All soils and most rocks are porous. The pores are often filled with air or water, but in situations of most economical or environmental interest they are filled with other fluids such as hydrocarbons or fluid-born chemical contaminants. Sound waves readily propagate through such solid/fluid mixtures and are therefore commonly used by geophysicists to locate economically desirable fluids or to attempt to map the underground migration of environmentally undesirable fluids. Images of underground velocity distributions may be obtained either by surface seismic reflection surveying or by cross-borehole transmission tomography. These sound velocity maps can often be obtained with little concern for their ultimate interpretation. But accurate interpretation of fluid content requires careful analysis of the dependence of wave velocity and attenuation on the pore-fluid distribution.
In the context of elasticity theory, measurements of shear and compressional velocities are commonly used to invert for shear and bulk moduli. However, at frequencies typical of seismic exploration, sound waves propagating through solid/fluid mixtures are governed by Biot's equations of poroelasticity (Biot, 1956a). Since the fluid constituents are not just incidental contaminants but rather the feature of primary interest in these problems, it is essential for the success of this effort that both the forward and inverse problems of poroelasticity be understood to a much higher degree than has been previously possible. Since fluid-filled porous media contain multiple constituents, poroelasticity theory contains more parameters than elasticity theory and therefore inversion for constituent moduli requires more measurements. Unrealistic simplifications must be avoided or those physical effects resulting from the presence of the fluid and from inhomogeneities in the solid may be misinterpreted -- resulting in lost opportunities to discover resources or incorrect assessments of hazards.
The Biot-Gassmann theory of poroelasticity has been highly successful in explaining ultrasonic data on fluid-saturated porous packings of sintered glass beads (Plona, 1980; Berryman, 1980a; Johnson et al., 1982). However, it is also known that systematic discrepancies exist for ultrasonic data on fluid-saturated rocks (Thomsen, 1985; Marion and Nur, 1991). The cause of these discrepancies is still unknown, but two of the more likely sources are: (1) inhomogeneities in the rocks and (2) high-frequency dispersion. Gassmann's equation (Gassmann, 1951) -- one of the key results in the theory -- is strictly valid only for homogeneous porous media and for static deformations; thus, improvements in the analysis of ultrasonic data may arise from a careful reexamination of the limitations, implications, and possibility of extensions of Gassmann's result.
Brown and Korringa (1975) have shown that Gassmann's equation for the saturated bulk modulus can be generalized for an inhomogeneous porous medium by introducing two more bulk moduli -- one for the solid deformation (Ks) and one for the pore deformation (). For the forward problem, Berryman and Milton (1991) have shown that these constants can be determined exactly from bulk moduli of the constituents for a porous frame composed of just two minerals in welded contact. Berryman and Milton (1992) have shown further that exact results for these constants can also be obtained for two minerals together with cracks if the thermal expansion coefficients of the constituents and the porous frame are also known.
In the next section, we present the Biot-Gassmann equations of poroelasticity, including a discussion of differences between Gassmann's specialized result for the coefficients and the general results of Brown and Korringa. Then, we discuss the inverse problem for the Biot-Gassmann parameters.