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.

The differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.

Shales are complex porous materials, normally consisting of percolating and interpenetrating fluid and solid phases. The solid phase is generally comprised of several mineral components and forms an intricate and anisotropic microstructure. The shape, orientation, and connection of the two phases control the anisotropic elastic properties of the composite solid. We develop a theoretical framework that allows us to predict the effective elastic properties of shales. Its usefulness is demonstrated with numerical modeling and by comparison with established ultrasonic laboratory experiments. The theory is based on a combination of anisotropic formulations of the self‐consistent (SCA) and differential effective‐medium (DEM) approximations. This combination guarantees that both the fluid and solid phases percolate at all porosities. Our modeling of the elastic properties of shales proceeds in four steps. First, we consider the case of an aligned biconnected clay‐fluid composite composed of ellipsoidal inclusions. Anisotropic elastic constants are estimated for a clay‐fluid composite as a function of the fluid‐filled porosity and the aspect ratio of the inclusions. Second, a new processing technique is developed to estimate the distribution of clay platelet orientations from digitized scanning electron microphotographs (SEM). Third, the derived clay platelet distribution is employed to estimate the effective elastic parameters of a solid comprising clay‐fluid composites oriented at different angles. Finally, silt minerals are included in the calculations as isolated spherical inclusions.

This paper studies the effect on the overall properties of a cracked solid of the existence of connections between otherwise isolated cracks and of small-scale porosity within the 'solid' material. The intention is to provide effective medium models for the calculation of elastic wave propagation with wavelengths greater than the dimensions of the cracks. The method follows that of earlier papers in which the overall elastic properties are directly related to parameters governing the microstructure, such as crack number density and the mean radius and spacing distance of the cracks. Expressions derived by the method of smoothing are evaluated to second order in the number density of cracks, thereby incorporating crack-crack interactions through both the strain field in the solid and the flow field of fluids in the pores.Flow of interstitial liquids tends to weaken the material; the limit of zero flow is equivalent to isolating the cracks and the limit of free flow is equivalent to dry (gasfilled) cracks. It also introduces additional attenuation. The inclusion of small-scale porosity gives a model of 'equant porosity' which is more closely constrained by the details of crack dynamics than earlier models.

Seismic signals provide information about the underlying moment tensor which, in turn, may be interpreted in terms of source mechanism. This paper is concerned with a two-dimensional graphical display of all possible relative sizes of the three principal moments; it provides a method of representing the probability density of these relative sizes deduced from a given set of data. Information provided by such a display, together with that relating to the orientation of the principal moments, provides as full a picture of the moment tensor as possible apart from an indication of its absolute magnitude. As with the compatibility plot, which was previously introduced to portray probability measures for different forms of P wave seismogram given a presumed source type, this "source type plot" for display of the principal moments is constructed to be "equal area" in the sense that the a priori probability density of the moment ratios is uniform over the whole plot. This a priori probability is based on the assumption that, with no information whatsoever concerning the source mechanism, each principal moment may independently take any value up to some arbitrary upper limit of magnitude, with equal likelihood. Although we have in mind the study of teleseismic relative amplitude data, the ideas can, in principle, be applied quite generally. The aim is to be able to display the degree of constraint imposed on the moment tensor by any set of observed data; estimates of the sizes of the principal moments together with their errors, when displayed on the source type plot, show directly the range of moment tensors compatible with the data.