INTRODUCTION

Fractures can play a key role in many reservoirs. Cracks and/or fractures increase rock compliance (reduce stiffness), and thus lower wave speeds. If the distribution in space and/or orientation of fractures present is not isotropic, then significant anisotropy can be observed in seismic data. Aligned vertical fractures in particular are important as they can lead to azimuthal dependence (i.e., so that results depend on the direction in which any linear surface seismic array has been emplaced). Fracture-induced effects are also sensitive to fluids within the fractures. In particular, gas or air will have little effect, while liquids in fractures can stiffen them so much that liquid-saturated fractures are nearly as stiff as the surrounding rock. For partially saturated cracks and fractures, the fracture will be almost as compliant as a gas-filled fracture until 90 to 95% or more of the fracture volume is filled with liquid. Fractures having patchy saturation (separate and distinct pockets of gas and liquid) (White, 1975; Berryman et al., 1988; Endres and Knight, 1989; Mavko and Nolen-Hoeksema, 1994; Dvorkin and Nur, 1998; Johnson, 2001; Berryman et al., 2002a; Berryman, 2004) can also behave differently from any of the other cases mentioned.

Fluids such as oil, gas, water, or CO₂ are often involved in many of the problems of most practical interest. Resolution of various practical and scientific issues in the earth sciences (Wawersik et al., 2001) depends on knowledge of fluid properties underground, and also how the fluids move. In environmental cleanup applications, the contaminants to be removed from the earth are typically liquids such as gasoline or oil, or ground water contaminated with traces of harmful chemicals. In commercial oil and gas exploration, the fluids of interest are hydrocarbons in liquid or gaseous form. In analysis of the earth structure, partially melted rock is key to determining temperature and local changes of structure in the Earth's mantle. In all these cases the tool most commonly used to analyze the fluid content is measurements of seismic (compressional and shear) wave velocities in the earth. Depending on the application, the sources of these waves may be naturally occurring such as earthquakes, or man-made such as reflection seismic surveys at the surface of the earth, or ship-based survey methods over the ocean, vertical seismic profiling from boreholes to surface, or still more direct (but higher frequency) measurements using logging tools in either shallow or deep boreholes.

In many of the cases mentioned a variety of possible explanations for the observed wave velocity and attenuation discrepancies between theory and experiment have been put forward, including viscoelastic effects (velocity decrement due to frequency-dependent attenuation), fluid-enhanced softening of intragranular cementing materials, chemical changes in wet clays that alter mechanical properties, etc. Providing some of the analytical and computational tools needed for treating these difficult problems as well as others for various applications is one of the goals of the work presented here.

A review article by Berryman (1995) summarized the state of the art in effective medium theories as applied to heterogeneous rocks and rock/fluid mixtures. I assume throughout that this material is available to or already known to the reader and will, therefore, not attempt to repeat this review already covered in the AGU Handbook. Then, I can concentrate on more recent developments that are the specific focus of the paper.