Statistics of resonances in one-dimensional disordered systems

The paper is devoted to the problem of resonances in one-dimensional disordered systems. Some of the previous results are reviewed and a number of new ones is presented. These results pertain to different models (continuous as well as lattice) and various regimes of disorder and coupling strength. In particular, a close connection between resonances and the Wigner delay time is pointed out and used to obtain information on the resonance statistics.Comment: Submitted to the special issue, in memory of Y. Levinson, of the Lithuanian Journal of Physic

Comparison of the low-frequency predictions of Biot's and de Boer's poroelasticity theories with Gassmann's equation

Predictions of Biot's theory (BT) of poroelasticity (J. Acoust. Soc. Am. 28, 168 (1956)) and de Boer's theory of porous media (TPM) (Theory of Porous Media (Springer, Berlin, 2000)) for the low-frequency bulk modulus of a fluid-saturated porous medium are compared with the Gassmann equation (Vierteljahrsschr. Naturforsch. Ges. Zur. 96, 1 (1951)). It is shown that BT is consistent with the Gassmann equation, whereas TPM is not. It is further shown that the bulk modulus of a suspension of solid particles in a fluid as predicted by TPM is only correct if the particles are incompressible

Effect of fluid viscosity on elastic wave attenuation in porous rocks

Attenuation and dispersion of elastic waves in fluidsaturated rocks due to pore fluid viscosity is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers.Waves in periodic layered systems at low frequencies can be studied using an asymptotic analysis of Rytov's exact dispersion equations. Since the wavelength of the shear wave in the fluid (viscous skin depth) is much smaller than the wavelength of the shear or compressional waves in the solid, the presence of viscous fluid layers requires a consideration of higher-order terms in the low-frequency asymptotic expansions.This expansion leads to asymptotic lowfrequency dispersion equations. For a shear wave with the directions of propagation and of particle motion in the bedding plane, the dispersion equation yields the low-frequency attenuation (inverse quality factor) as a sum of two terms which are both proportional to frequency omega] but have different dependencies on viscosity [eta]:one term is proportional to [omega]/[eta], the other to [omega][eta]:.The low-frequency dispersion equation for compressional waves allows for the propagation of two waves correspondingto Biot's fast and slow waves. Attenuation of the fast wave has the same two-term structure as that of the shear wave. The slow wave is a rapidly attenuating diffusion-type wave, whose squared complex velocity again consists of two terms which scale with i[omega]/[eta]and i[omage][eta]. For all three waves, the terms proportional to [eta] are responsible for the viscoelastc phenomena (viscous shear relaxation), whereas the terms proportional to [eta]to negative 1 account for the visco-inertial (poroelastic) mechanism of Biot's type.Furthermore, the characteristic frequencies of visco-elastic [omega sub v] and poroelastic [omega sub b] attenuation mechanisms obey the relation [omega sub v][omega sub b]=A[omega sub r squared], where [omega sub r]is the resonant frequency of the layered system, and A is a dimensionless constant of order 1. This result explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic theories that imply [omega]<< [omega sub r]. The poroelastic mechanism dominates over the visco-elastic one when the frequency-indepenent parameter B=[omega sub b]/[omega sub v]=12 [eta squared]/[mu sub s][rho sub f][h sub f squared]<<1and vice versa, where [h sub f]is the fluid layer thickness, [rho sub f] the fluid density, and [mu sub s] represents the shear modulus of the solid

Fluid substitution in heavy oil rocks

Heavy oils are defined as having high densities and extremely high viscosities. Due to their viscoelastic behavior the traditional rock physics based on Gassmann theory becomes inapplicable. In this paper, we use effective-medium approach known as coherent potential approximation or CPA as an alternative fluid substitution scheme for rocks saturated with viscoelastic fluids. Such rocks are modelled as solids with elliptical fluid inclusions when fluid concentration is small and as suspensions of solid particles in the fluid when the solid concentration is small. This approach is consistent with concepts of percolation and critical porosity, and allows one to model both sandstones and unconsolidated sands. We test the approach against known solutions. First, we apply CPA to fluid-solid mixtures and compare the obtained estimates with Gassmann results. Second, we compare CPA predictions for solid-solid mixtures with numerical simulations. Good match between the results confirms the applicability of the CPA scheme. We extend the scheme to predict the effective frequency- and temperature-dependent properties of heavy oil rocks. CPA scheme reproduces frequency-dependent attenuation and dispersion which are qualitatively consistent with laboratory measurements and numerical simulations. This confirms that the proposed scheme provides realistic estimates of the properties of rocks saturated with heavy oil