Deriving Biot-Gassmann relationship by inclusion-based method

The quasi-static theory of poroelasticity presented by Biot and Gassmann provides a relationship between the drained and undrained elastic constants of an isotropic fluid-saturated porous material in terms of the porosity of the material, bulk modulus of the solid grains, and bulk modulus of the pore fluid. We have developed an alternative approach to derive the Biot-Gassmann (BG) relationship while including the effects of the pore microstructure. First, the Eshelby transformation is used to express the local inclusion/pore strain tensor in terms of the applied strain tensor and reference material elastic properties by the superposition of a void strain and a perturbation term due to induced inclusion stress. Second, the inclusion strain expression and Hill’s average principles are combined with the Mori-Tanaka/KusterToksöz scheme to obtain inclusion-stress-dependent effective elastic moduli of porous materials. For an isolated pore system, the effective modulus tensor corresponds to the original MoriTanaka/Kuster-Toksöz’s expression. Although for communicating pore system, it is proven to satisfy the BG relation. In the second case, the deformation is assumed to occur so slowly that the infiltrating fluid mass has sufficient time to diffuse between material elements and, consequently, the pore fluid pressure is equilibrated within the whole pore system. It is noteworthy that we arrive at a BG relationship without applying reciprocity theorem and that the porous material effective strain is defined from Hill’s principles instead of solid phase average strain. A potential application of the stress-independent effective modulus is to help develop a dynamical modulus model of rock physics for a specific pore microstructure.