Hydraulic permeabilities of fiber networks are of considerable interest in a wide variety of applications such as paper production, filtration, fibrous beds for manufacturing processes and transport in biological systems, and for that reason they have been studied extensively. There is little work, however, on permeability calculations in three-dimensional random networks. Computational power is now sufficient to calculate permeabilities directly by constructing artificial fiber networks and simulating flow through them. Even with today's high-performance computers, however, such an approach would be infeasible for large simulations. It is therefore necessary to develop a correlation based on fiber volume fraction, radius and orientation, preferably by incorporating previous studies on isotropic or structured networks.

In the present work, two methodologies for permeability calculations in three-dimensional isotropic and oriented fiber networks will be presented. Fiber networks are generated stochastically and served as a basis of comparison for the two methods. In the first approach, finite element models of the networks are generated, and the Stokes equations are solved directly. The predictions of this direct method are considered to be exact. Subsequently, a volume-averaging method is employed, and the permeability is determined by adding the contribution of each fiber to the total network drag based on existing correlations for drug coefficients for flow parallel and perpendicular to a single fiber or an array thereof.

We will show that when drag coefficients for spatially periodic arrays are used the results of the volume-averaging method agree well with the direct finite element calculations. On the contrary, the use of drag coefficients for isolated fibers overpredicts the permeability for the volume fraction range that is employed. We conclude that a weighted combination of drag coefficients for spatially periodic arrays of fibers could be used as a good approximation for fiber networks, which further implies that the effect of the fiber volume fraction and orientation on the permeability of fiber networks is more important than the effect of local network structure. We will also show that there is a strong dependence of the permeability on the network orientation.