J. A. Hudson

Anisotropic effective‐medium modeling of the elastic properties of shales

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 inclusion...

Elastic properties of hydrate‐bearing sediments using effective medium theory

Accurate and detailed models of the seismic velocity structure of gas hydrate-bearing sediments may be determined by careful analysis of controlled source seismic data. However, interpretation of these velocities in terms of hydrate saturation of the pore space has hitherto relied on semiempirical formulas and/or simple effective medium theory. We develop a rigorous theoretical scheme to relate the seismic properties of a clay-rich hydrate-bearing sediment to its porosity, mineralogy, microstructural features and hydrate saturation. We consider separately the two possible end-members for the distribution of hydrate in the pore space: (1) hydrates are unconnected and located in the pore voids without appreciable grain contact and (2) connected hydrates are forming cement binding around the grains. The scheme is transversely isotropic, to allow for anisotropy due to alignment of clay platelets, and is based on a combination of a self-consistent approximation, a differential effective medium theory, and a method of smoothing for crystalline aggregates. We have applied the scheme to lithological and seismic velocity data from Ocean Drilling Program Site 995 on the Blake Ridge (southeastern U.S. continental margin) to make estimates of the hydrate saturation. It was found that the hydrates are probably unconnected, and their volume concentration varies between approx. 0% at 100 m below the seabed and approx. 9% at 400 m depth, just above the “bottom simulating reflector”, if the clay platelet orientation distribution resembles the function we have used.

Equivalent medium representation of fractured rock

A similarity may be found between various approaches for determining the effects of parallel fractures or aligned cracks on seismic wave propagation at wavelengths that are long compared with the scale length of the cracks. Fractures can be modeled using an empirical linear slip condition; however, natural fracture surfaces can also be simulated directly as planar distributions of small isolated areas of slip (cracks) (model 1) or, conversely, as planar distributions of imperfect interfacial contacts (model 2). An alternative is plane surfaces separated by thin continuous layers of viscous fluid or a soft material (model 3). We present analytic expressions for the fracture compliances for these three models and, using these analytic results, compute the effective compliances and stiffnesses of the fractured material. As a result, it is possible to relate the measured compliances or stiffnesses directly to the statistics of the microstructural details of a fracture, given appropriate a priori information on the fracture surfaces. The results for model 1 are equivalent to a volume distribution of cracks as studied by Hudson [1980, 1981] for small crack density; the results for model 2 are basically the same as those given by White [1983] for a packing of spheres; and finally, the results for model 3 are in agreement with those given by Backus [1962] for combinations of two constituent layers. These results can be extended to the case of nonaligned fractures and to allow for fluid flow between cracks and into a porous matrix rock. Finally, it is shown that the ratio of the normal to shear fracture compliance is a good indicator of the properties of the fracture infill.

Diffraction of seismic waves by cracks with application to hydraulic fracturing

The authors describe a method of modeling seismic waves interacting with single liquid-filled large cracks based on the Kirchhoff approximation and then apply it to field data in an attempt to estimate the size of a hydraulic fracture. They first present the theory of diffraction of seismic waves by fractures using a Green`s function representation and then compute the scattered radiation patterns and synthetic seismograms for fractures with elliptical and rectangular shapes of various dimensions. It is shown that the characteristics of the diffracted wavefield from single cracks are sensitive to both crack size and crack shape. Finally, they compare synthetic waveforms to observed waveforms recorded during a hydraulic fracturing experiment and are able to predict successfully the size of a hydraulically induced fracture (length and height). In contrast to previously published work based on the Born approximation, the authors model both phases and amplitudes of observed diffracted waves. The modeling has resulted in an estimation of a crack length 1.1 to 1.5 times larger than previously predicted, whereas the height remains essentially the same as that derived using other techniques. This example demonstrates that it is possible to estimate fracture dimensions by analyzing diffracted waves.

A calculus for finely layered anisotropic media; discussion and reply

The calculation of the overall anisotropic response of materials with variations in properties on a scale short compared to the wavelength of the imposed stressfield is complicated but, when the variations in structure are one‐dimensional, a straightforward averaging technique exists. This method, originated by Riznichenko (1949) and Postma (1955), and further developed by Backus (1962), has been cast into a convenient matrix form by Helbig and Schoenberg (1987). Schoenberg and Douma (1988) have shown that distributions of parallel cracks may also be incorporated into the system by representation as layers in the limit of vanishing thickness and stiffness. Finally, Schoenberg and Muir (1989) cast the whole calculation into a form in which the rules of group theory apply, and suggest that distributions of intersecting sets of parallel cracks in fine layering can be manipulated in this way.

Seismic wave propagation in media with interconnected cracks and pores

The effects of movement of interstitial fluids within a cracked solid are examined. We consider two distinct mechanisms: flow through connections between otherwise isolated cracks, and diffusion into a porous matrix material (Hudson, Liu & Crampin 1996). These models are used to obtain the overall effective elastic constants for a medium whose parameters are considered geologically realistic. We determine the phase velocities and attenuations for seismic waves passing through such a rock and analyse their dependence on the permeability and the background porosity. We find that the properties of and waves are very sensitive to the transport of fluids through interconnected pathways. The limit of zero flow is equivalent to isolated fluidfilled cracks, the opposite extreme of free flow is that of dry or gas-filled cracks. The transitional behaviour which is observed occurs over a range of permeabilities that encompasses those of most oil and gas bearing lithologies. The introduction of fluid flow causes Pwaves to travel slower in the direction across the cracks than parallel to them, as expected intuitively. There is also a marked increase in the attenuation from negligible values to the order of 1. Similar effects arise from the introduction of an equant porosity.

Quantitative determination of hydraulic properties of fractured rock using seismic techniques

Abstract There have been significant advances over the last ten years in the use of the seismic anisotropy concept to characterize subsurface fracture systems. Measurements of seismic anisotropy are now used to deduce quantitative information about the fracture orientation and the spatial distribution of fracture intensity. Analysis of the data is based upon various equivalent medium theories that describe the elastic response of a rock containing cracks or fractures in the long wavelength limit. Conventional models assume scale/frequency independence and hence cannot distinguish between micro-cracks and macrofractures. The latter, however, control the fluid flow in many oil/gas reservoirs, as the fracture size and spacing (hence fracture storability) are essential parameters for reservoir engineers. Recently, a new equivalent medium theory for modelling of wave propagation in media with multi-scale fractures has been presented. The model predicts velocity dispersion and attenuation due to a squirt-flow mechanism at two different scales: the grain scale (micro-cracks and equant matrix porosity) and formation-scale fractures. Application of this model to field data shows that fracture density and fracture size can be inverted successfully from the frequency dependence of the time delay between split shear waves. The derived fracture length matches independent observations from borehole data. This paper presents the results of the latest development in the seismic characterization of natural fractures, with an emphasis on the quantitative determination of fracture sizes.

Numerical modeling of seismic wave propagation in media with distributed inclusions

We describe the 2-D elastodynamic boundary element method (BEM) in this paper and apply it to solve scattering problems. The method is based on the integral representation of a scattered wave eld by placing a ctitious source distribution on the surface of the scattering objects or inclusions. The ctitious sources can be determined by matching the appropriate boundary conditions at the bounding surfaces of the inclusions. This method, known as an indirect BEM, has the capability to calculate full wave elds including multiple scattering. The accuracy of the method is assured by comparing the BEM results with those calculated using other methods. We present some numerical examples of scattering of seismic waves by several di erent distributions of inclusions in order to demonstrate the versatility of the method.

Modelling Frequency-dependent Anisotropy Due to Fluid Flow In Bed Limited Cracks

Evidence from a number of measurements support the idea that anisotropy, or shear-wave splitting, exhibits a frequency dependence, that is generally attributed to properties of the microstructure of the rock. This effect is generally assumed to be the result of scattering from oriented inclusions within the rock mass, however there are a number of competing mechanisms that may give rise to this observed frequency dependence. The scale length of the inclusions must be much smaller than the wavelength at which the measurements where conducted, in order for their presence to be observed as an effective anisotropy, and may therefore be insufficient to account for a significant frequency dependence. An alternative mechanism resulting in frequency dependence is the transfer of fluid between the inclusions, assumed to be fluid filled. Using an established model, it is demonstrated that this fluid effect is potentially significant enough to explain observed frequency dependence.

Pressure induced anisotropy of interconnected cracks

An effective medium theory, based on the method of smoothing and incorporating a transfer of fluid between connected cracks via non-compliant pores, is used to derive an expression for the effective elastic parameters of the material, to first order in crack density . This expression involves a dependence on both the applied stress and the fluid pressure, and is used to determine the effects on the anisotropy of the effective medium of the applied stress and fluid pressure. A biaxial compressive stress is applied to an isotropic crack distribution to determine the anisotropy of the (transversely isotropic) effective medium as a function of differential pressure. As a result of competing processes, the theory predicts that there is a pressure at which the anisotropy reaches a maximum value before the properties of the effective medium decay, under increasing stress, to those of the uncracked matrix.