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

SUMMARY Gassmann's fluid substitution formulae for bulk and shear moduli were originally derived for the quasi‐static mechanical behaviour 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 heterogeneity anywhere is locally anisotropic. On the other hand, a well‐known way of generating anisotropy in the earth is through fine layering. Then, Backus' averaging of the mechanical behaviour of the layered isotropic media at the microscopic level produces anisotropic mechanical behaviour at the macroscopic level. For our present purposes, the Backus averaging concept can also be applied to fluid‐saturated porous media, thereby permitting us to study how deviations from Gassmann's predictions could arise in an elementary fashion. We consider both open‐pore and closed‐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 find that two choices, G (1) eff and G (2) eff , stand out and that they satisfy a pair of product formulae 6 K V G (1) eff = 6 K R G (2) eff =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 an isotropic polycrystalline system composed of simple layered systems randomly oriented at the microscale, K V and K R are the upper and lower bounds respectively on the bulk modulus, and G (2) eff and G (1) eff are the upper and lower bounds, respectively, on the G eff of interest here. For poroelasticity, we find that G (2) eff = ( C 11 + C 33 − 2 C 13 − C 66 )/3 exhibits many of the expected/desired properties of an effective shear modulus, exhibiting dependence on fluid properties. In particular, it is dependent on the fluctuations in the layer shear moduli, and also is a monotonically increasing function of Skempton's coefficient B of pore‐pressure build‐up, which is itself a measure of the pore fluid's ability to stiffen the porous material in compression. Moreover G (2) eff also reduces exactly to one‐half of the quasi‐shear eigenvalue in the special case when the bulk modulus of all the layers is the same.