A finite element mixture model for hierarchical porous media

A finite element description of fluid flow through a deforming porous solid, with a hierarchical structure of pores, has been developed and implemented in the finite element software package DIANA. Several standard element types can be used for 2-D, axisymmetric and 3-D finite deformation analysis. The hierarchy is dealt with as an extra dimension, quantified by a parameter x₀. Both spatial and hierarchical fluid flow is described by a Darcy equation. Fluid pressure and hydrostatic solid pressure are related via an elastic fluid-solid interface. The state of the fluid, the Darcy permeability tensor and the elastic interface depend on both spatial position and hierarchical level. Discretization and integration of fluid related quantities are split into a spatial and a hierarchical part. The degrees of freedom of the finite element model are the displacements of the solid, the hydrostatic pressure and a number of fluid pressures on different hierarchical levels. Blood-perfused biological tissue can be regarded as a hierarchical porous solid, where the fluid represents the blood and the hierarchy corresponds to the tree-like vascular structure. As an example, a simulation of a contracting, blood-perfused skeletal muscle is presented.