Sergei A. Shapiro

Dynamic poroelasticity of thinly layered structures

Abstract Compressional seismic P -waves, propagating in poroelastic, fluid saturated, laminated sediments are strongly affected by the medium heterogeneity. Here, simple analytical expressions for the P -wave phase velocity and attenuation coefficient are derived. Both are functions of frequency and statistical medium parameters such as correlation lengths and variances. The theoretical results are compared with results from numerical simulations and show good agreement. In heterogeneous media, impedance fluctuations lead to poroelastic scattering ; variations of the layer compressibilities cause inter-layer flow (a 1-D macroscopic local flow) . From the seismic frequency range (10–100 Hz) up to ultrasonic frequencies, attenuation due to heterogeneity is strongly enhanced compared to homogeneous Biot models. The new theory automatically includes different asymptotic approximations, such as poroelastic Backus averaging in the quasi-static and the no-flow limit, geometrical optics, and intermediate frequency ranges.

Seismic signatures of permeability in heterogeneous porous media

In homogeneous poroelastic systems, the permeability tensor practically does not influence propagating seismic waves in the low frequency range (0–1000 Hz; see, e.g., Schmitt, 1989; Gelinsky and Shapiro, 1996). In this paper, we show that this situation changes in heterogeneous systems such as, layered or fractured sediments. Due to the heterogeneities of poroelastic structures, the attenuation of P-waves is influenced by the permeability in an enhanced way. We show, however, that such a “seismic permeability” can differ very strongly from the hydraulic permeability.

Poroelastic Backus averaging for anisotropic layered fluid- and gas-saturated sediments

A homogeneous anisotropic effective-medium model for saturated thinly layered sediments is introduced. It is obtained by averaging over many layers with different poroelastic moduli and different saturating fluids. For a medium consisting of a stack of vertically fractured horizontal layers, this effective medium is orthorhombic. We derive the poroelastic constants that define such media in the long-wavelength limit as well as the effective largescale permeability tensor. The permeability shows strong anisotropy for large porosity fluctuations. We observe pronounced effects that do not exist in purely elastic media. At very low frequencies, seismic waves cause interlayer flow of pore fluid across interfaces from more compliant into stiffer layers. For higher frequencies, the layers behave as if they are sealed, and no fluid flow occurs. The effective-medium velocities of the quasi-compressional waves are higher in the noflow than in the quasi-static limit. Both are lower than the high-frequency, i.e., ray-theory limit. Partial saturation affects the anisotropy of wave propagation. In the no-flow limit, gas that is accumulated primarily in the stiffer layers reduces the seismic anisotropy; gas that is trapped mainly in layers with a more compliant frame tends to increase the anisotropy. In the quasi-static limit, local flow keeps the anisotropy constant independent of partial saturation effects. For dry rock, no-flow and quasi-static velocities are the same, and the anisotropy caused by layering is controlled only by fluctuations of the layer shear moduli. If the shear stiffness of all layers is the same and only the compressive stiffness or saturation varies, only the ray-theory velocity exhibits anisotropy.

Elastic waves in finely layered sediments: The equivalent medium and generalized O’Doherty‐Anstey formulas

We study the influence of elastic 1-D inhomogeneous random media (e.g., finely layered media with variable density and shear and compressional velocities) on the kinematics and dynamics of the transmitted obliquely incident P‐ and SV‐plane waves. Multiple scattering (resulting in localization and spatial dispersion of the elastic wavefield) is the main physical effect controlling the properties of the wavefield in such media. We analyze the wave propagation assuming the fluctuations of velocities and density to be small (of the order of 20% or smaller). We obtain explicit analytic solutions for the attenuation coefficient and phase velocity of the transmitted waves. These solutions are valid for all frequencies. They agree very well with results of numerical modeling. Our theory shows that fine elastic multilayering is characterized by a frequency‐dependent anisotropy. At typical acquisition frequencies this anisotropy differs significantly from the low‐frequency anisotropy described by the well‐known Bac...

An approach to upscaling for seismic waves in statistically isotropic heterogeneous elastic media

For complex heterogeneous models, smoothing is often required for imaging or forward modeling, e.g., ray tracing. Common smoothing methods average the slowness or squared slowness. However, these methods are unable to account for the difference between scattering caused by fluctuations of the different elastic moduli and density. Here, we derive a new smoothing method that properly accounts for all of the parameters of an isotropic elastic medium. We treat the geophysical problem of optimum smoothing of heterogeneous elastic media as a problem of moving-window upscaling of elastic media. In seismology, upscaling a volume of a heterogeneous medium means replacing it with a volume of a homogeneous medium. In the low-frequency limit, this replacement should leave the propagating seismic wavefield approximately unchanged. A rigorous approach to upscaling is given by homogenization theory. For randomly heterogeneous models, it is possible to reduce the problem of homogenization to calculating the coherent wavefield (mean field) in the low-frequency limit. After deriving analytical expressions for the coherent wavefield in weakly heterogeneous and statistically isotropic random media, we obtain a smoothing algorithm. We apply this algorithm to random media and to deterministic models. The smoothing algorithm is frequency dependent, i.e., for different dominant frequencies, different smooth versions of the same medium should be considered. Several numerical examples using finite differences demonstrate the advantages of our approach over common smoothing schemes. In addition, using a numerical eikonal equation solver we show that, in the case of complex heterogeneous media, appropriate initial smoothing is important for high-frequency modeling.

AVO correction for scalar waves in the case of a thinly layered reflector overburden

The reflection response of a seismic target is significantly affected by a thinly layered overburden, which creates velocity anisotropy and a transmission loss by scattering attenuation. These effects must be taken into account when imaging a target reflector and when inverting reflection coefficients. Describing scalar wave (i.e., acoustic wave or SH‐wave) propagation through a stack of thin layers by equivalent‐medium theory provides a simple generalized O’Doherty‐Anstey formula. This formulation is defined by a few statistical parameters that depend on the 1-D random fluctuations of the reflector overburden. The formula has been combined with well‐known target‐oriented and amplitude‐preserving migration/inversion algorithms and amplitude variation with offset (AVO) analysis procedures. The application of these combined procedures is demonstrated for SH‐waves in an elastic thinly‐layered medium. These techniques offer a suitable tool to compensate for the thin‐layer influence on traveltimes and amplitud...

Ultrasonic signal analysis to monitor damage development in short fiber-reinforced polymers

This paper demonstrates that signal processing of digitized ultrasonic B-scan data can be applied to quantify damage in composites. The studies were carried out on polypropylene samples with short glass-fiber reinforcement. The specimens were characterized with B-scans before and after being subjected to various levels of applied strain in tension tests. The B-scans were analyzed with respect to velocity, damping and statistical properties, with the following findings. (i) The velocity decreases with the amount of damage due to the loss of stiffness in the material. The decrease is independent of frequency. (ii) The damping of the ultrasonic beam results mainly from intrinsic effects (α ~ ƒ) which can be attributed to the high mechanical loss factor of the viscoelastic polymer matrix. In the frequency range from 2 to 7 MHz Rayleigh scattering at fibers or cracks does not contribute significantly. (iii) The statistical analysis based on the ‘meanfield theory’ evaluates the coherence (or incoherence) of the wavefront. Regions with a higher degree of damage display more incoherence which can be attributed to stronger fluctuations of the elastic constants in this case. Results obtained with light microscopy are presented to better illustrate specific properties of the material under investigation.

Seismic characterization of statistical properties of fractured composite materials

A wavefield propagating in a scattering medium can be split up into coherent and incoherent parts, whose intensity changes depending on traveldistance and frequency. Looking at the statistics of the transmitted wavefield, we have found a quantity that allows to characterize a medium and estimate its statistical parameters. Laboratory measurements were carried out on FRC (fibre reinforced composite) specimens serving as a model for heterogeneous rocks with cracks. The underlying theory (Rytov approximation) neglects backscattering and assumes wavelengths shorter than the characteristic length of inhomogeneities. We introduce the parameter being a measure for the ratio of incoherent to coherent wavefield intensity. Applying these concepts to real data obtained from ultrasonic measurements, theoretical predictions for the relation between and frequency are confirmed. One expects a quadratic power law, resulting in a straight line in a double-logarithmic diagram. This suggests scattering at large-scale inhomogeneities rather than at small individual structures (cracks). The evaluation of P-wave data from upper crust events shows a similar behaviour. From the calculation of ( it is possible to infer medium properties, such as the variance and the correlation length of medium parameters. As a result, we have found a significant and robust parameter extracted from a wavefield that allows a qualitative and quantitative statistical characterization of the medium penetrated by the wave.

Seismic wave scattering by an inclusion in a fluid‐saturated porous medium

The problem of the scattering of a seismic wave by a small (compared to the wavelength of the fast compressional wave) porous inclusion placed in another porous medium is studied using the Born approximation. The mechanical behavior of both host and inc lus ion mate r ia l s i s desc r ibed by the low-frequency version of Biot s theory. The resu l t s depend s ign i f ican t ly on the ra t io o f the wavelength of the slow compressional wave and the inhomogeneity size. For small values of th i s ra t io resu l t s a re in agreement wi th exact results of Berryman (1985), derived for spher ica l inc lus ion . In the opposite case new results are obtained. They can be used to estimate compressional wave velocity and attenuation in a porous medium containing randomly d i s t r ibu ted inc lus ions . The frequency dependence of attenuation is found to be in agreement wi th the resu l t s fo r randomly layered porous materials.

Slow Compressional Wave in Porous Media - Finite Difference Simulations on Micro-Scale

We perform wave propagation simulations in porous media on microscale in which a slow compressional wave can be observed. Since the theory of dynamic poroelasticity was developed by Biot (1956), the existence of the type II or Biots slow compressional wave (SCW) remains the most controversial of its predictions. However, this prediction was confirmed experimentally in ultrasonic experiments. The purpose of this paper is to observe the SCW by applying a recently developed viscoelastic displacement-stress rotated staggered finite-difference (FD) grid technique to solve the elastodynamic wave equation. To our knowledge this is the first time that the slow compressional wave is simulated on first principles.