Surface-wave propagation in a cracked poroelastic half-space lying under a uniform layer of fluid

The effect of cracks on the elastic properties of an isotropic elastic solid is studied when the cracks are saturated with a soft fluid. A polynomial equation in effective Poisson's ratio is obtained, whose coefficients are functions of Poisson's ratio of the uncracked solid, crack density and saturating fluid parameter. Elastic and dynamical constants used in Blot's theory of wave propagation in poroelastic solids are modified for the introduction of cracks. The effects of cracks on the velocities of three types of waves are observed numerically. The frequency equation is derived for the propagation of Rayleigh-type surface waves in a saturated poroelastic half-space lying under a uniform layer of liquid. Dispersion curves for a particular model of oceanic crust containing cracks are plotted. The effects of variations in crack density and saturation on the phase and group velocity are also analysed.     

Non-linear elastic behaviour of damaged rocks

The pervasive damage of rocks by microcracks and voids strongly affects their macroscopic elastic properties. To evaluate the damage effects, we derive here the macroscopic stress-strain relations for a 3-D elastic solid with non-interacting cracks embedded inside a homogeneous matrix. The cracks considered are oriented either perpendicular to the maximum tension axis, or perpendicular to the maximum compression axis. In the first case they dilate during loading and in the second case they contract during loading. We derive a solution for the elastic energy of this rock following the self-consistent scheme of Budiansky & O'Connell (1976). The solution describes the stress-strain relations in terms of Λ^d and μ^d, which are the modified Lame constants, and an additional parameter Λ. The latter accounts for the non-linear behaviour of the solid and is related to crack density. The solution predicts a non-linear elastic rheology even for an infinitesimal strain of ɛ < 0.001, and abrupt change in the elastic moduli when the loading reverses from uniaxial compression to uniaxial tension.
We use the derived solution to analyse rock-mechanics experiments in which beams of Indiana limestone were deformed under four-point loading. This configuration provides simultaneously the apparent tensile and compressive moduli for small strains. The apparent moduli fit the effective elastic moduli calculated with the present damage model well, including the differences between tensile and compressive moduli.     

Wave speeds and attenuation of elastic waves in material containing cracks

Summary. Expressions now exist from which may be calculated the propagation constants of elastic waves travelling through material containing a distribution of cracks. The cracks are randomly distributed in position and may be randomly orientated. The wavelengths involved are assumed to be large compared with the size of the cracks and with their separation distances so that the formulae, based on the mean taken over a statistical ensemble, may reasonably be used to predict the properties of a single sample. The results are valid only for small concentrations of cracks.
Explicit expressions, correct to lowest order in the ratio of the crack size to a wavelength, are derived here for the overall elastic parameters and the overall wave speeds and attenuation of elastic waves in cracked materials where the mean crack is circular, and the cracks are either aligned or randomly orientated. The cracks may be empty or filled with solid or fluid material. These results are achieved on the basis of simply the static solution for an ellipsoidal inclusion under stress.
The extension to different distributions of orientation or to mixtures of different types of crack is quite straightforward.     

Effective anisotropic elastic constants for wave propagation through cracked solids

Summary. Theoretical developments of Hudson demonstrate how to calculate the variations of velocity and attenuation of seismic waves propagating through solids containing aligned cracks. The analysis can handle a wide variety of crack configurations and crack geometries. Hudson associates the velocity variations with effective elastic constants. In this paper we associate the variation of attenuation with the imaginary parts of complex effective elastic constants. These complex elastic constants permit the simulation of wave propagation through two-phase materials by the calculation of wave propagation through homogeneous anisotropic solids.     

Scattering of waves in elastic media containing passive and active cracks

Scattering of wavefields in a 3-D medium that includes passive and/or active structures, is numerically solved by using the boundary integral equation method (BIEM). The passive structures are velocity anomalies that generate scattered waves upon incidence, and the active structures contain endogenous fracture sources, which are dynamically triggered by the dynamic load due to the incident waves. Simple models are adopted to represent these structures: passive cracks act as scatterers and active cracks as fracture sources. We form cracks using circular boundaries, which consist of many boundary elements. Scattering of elastic waves by the boundaries of passive cracks is treated as an exterior problem in BIEM. In the case of active cracks, both the exterior and interior problems need to be solved, because elastic waves are generated by fracturing with stress drop, and the growing crack boundaries scatter the incident waves from the outside of the cracks. The passive cracks and/or active cracks are randomly distributed in an infinite homogeneous elastic medium. Calculations of the complete waveform considering a single scatter show that the active crack has weak influence on the attenuation of first arrivals but strong influence on the amplitudes of coda waves, as compared with those due to the passive crack. In the active structures, multiple scattering between cracks and the waves triggered by fracturing strongly affect the amplitudes of first arrivals and coda waves. Compared to the case of the passive structures, the attenuation of initial phase is weak and the coda amplitudes decrease slowly.     

Seismic-wave propagation through a cracked solid: polarization as a possible dilatancy diagnostic

summary . A new technique is presented for modelling the elastic constants of cracked structures with application to systems with weak concentrations of parallel cracks, and of simple biplanar and triplanar cracks. The velocities and V_p/V_s ratios of these anisotropic structures are used to provide quantitative models for some earthquake precursors. These results indicate the great importance of crack geometry to the behaviour of precursors. The velocities of saturated cracks appear to favour the dilatancy-diffusion model of precursory phenomena. Synthetic seismograms are calculated for propagation through possible dilatancy zones. The seismograms show some characteristic features which may be useful for the investigation of earthquake dilatancy.     

Dynamic nucleation process of shallow earthquake faulting in a fault zone

Two distinct phases are commonly observed at the initial part of seismograms of large shallow earthquakes: low-frequency and low-amplitude waves following the onset of a P wave ( P ₁) are interrupted by the arrival of the second impulsive phase P₂ enriched with high-frequency components. This observation suggests that a large shallow earthquake involves two qualitatively different stages of rupture at its nucleation.
We propose a theoretical model that can naturally explain the above nucleation behaviour. The model is 2-D and the deformation is assumed to be anti-plane. A key clement in our model is the assumption of a zone in which numbers of pre-existing cracks are densely distributed; this cracked zone is a model for the fault zone. Dynamic crack growth nucleated in such a zone is intensely affected by the crack interactions, which exert two conflicting effects: one tends to accelerate the crack growth, and the other tends to decelerate it. The accelerating and decelerating effects are generally ascribable to coplanar and non-coplanar crack interactions, respectively. We rigorously treat the multiple interactions among the cracks, using the boundary integral equation method (BIEM), and assume the critical stress fracture criterion for the analysis of spontaneous crack propagation.
Our analysis shows that a dynamic rupture nucleated in the cracked zone begins to grow slowly due to the relative predominance of non-coplanar interactions. This process radiates the P₁ phase. If the crack continues to grow, coalescence with adjacent coplanar cracks occurs after a short time. Then, coplanar interactions suddenly begin to prevail and crack growth is accelerated; the P₂ phase is emitted in this process. It is interpreted that the two distinct phases appear in the process of the transition from non-coplanar to coplanar interaction predominance.     

Numerical simulation of the propagation of P waves in fractured media

We study the propagation of P waves through media containing open fractures by performing numerical simulations. The important parameter in such problems is the ratio between crack length and incident wavelength. When the wavelength of the incident wavefield is close to or shorter than the crack length, the scattered waves are efficiently excited and the attenuation of the primary waves can be observed on synthetic seismograms. On the other hand, when the incident wavelength is greater than the crack length, we can simulate the anisotropic behaviour of fractured media resulting from the scattering of seismic waves by the cracks through the time delay of the arrival of the transmitted wave. The method of calculation used is a boundary element method in which the Green's functions are computed by the discrete wavenumber method. For simplicity, the 2-D elastodynamic diffraction problem is considered. The rock matrix is supposed to be elastic, isotropic and homogeneous, while the cracks are all empty and have the same length and strike direction. An iterative method of calculation of the diffracted wavefield is developed in the case where a large number of cracks are present in order to reduce the computation time. The attenuation factor Q ⁻¹ of the direct waves passing through a fractured zone is measured in several frequency bands. We observe that the attenuation factor Q ⁻¹ of the direct P wave peaks around kd = 2, where k is the incident wavenumber and d the crack length, and decreases proportionally to ( kd ) ⁻¹ in the high-wavenumber range. In the long-wavelength domain, the velocity of the direct P wave measured for two different crack realizations is very close to the value predicted by Hudson's theory on the overall elastic properties of fractured materials.     

A theoretical approach to the propagation of interacting cracks

We propose a scheme to compute interaction effects between two randomly oriented cracks under compressive stresses and we discuss the role crack interactions play in the crack coalescence process. Stress intensity factors are computed by using an iterative technique based on the method of successive approximations. Once crack propagation occurs, curved wing cracks grow from the initial crack tips. The stress intensity factors at the wing crack tips are calculated as the sum of two terms: a component for a single wing crack subjected to both the applied stresses and the interaction effect, and a component due to the sliding of the initial crack. We have applied our procedure to various crack geometries. Our results show that interaction effects act on the crack propagation path. For cracks under tension, our approach correctly predicts the curving, hook-shaped paths of interacting cracks that have been observed in various materials. For en echelon compressive cracks, interaction effects depend on the geometry of stepping. For right-stepping cracks, no mode I crack coalescence occurs. A mixedmode propagation criterion may be introduced to check whether coalescing secondary shear fractures initiate. For left-stepping cracks, depending on whether or not there is overlapping, crack coalescence is achieved by tension wing cracks at the inner crack tips. Without overlapping, the growing wing cracks delimit a region where a tensile secondary fracture may develop and lead to coalescence. These results are consistent with previous work and show that our procedure may be now extended to a population of cracks.     

Earthquake prediction: a new physical basis

Summary. Subcritical crack growth in the laboratory occurs slowly but progressively in solids subjected to low stresses at low strain rates. The cracks tend to grow parallel to the maximum compressive stress so that, when this stress is aligned over a large enough region, the cracks will also be aligned and possess effective seismic anisotropy. It is suggested that such subcritical crack growth produces extensive-dilatancy anisotropy (EDA) throughout earth-quake preparation zones. This process is a possible driving mechanism for earthquake precursors observed at substantial distances from impending focal zones, and provides, in the shear-wave splitting which has been observed in several seismic regions, a possible technique for monitoring the build-up of stress before earthquakes.     

The basis for earthquake prediction

Summary. Recent advances in understanding the behaviour of shear waves propagating in the crust make the routine prediction of earthquakes seem practicable. Accumulating evidence suggests that most of the Earth's crust is pervaded by distributions of fluid-filled cracks and microcracks that are aligned by the contemporary stress-field so that the cracked rockmass is effectively anisotropic to seismic waves. This causes shear-waves to split, and shear-wave splitting is observed whenever shear-waves propagating along suitable raypaths in the crust are recorded by three-component instruments. These distributions of cracks are known as extensive-dilatancy anisotropy or EDA. Many characteristics of the crack- and stress-geometry can be monitored by analyzing shear-waves propagating through the cracked rockmass. Observations of temporal variations of the behaviour of shear-wave splitting in seismic gaps confirm these hypotheses, and suggest that stress changes before earthquakes may be monitored by analyzing shear-waves. In particular, monitoring earthquake preparation zones with three-component shear-wave vertical-seismic-profiles could lead to techniques for the routine prediction of earthquakes.     

Attenuation behaviour of tuffaceous sandstone and granite during microfracturing

Laboratory measurements of ultrasonic wave propagation in tuffaceous sandstone (Kimachi, Japan) and granite (Iidate, Japan) were performed during increasing fracturing of the samples. The fracturing was achieved by unconfined uniaxial compression up to and beyond the point of macrofracture of the specimen using a constant low strain rate. The observed variation of wave velocity (up to 40 per cent) due to the development of micro- and macrofractures in the rock is interpreted by rock models relating velocity changes to damage and crack density. The calculated density of the newly formed cracks reaches higher values for the sandstone than for the granite. Using the estimated crack densities, the attenuation behaviour is interpreted in terms of different attenuation mechanisms; that is, friction and scattering. Rayleigh scattering as described by the model of Hudson (1981 ) may explain the attenuation qualitatively if the largest plausible crack dimensions are assumed in modelling.     

Seismic waves in rocks with fluids and fractures

Seismic wave propagation through the earth is often strongly affected by the presence of fractures. When these fractures are filled with fluids (oil, gas, water, CO₂, etc.), the type and state of the fluid (liquid or gas) can make a large difference in the response of the seismic waves. This paper summarizes recent work on methods of deconstructing the effects of fractures, and any fluids within these fractures, on seismic wave propagation as observed in reflection seismic data. One method explored here is Thomsen's weak anisotropy approximation for wave moveout (since fractures often induce elastic anisotropy due to non-uniform crack-orientation statistics). Another method makes use of some very convenient crack/fracture parameters introduced previously that permit a relatively simple deconstruction of the elastic and wave propagation behaviour in terms of a small number of crack-influence parameters (whenever this is appropriate, as is certainly the case for small crack densities). Then, the quantitative effects of fluids on these crack-influence parameters are shown to be directly related to Skempton's coefficient B of undrained poroelasticity (where B typically ranges from 0 to 1). In particular, the rigorous result obtained for the low crack density limit is that the crack-influence parameters are multiplied by a factor  (1 − B )  for undrained systems. It is also shown how fracture anisotropy affects Rayleigh wave speed, and how measured Rayleigh wave speeds can be used to infer shear wave speed of the fractured medium in some cases. Higher crack density results are also presented by incorporating recent simulation data on such cracked systems.     

Elastic wave propagation in model sediments —I

Summary. A fluid-saturated packing of like elastic spheres is used as a model of an oceanic sediment and a method is presented for calculating the effective velocities of elastic waves in such a medium. In particular the method is applied to low-frequency waves travelling vertically down a cubic packing, saturated with an inviscid fluid and initially at rest under a uniform compressive force. It is found that two waves propagate and moreover, that their velocities are not related through the usual equations of classical elasticity to the effective elastic moduli for static deformation of the packing. For a dry packing, there is found to exist a 'cut-off' frequency above which the wave decays with depth. An extension of the method to slightly viscous fluids is also given.     

Partial derivatives of surface wave phase velocities for flat anisotropic models

Summary. Surface wave behaviour in flat anisotropic structures is first illustrated by performing an exact computation on a simple two-layer model. The variational procedure of Smith & Dahlen is then used to compute the partial derivatives of surface wave phase velocities with respect to the elastic parameters in more realistic earth models. Linear relationships between the partial derivatives for a general anisotropic structure and those for a transversely isotropic structure are derived. When considering waves propagating in a fixed direction, there are only four independent derivatives for Rayleigh waves, and two for Love waves. To avoid the lack of resolution in an inverse method, we propose to use physically constrained models. These results are illustrated by using a model with hexagonal symmetry and a symmetry axis oriented either vertically or horizontally. Quasi-Love- and quasi-Rayleigh-wave partial derivatives are computed for both axis orientations. Modes up to the second overtone and periods ranging between 45 and 130 s have been considered. Finally, anomalies of phase velocity are computed in an oceanic model made of 1/6 oriented olivine crystals with horizontal or vertical preferred orientations of the a -axis.     

Seismic body waves in anisotropic media: reflection and refraction at a plane interface

Summary. A formulation is derived for calculating the energy division among waves generated by plane waves incident on a boundary between generally anisotropic media. A comprehensive account is presented for P, SV and SH waves incident from an isotropic half-space on an orthorhombic olivine half-space, where the interface is parallel to a plane of elastic symmetry. For comparison, a less anisotropic medium having transverse isotropy with a horizontal axis of symmetry is also considered. The particle motion polarizations of waves in anisotropic medium differ greatly from the polarizations in isotropic media, and are an important diagnostic of the presence of anisotropy. Incident P and SV waves generate quasi- SH waves, and incident SH waves generate quasi- P and quasi- SV waves, often of considerable relative magnitude. The direction of energy transport diverges from the propagation direction.     

Reflection—refraction of general P-and type-I S-waves in elastic and anelastic solids

Summary. The reflection and refraction of general (homogeneous or inhomo-geneous) plane P and type-I S ( SV ) body waves incident on plane boundaries are considered for general linear viscoelastic solids. Reflection—refraction laws, physical characteristics of the waves, and the nature of critical angles are examined in detail at welded boundaries and a free surface. General visco-elasticity with no low-loss approximations predicts that contrasts in intrinsic absorption at boundaries give rise to inhomogeneous reflected and refracted waves with elliptical particle motions, velocities and maximum attenuations that vary with frequency and angle of incidence, energy propagation at speeds and directions different from phase propagation, phase propagation that in general is parallel to the boundary for at most one angle of incidence, and reflection—transmission coefficients dependent on energy flow due to wave interaction. None of these physical characteristics are predicted for waves incident on boundaries that respond instantaneously.     

Characterization of crack distribution: fabric analysis versus ultrasonic inversion

We analyse the relation between rock fabric, expressed by the preferred orientation of rock-forming minerals and microcracks, and elastic anisotropy of crystalline rock from the KTB pilot well. Detailed analyses of mineralogical composition, textures and microcrack fabrics were performed. In addition, ultrasonic velocity measurements of spherical samples in several directions were carried out at various confining pressures, and inverted in terms of the complete set of 21 elastic constants. By comparing the elastic tensors of the rocks at the final confining pressure (at which most of the microcracks are closed) with those at a lower pressure level, it is possible to separate the anisotropy induced by microcracks from that caused by mineral alignment. In contrast to previous work, no a priori knowledge of the type of anisotropy (triclinic, monoclinic, orthotropic etc.), or of the spatial orientation of the symmetry elements (planes, axes) of the cracked rock or of the intact rock is assumed. Furthermore, no restrictive assumptions on the orientation distribution function and the shape of the cracks are needed.
The results show that the elastic anisotropy characteristics, whether they are related to the microcracks or to the rock-forming minerals, are clearly correlated with the directly observed rock fabrics. We show that the symmetry directions of the mineral fabric and of microcrack fabric agree. A further result is that the microcrack-induced anisotropy dominates the other causes of anisotropy at confining pressures smaller than a few tens of megapascals, the situation being reversed at higher pressures. The laboratory data are quantitatively compared with sonic log data from the KTB well, showing the influence of pore fluids, effective pressure and crack density reduction on the anisotropy in situ .     

Ray amplitudes of seismic body waves in laterally inhomogeneous media

Summary . The most complicated part in the computation of ray amplitudes of seismic body waves in laterally inhomogeneous media with curved interfaces lies in the evaluation of the geometrical spreading. Geometrical spreading can be simply expressed in terms of the Jacobian J of the transformation from the Cartesian into ray coordinates. Several systems of ordinary differential equations to compute the function J are suggested. For general three-dimensional media, in which the velocity changes with all the three spatial coordinates, a system of three non-linear ordinary differential equations of the first order is derived. If the velocity does not depend on one coordinate, the system of equations reduces to only one non-linear differential equation. The initial conditions for these differential equations at point (or line) source and at points of intersection of the ray with curved interfaces are presented.     

Shear-wave polarization anisotropy in the Pacific Basin

Summary. Inversion of 295. Love- and Rayleigh-wave phase travel times across the Pacific Basin has yielded a structure which has a channel that is anisotropic with respect to the polarization of shear waves. The velocity of SH waves is approximately 4.24 km/s, and the velocity of SV waves is approximately 4.10 km/s in the low-velocity channel. The lid to the channel is isotropic with respect to the polarization of S waves and the velocity is approximately 4.60 km/s. The lid to the low-velocity channel increases in thickness with lithospheric age at the expense of the channel, and its thickness is apparently still increasing at a sea-floor age of 150 Myr.
These results can be explained in terms of a model with both randomly-and preferentially-oriented, liquid-filled cracks in the channel. In the model, it is assumed that the liquid-filled cracks are due to partial melting in the channel, and that any preferred orientation is caused by a shear-flow gradient resulting from differential motion between the lid and the deeper parts of the mantle.