Modeling nonlinear phenomena in deforming fluid-saturated porous media using homogenization and sensitivity analysis concepts

Homogenization of heterogeneous media with nonlinear effects leads to computationally complex problems requiring updating local microstructures and solving local auxiliary problems to compute characteristic responses. In this paper we suggest how to circumvent such a computationally expensive updating procedure while using an efficient approximation scheme for the local homogenized coefficients, so that the complexity of the whole two-scale modeling is reduced substantially. We consider the deforming porous fluid saturated media described by the Biot model. The proposed modeling approaches are based on the homogenization of the quasistatic fluid–structure interaction whereby differentiation with respect to the microstructure deformation is used as a tool for linearization. Assuming the linear kinematics framework, the physical nonlinearity in the Biot continuum is introduced in terms of the deformation-dependent material coefficients which are approximated as linear functions of the macroscopic response. These functions are obtained by the sensitivity analysis of the homogenized coefficients computed for a given geometry of the porous structure which transforms due to the local deformation. The deformation-dependent material coefficients approximated in this way do not require any solving of local microscopic problems for updated configurations. It appears that difference between the linear and nonlinear models depends significantly on the specific microstructure of the porous medium; this observation is supported by numerical examples.