Let consider a porous medium made of a solid frame and a pore space filled by fluid. The coupled problem between fluid flow and mechanical deformation can be effectively modeled by the Biot-Gassmann-Coussy theory (Biot, 1941; Gassmann, 1951; Coussy, 2004). It is a basic subject of the poromechanics that is largely applied to model underground structures such as petroleum reservoirs and wells, tunnels or buildings' foundations, etc.

For an isotropic porous medium, the main parameters those are required for a standard poromechanical coupled simulation are the drained bulk and shear elastic moduli (or the drained Young modulus and Poisson ratio), the Biot' coefficient and the Biot' modulus. The effective drained elastic moduli can be estimated by the classical homogenization theory while the Biot' parameters can be computed via the Biot-Gassmann formulas.

It is of interest to remark that the compressional and shear velocities of a sonic wave that passes through a saturated porous medium can also be computed using the undrained elastic moduli and the bulk density. The undrained elastic moduli can be calculated using the drained moduli and the Biot's parameter. The bulk density is a simple volumetric average between the densities of the solid and fluid phases. Such a relationship between sonic wave velocities and the elastic moduli offers a possibility to predict, by an inverse ways, the poroelastic properties and the porosity of a porous medium using sonic measurement.

An example of simulating the effective poroelastic and sonic properties of a porous medium is given below. Here, the Hashin-Shtrikman's upper bound (Hashin and Shtrikman, 1963) is employed to compute the drained bulk and shear elastic moduli.

References

Biot, M. A. (1941). General theory of three‐dimensional consolidation. Journal of applied physics, 12(2), 155-164.

Gassmann, F. (1951). Elastic waves through a packing of spheres. Geophysics, 16(4), 673-685.

Hashin, Z., & Shtrikman, S. (1963). A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids, 11(2), 127-140.

Coussy, O. (2004). Poromechanics. John Wiley & Sons.