving loads on their surface. The road pavement can be modeled as a beam, a plate, or the top layer of a layered soil medium. The foundation soil can be modeled as a system of elastic springs and dashpots or a homogeneous or layered half-space. The material behavior of the pavement can be elastic or viscoelastic, while that of the foundation layers elastic, viscoelastic, water-saturated poroelastic or even inelastic. The loads are concentrated or distributed of finite extent, may vary with time and move with constant or variable speed. The analysis is done by analytical, analytical/numerical and purely numerical methods, such as finite element and boundary element methods, under conditions of plane strain or full three-dimensionality. A number of representative examples is presented in order to illustrate the problem and the methods of analysis, demonstrate the dynamic effects of moving loads on the layered soil medium and indicate the implications of the results on road and airport pavement design.     

基于Kelvin-Voigt黏弹性骨架的含非饱和流体孔隙介质BISQ模型(英文)

本文综合考虑了在波传播过程中孔隙介质的三种重要力学机制——"Biot流动机制一squirt流动机制-固体骨架黏弹性机制",借鉴等效介质思想,将含水饱和度引入波动力学控制方程,并考虑了不同波频率下孔隙流体分布模式对其等效体积模量的影响,给出了能处理含粘滞性非饱和流体孔隙介质中波传播问题的黏弹性Biot/squirt(BISQ)模型。推导了时间-空间域的波动力学方程组,由一组平面谐波解假设,给出频率-波数域黏弹性BISQ模型的相速度和衰减系数表达式。基于数值算例分析了含水饱和度、渗透率与频率对纵波速度和衰减的影响,并结合致密砂岩和碳酸盐岩的实测数据,对非饱和情况下的储层纵波速度进行了外推,碳酸盐岩储层中纵波速度对含气饱和度的敏感性明显低于砂岩储层。     

Three‐dimensional models of reservoir sediment and effects on the seismic response of arch dams

The important effects of bottom sediments on the seismic response of arch dams are studied in this paper. To do so, a three‐dimensional boundary element model is used. It includes the water reservoir as a compressible fluid, the dam and unbounded foundation rock as viscoelastic solids, and the bottom sediment as a two‐phase poroelastic domain with dynamic behaviour described by Biot's equations. Dynamic interaction among all those regions, local topography and travelling wave effects are taken into account. The results obtained show the important influence of sediment compressibility and permeability on the seismic response. The former is associated with a general change of the system response whereas the permeability has a significant influence on damping at resonance peaks. The analysis is carried out in the frequency domain considering time harmonic excitation due to P and S plane waves. The time‐domain results obtained by using the Fourier transform for a given earthquake accelerogram are also shown. The possibility of using simplified models to represent the bottom sediment effects is discussed in the paper. Two alternative models for porous sediment are tested. Simplified models are shown to be able to reproduce the effects of porous sediments except for very high permeability values. Copyright © 2004 John Wiley & Sons, Ltd.     

On results of the surface wave analyses in poroelastic media by means of the Simple Mixture Model and the Biot model

This note is devoted to an overview of the results on the number, velocity and attenuation of surface waves in two-component saturated poroelastic media. The author's numerical results which have been obtained using the Simple Mixture Model are compared to some results of other authors who carried out either an analysis by means of the classical Biot model for poroelastic media or experiments. It is shown that the Simple Mixture Model is a simplification of the Biot model. Due to its simpler form the Simple Mixture Model is more suitable for a complex analysis which depends on several parameters. Thus, the Simple Mixture Model is the only model for which, simultaneously, either the dependence on the frequency, ω, and the dependence on the bulk permeability, π (boundary porous medium/vacuum) or the dependence on the frequency, ω, and the dependence on the surface permeability, α (boundary porous medium/fluid) has been considered.     

Semi-analytical Solution for Viscoelastic Relaxation in Eccentrically-nested Spheres Induced by Surface Toroidal Traction

A semi-analytical solution to the 2-D forward modelling of viscoelastic relaxation in a heterogeneous sphere induced by a surface toroidal force is derived. The model consists of a concentrically-nested elastic lithosphere, a viscoelastic mantle, and an eccentrically-nested viscoelastic core. Since numerical codes based on finite-element or spectral-finite-difference techniques for modelling viscoelastic relaxation in a spherical geometry in the presence of lateral viscosity variations are becoming more popular, reliable examples for testing and validating such codes are essential. The eccentrically-nested sphere solution has been tested by comparing it with two distinct results: The analytical solution for viscoelastic relaxation in concentrically-nested spheres and the time domain, spectral finite-element numerical solution for viscoelastic relaxation in eccentrically-nested spheres, with excellent agreement being obtained.     

Scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space, modeled using the indirect boundary element method

The scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space is modeled using the indirect boundary element method in the frequency domain. The free-field responses are calculated to determine the displacements and stresses at the surface of the canyon, and fictitious distributed loads are then applied at the surface of the canyon in the free field to calculate the Green's functions for displacements and stresses. The amplitudes of the fictitious distributed loads are determined from the boundary conditions, and the displacements arising from the waves in the free field and from the fictitious distributed loads are summed to obtain the solution. The effects of fluid saturation, boundary conditions, porosity, and soil layers on the surface displacement amplitudes and phase shifts are discussed, and some useful conclusions are obtained. It is shown that the surface displacement amplitudes due to saturation and boundary conditions, different porosities, or the presence of a soil layer can be very dissimilar, and large phase shifts can be observed. The resulting wavelengths for an undrained saturated poroelastic medium are slightly longer than those for a drained saturated poroelastic medium; and are longer for a drained saturated poroelastic medium than those for a dry poroelastic medium. As porosity increases, the wavelengths become longer; and a layered half-space produces longer wavelengths than a homogeneous half-space.     

昆仑山Ms 8.1级地震震后变形场数值模拟与成因机理探讨

2001年11月14日,在青海和新疆交界处发生了昆仑山M_s8.1级强烈地震,GPS后观测显示,此次地震震后形变不仅在断裂南北两侧存在很大的差异,而且在短时间调整后断裂南北两侧表现为同向运动.本文以观测的地震形变为约束,通过有限元数值模拟分析昆仑山地震震后形变的物理机制.建立有关的有限元虚功方程,通过有限元数值方法模拟震后形变,从理论上分析介质的非均匀性、黏滞性松弛、流体调整对震后形变的影响.采用网格搜索确定昆仑断裂南北两侧下地壳的黏滞系数分别为5.0×10¹⁷Pa·s, 9.0×10¹⁸Pa·s左右,正是这十余倍的差异引起了断裂两侧震后形变的非对称性和同向运动,这一差异既是长期地质作用的结果,又是现代地球动力学环境的决定因素之一.通过数值模拟定性讨论了断裂北侧地表形变在震后短期内的调整,对于靠近断裂附近的测点可能是黏弹性松弛和孔隙流体调整共同作用的结果,所以在分析短期震后形变时综合考虑黏弹松弛和孔隙流体调整是很有必要的.