On microstructural tests for poroelastic materials and corresponding Gassmann-type relations

Abstract

This paper considers a transition from the microstructure to a linear thermodynamic model of poroelastic media that yields Biot's model. It investigates a two–component poroelastic linear model in which a constitutive dependence on the porosity gradient is incorporated, and this is compared with the classical Biot's model without added mass effects. It analyses three microstructural tests (Gedankenexperiments)—jacketed undrained, jacketed drained, and unjacketed—and derive a generalisation of classical Gassmann relations between macroscopic material parameters and microstructural (true) compressibility moduli of the solid, and of the fluid. Dependence on the porosity is particularly exposed owing to its importance in acoustic applications of the model. In particular it is shown that Gassmann relations follow as one of two physically justified solutions of the full set of compatibility relations between microstructure and the macroscopic model. In this solution the coupling to the porosity gradient is absent. Simultaneously, the paper demonstrates the second solution, which lies near the Gassmann results but admits the coupling. In both models couplings are weak enough to admit, within the class of problems of acoustic wave analysis, an approximation by a ‘simple mixture’ model in which coupling of stresses is fully neglected.