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.

4/29/2001