The elastic moduli of rock are dependent on its mineral properties, porosity distribution, and state of saturation. Two major theoretical approaches have been developed to address the problem of estimating elastic moduli from knowledge of rock constituents and microstructure. Effective medium theories, which include the classical bounds of Voigt (1928) and Reuss (1929) and Hashin and Shtrikman (1961,1962) as well as estimates obtained from self-consistent theories [e.g., Hill (1965), Budiansky (1965), and Berryman (1980a,b)], require a parameter characterizing the pore distribution. Alternatively, poroelastic constitutive equations (Biot, 1941; Gassmann, 1951) are phenomenological and do not require characterization of matrix and pore space geometry. However, they contain the fundamental discrepancy that shear modulus is always independent of saturation state (Berryman, 1999). Although the lack of a shear dependence on saturating fluid bulk modulus can be correct for special microgeometries and very low modulation frequencies, this predicted lack of dependence is often contradicted by high frequency experiments (above kHz), and especially so in rocks with crack porosity. As a result, Biot's theory has been modified in various ways. For example, Mavko and Jizba (1991) partition porosity into ``soft'' and ``stiff'' porosity fractions to account for the change of both bulk modulus and shear modulus with fluid saturation.
Recent comprehensive reviews of the literature on analysis of cracked elastic materials include Kachanov (1992), Nemat-Nasser et al. (1993), and Kushch and Sangani (2000), as well as the textbook by Nemat-Nasser and Hori (1993). Some of the notable work on dry cracked solids using techniques similar to those that will be employed here includes Zimmerman (1985), Laws and Dvorak (1987), Hashin (1988), and Sayers and Kachanov (1991). Pertinent prior work on both dry and saturated cracked rocks includes Walsh (1969), Budiansky and O'Connell (1976), O'Connell and Budiansky (1974, 1977), Henyey and Pomphrey (1982), Hudson (1981, 1986, 1990), and Mavko and Jizba (1991).
The purpose of this paper is to obtain approximate analytical results for the elastic moduli of dry and fully-saturated cracked rock based on Differential Effective Medium (DEM) theory (Bruggeman, 1935; Cleary et al., 1980; Walsh, 1980; Norris, 1985; Avellaneda, 1987). Penny-shaped cracks have been used extensively to model cracked materials (Walsh, 1969; Willis, 1980; Kachanov, 1992; Smyshlyaev et al., 1993), but the penny-shaped crack model is itself an approximation to Eshelby's results (Eshelby, 1957; Wu, 1966) for oblate spheroids having small aspect ratio. In order to obtain some analytical formulas that are then relatively easy to analyze, a further simplifying assumption is made here that certain variations in Poisson's ratio with change of crack porosity can be neglected to first order. The consequences of this new approximation are checked by comparison with numerical computations for the fully coupled equations of DEM. The agreement between the analytical approximation and the full DEM for cracked rock is found to be quite good over the whole range of computed porosities. Justification for the approximation is provided in part by an analysis of the actual variation of Poisson's ratio and some further technical justifications are also provided in two appendices.
For simplicity, the main text of the paper treats materials having only crack porosity, and we consider these models to be granite-like idealizations of rock. A third appendix shows how the results of the main text change if the model is treated instead as a sandstone-like material having finite stiff porosity in addition to the soft, crack porosity.