Double poroelasticity derived from the microstructure

Abstract

AbstractWe derive the balance equations for a double poroelastic material which comprises a matrix with embedded subphases. We assume that the distance between the subphases (the local scale) is much smaller than the size of the domain (the global scale). We assume that at the local scale both the matrix and subphases can be described by Biot’s anisotropic, heterogeneous, compressible poroelasticity (i.e. the porescale is already smoothed out). We then decompose the spatial variations by means of the two-scale homogenization method to upscale the interaction between the poroelastic phases at the local scale. This way, we derive the novel global scale model which is formally of poroelastic-type. The global scale coefficients account for the complexity of the given microstructure and heterogeneities. These effective poroelastic moduli are to be computed by solving appropriate differential periodic cell problems. The model coefficients possess properties that, once proved, allow us to determine that the model is both formally and substantially of poroelastic-type. The properties we prove are a) the existence of a tensor which plays the role of the classical Biot’s tensor of coefficients via a suitable analytical identity and b) the global scale scalar coefficient $$\bar{\mathcal {M}}$$ M ¯ is positive which then qualifies as the global Biot’s modulus for the double poroelastic material.