On Computational Homogenization of Fluid-filled Porous Materials

Doctoral thesis, 2015

Porous materials are present in many natural as well as engineered structures. Engineering examples include filters, sanitary products and foams while examples of natural occurrences are oil reservoirs and biological tissue. These materials possess a strongly heterogeneous microstructure consisting of a contiguous solid skeleton, more or less saturated with fluid. The scale of the substructural features is normally much smaller than that of the engineering structure. For instance, groundwater flow takes place at a length scale of kilometers while the pores and channels where the fluid is transported have a length scale of millimeters. Thus, taking the complete microstructure into consideration when performing analysis on such structures is simply too computationally demanding. Traditionally, computations on porous materials are performed using phenomenological models, the simplest one being the linear Darcy's law which relates the seepage and pressure gradient. However, this thesis concerns the modeling of porous materials using homogenization, where the macroscale properties are derived from the subscale. This technique can either be used to calibrate existing phenomenological models or in a fully concurrent setting where a subscale model replaces the macroscale material model in each Gauss point in an FE-setting. The latter constitutes the so-called FE2-approach. The obvious drawback of FE2 is that the computational cost, while smaller than the fully resolved case, is still high. Due to its mathematical and physical consistency, the method used is the Variationally Consistent Homogenization method. The ultimate goal of this work is to predict the mechanical behaviour of a two-phase material consisting of fluid that flows through a deformable open-pore solid. An important feature is that interaction between the solid and the fluid phases is taken into account. It is required that the modeling is performed in 3D, since the solid phase in a 2D model of a porous material is not connected and can, therefore, not sustain mechanical loading. The issue of imposing periodic boundary conditions on a unstructured, non-periodic mesh is addressed. Numerical results include the assesment of how the pore characteristics affect the macroscopic permeability, comparison of solutions pertaining to the fully resolved problem versus the homogenized problem, performance of weakly periodic boundary conditions and the interaction of fluid and deforming solid.