In this paper, we present methods that directly model the random structure of porous media using Voronoi tessellations. Three basic structures were generated and they correspond to porous medium geometries with intersecting fractures (granular), interconnected tubes (tubular), and fibers (fibrous). Fluid flow through these models was solved by a massively parallelized lattice Boltzmann code. We established the porosity-permeability relations for these basic geometry models. It is found that, for granular and tubular geometries, the specific surface area is a critical structural parameter that can bring their porosity-permeability relations together under a unified Kozeny-Carman equation. A connected fracture network, superimposed on the basic Voronoi structure, increases the dimensionless permeability relative to the Kozeny-Carman equation; isolated large pores (vugs), on the other hand, decreases the dimensionless permeability relative to the Kozeny-Carman equation. The Kozeny-Carman equation, however, cannot distinguish a heterogeneous structure with an embedded partially penetrating fracture. The porosity-permeability relation for fibrous geometries in general agrees with those established for Simple-cubic, body-centered cubic, and face-centered cubic models. In the dilute limit, however, the dependence on the solid fraction is weaker in Voronoi geometries, indicating weaker hydrodynamic interactions among randomly interconnected fibers than those in the idealized models. (C) 2015 Elsevier Ltd. All rights reserved.