Poroelastic shear modulus dependence on pore-fluid properties arising in a model of thin isotropic layers

Gassmann's fluid substitution formulas for bulk and shear moduli were originally derived for the quasi-static mechanical behavior of fluid-saturated rocks. It has been shown recently that it is possible to understand deviations from Gassmann's results at higher frequencies when the rock is heterogeneous, and in particular when the rock heterogene-ity anywhere is locally anisotropic. On the other hand, a well-known way of generating anisotropy in the earth is through fine (compared to wavelength) layering. Then, Backus' averaging of the mechanical behavior of the layered isotropic media at the microscopic level produces anisotropic mechanical and seismic behavior at the macroscopic level. For our present purposes, the Backus averaging concept can also be applied to fluid-saturated porous media, and thereby permits us to study how and what deviations from Gassmann's predictions could arise in an elementary fashion. We consider both closed-pore and open-pore boundary conditions between layers within this model in order to study in detail how violations of Gassmann's predictions can arise. After evaluating a number of possibilities, we determine that energy estimates show unambiguously that one of our possible choices – namely, G (2) e f f = (C 11 + C 33 − 2C 13 − C 66)/3 – is the correct one for our purposes. This choice also possesses the very interesting property that it is one of two sets of choices satisfying a product formula 6K V G e f f = ω + ω − , where ω ± are eigenvalues of the stiffness matrix for the pertinent quasi-compressional and quasi-shear modes. K R is the Reuss average for the bulk modulus, which is also the true bulk modulus K for the simple layered system. K V is the Voigt average. For a polycrystalline system composed at the microscale of simple layered systems randomly oriented in space, K V and K R are the upper and lower bounds respectively on the bulk modulus, and G (2) e f f and G (1) e f f are the upper and lower bounds respectively on the G e f f of interest here. We find that G (2) e f f exhibits the expected/desired behavior, being dependent on the fluctuations in the layer shear moduli and also being a monotonically increasing function of Skempton's coefficient B of pore-pressure buildup, which is itself a measure of the pore fluid's ability to stiffen the porous material in compression.