A constitutive equation and generalized Gassmann modulus for multimineral porous media

We derive the time-domain stress-strain relation for a porous medium composed of n - 1 solid frames and a saturating fluid. The relation holds for nonuniform porosity and can be used for numerical simulation of wave propagation. The strain-energy density can be expressed in such a way that the two phases (solid and fluid) can be mathematically equivalent. From this simplified expression of strain energy, we analogize two-, three-, and n-phase porous media and obtain the corresponding coefficients (stiffnesses). Moreover, we obtain an approximation for the generalized Gassmann modulus. The Gassmann modulus is the bulk modulus of a saturated porous medium whose matrix (frame) is homogeneous. That is, the medium consists of two homogeneous constituents: a mineral composing the frame and a fluid. Gassmann's modulus is obtained at the low-frequency limit of Biot's theory of poroelasticity. Here, we assume that all constituents move in phase, a condition similar to the dynamic compatibility condition used by Biot, by which the P-wave velocity is equal to Gassmann's velocity at all frequencies. Our results are compared with those of the Berryman-Milton (BM) model, which provides an exact generalization of Gassmann's modulus to the three-phase case. The model is then compared to the wet-rock moduli obtained by static finite-element simulations on digitized images of microstructure and is used to fit experimental data for shaly sandstones. Finally, an example of a multimineral rock (n > 3) saturated with different fluids is given.