On Computational Homogenization of Flow in Porous Media
Porous materials are present in many natural as well as engineered structures. Engineering examples of porous materials include sandstone in which oil can be stored, earth dams, composite materials, air filters and sanitary products with fiber networks (e.g. diapers). The members of this class of materials possess a strongly heterogeneous substructure consisting of a ﬂuid contained in a solid matrix. As the scale of the substructural features is considerably smaller than the scale of the engineered component, taking the complete substructure into account when performing computations is virtually impossible due to limitations in computer power and memory.
The traditional approach to the modeling of seepage in a porous medium is to adopt a phenomenological model, the simplest one being the (linear) Darcy’s law, which is calibrated using experimental data. However, the absence of fundamental physical interpretation of the model implies that a change in the ﬂuid phase calls for new experiments.
This thesis concerns the modeling of porous media using a two-scale model, where a Stokes ﬂow is present on the heterogeneous subscale. Homogenization of the subscale problem is carried out on a Representative Volume Element (RVE) comprising the subscale phases. As a result, the single relevant balance equation on the macroscale is that of mass balance and Darcy’s permeability model is recovered. Both a priori homogenization (upscaling) and concurrent multiscale computations, in which the RVE problem is solved in each gausspoint on the macro level (FE2 ), are carried out. Hence, in the numerical simulations, both linear and nonlinear subscale ﬂows are considered.