Poroelasticity derived from the microstructure for intrinsically incompressible constituents

We provide a new derivation of the quasi-static equations of Biot's poroelasticity from the microstructure via the asymptotic (periodic) homogenisation method (AHM) by assuming intrinsic incompressibility of both an isotropic, linear elastic solid and a low Reynolds' number Newtonian fluid, in a small deformations regime. This is done by starting from a fluid-structure interaction (FSI) problem between the two phases at the pore scale, and by introducing both a solid and a fluid pressure, as both phases are equipped with an incompressibility constraint. Upscaling by the AHM then results in the expected Biot's equation at the macroscopic scale, with coefficients which are to be computed by solving non-standard periodic cell problems at the pore scale. These latter differ from the ones arising from classical derivations of poroelasticity via the AHM, which are typically obtained by assuming that the elastic phase is compressible, and are characterised by a saddle point structure which is inherited from the equations governing the original FSI problem. The proposed approach, which is new and cannot be derived as a particular case of existing formulations, means that the poroelastic governing equations for intrinsic incompressible phases are obtained without performing any "a posteriori" assumption on the macroscale coefficients, as these latter are typically employed based on physical arguments rather than following from a rigorous analysis of the properties of the pore scale cell problems. The advantages of the current formulation for incompressible solids are as follows. (a) The formulation is derived for two genuinely incompressible phases and in particular for an incompressible solid, which means that a reduced number of input parameters is required to compute the effective stiffness. In the case of pore scale isotropy, this means that only the shear modulus is to be provided. (b) The pore scale cell problems can be solved without approximating the pore scale elastic properties to that of an incompressible solid, i.e. in the case of isotropy, no additional errors are to be introduced by utilising approximate values of the Poisson's ratio (which in a compressible formulation can be close to, but not identical to, 0.5).