| Flow through porous media, in a geotechnical context, plays an important role in disciplines such as groundwater resources and oil geology. The ability to accurately predict fluid flow behavior and extraction capacity is of key importance, especially in pumping well applications such as in the petroleum industry. In general, most available numerical formulations are based on Darcy's law which is used to describe seepage flow, or Stokes' law which is used to describe channel flow. These approaches approximate the flow behavior through homogeneous media with either low or high permeability, respectively. Soils are naturally heterogeneous, and when inspected at a fine scale, often reveal multiple differing soil types, and thus, varying permeability values. Standard finite element methods when applied to the Darcy-Stokes problem lead to unstable formulations. Especially the equal order velocity-pressure interpolations violate the Babuska-Brezzi inf-sup condition. This thesis presents a stabilized method that accounts for the heterogeneity of soils at the fine-scales to more accurately predict fluid flow at the global scale. The structure of the stabilization parameter is derived from the fine-scale problem. The method yields a family of finite elements with equal and unequal order pressure-velocity combinations for application to the porous media flow problems. Several benchmark problems are presented to show the stability and convergence properties of the new method. |
