Heterogeneity of the earth plays a significant role in determining geophysical and geomechanical constants such as the bulk and shear moduli and the elastic and/or poroelastic wave speeds. The heterogeneities of importance may be due to fine layering (Postma, 1955; Backus, 1962) [layers being thin compared to seismic wavelength], due to partial or patchy saturation of pore fluids (White, 1975; Knight and Nolen-Hoeksema, 1990; Dvorkin et al., 1999; Johnson, 2001; Li et al., 2001), due to random positioning of joints and fractures (Berryman and Wang, 1995; Pride and Berryman, 2003; Pride et al., 2004), due to anisotropic stress distribution, etc. There have been many attempts to attack all of these problems, and the up-scaling methods employed have ranged from ad hoc to mathematically rigorous, and have had varying degrees of success in modeling field and laboratory data.
One of the main purposes of the present paper is to introduce a semi-analytical model of the earth, and especially of fluid-bearing earth reservoirs, that provides well-controlled estimates of the properties of most interest such as elastic/poroelastic constants, fluid permeability, etc. The concept is based on ``random polycrystals of porous laminates.'' Locally layered regions are treated as laminates and the poroelastic and other constants can be computed essentially exactly (i.e., within the assumed long wavelength limit and perfect layering of the laminate model) using Backus (1962) averaging for poroelastic constants (and similar methods for other parameters), in the long-wavelength or quasi-static limits. Then, since such layered materials are typically anisotropic (having hexagonal symmetry when the layers are isotropic), I assume that the earth is composed of a statistically isotropic jumble of such layered regions. The locally layered, anisotropic regions may be termed ``grains'' or ``crystals.'' Then, the overall behavior of this system can be determined/estimated using another method from the theory of composites: the well-known Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962). In this case the bounds of interest for the types of crystal symmetry that arise are those first obtained by Peselnick and Meister (1965) and later refined by Watt and Peselnick (1980). These bounds have been refined further recently by the author (Berryman, 2004b; 2005). In particular, these recent refinements provide sufficient insight into the resulting equations that self-consistent estimates (lying between the rigorous bounds) of the elastic constants can be formulated and very easily computed. I find that the Peselnick-Meister-Watt upper and lower bounds are already quite close together for this model material, so the resulting self-consistent estimates are very well constrained. The bounds then serve as error bars on the self-consistent model estimates.
The method being introduced can be applied to a wide variety of difficult technical issues concerning geomechanical constants of earth reservoirs. The one issue that will be addressed at length here is the question of how shear moduli in fully saturated, partially saturated, and/or patchy saturated porous earth may or may not depend on mechanical properties of the pore fluids. The well-known fluid substitution formulas of Gassmann (1951) [also see Berryman (1999)] show that -- for isotropic, microhomogeneous (single solid constituent) porous media -- the undrained bulk modulus depends strongly on a pore-liquid's bulk modulus, but the undrained shear modulus is not at all affected by changes in the pore-liquid modulus. Since the system we are considering violates Gassmann's microhomogeneity constraint as well as the the isotropy constraint in the vicinity of layer interfaces, I expect that the shear modulus will in fact depend on the fluid properties in this model (Mavko and Jizba, 1991; Berryman and Wang, 2001; Berryman et al., 2002b). The semi-analytical model presented here allows me to explore this issue in some detail, to show that overall shear modulus does depend on pore-fluid mechanics, and to quantify these effects.
The next section introduces the basic tools used later in the layer analysis. The third section reviews the Peselnick-Meister-Watt bounds and presents the new formulation of them. The fourth section summarizes the results needed from poroelastic analysis. The fifth section presents the main new results of the paper, including four distinct scenarios that help elucidate the behavior of the overall shear modulus and compare it to that of the bulk modulus. The final section summarizes our conclusions. Appendix A provides a brief proof of one of the results used in the text concerning the behavior of the effective stress coefficient for patchy saturation. Appendix B shows that Hill's equation should be used cautiously in analysis of heterogeneous reservoirs.