Numerical tools have become ubiquitous in design of manufactured porous materials. Many methods have been developed for imaging, reconstruction, material property estimation, and generation of materials in a virtual environment with the ultimate goal of understanding the connection between the synthesis process, material microstructure, and material properties. In previous works, we presented a new random field-based generation technique called adjustable level cut filtered Poisson field (ALCPF). We paired the ALCPF technique with a flow simulation method, the pore topology method (PTM), to compute material properties and verify that targets have been attained. Building on our earlier work where we demonstrated the ability of ALCPF to efficiently generate a wide variety of homogeneous microstructures, we pursue three new goals. First, we extend ALCPF to produce heterogeneous asymmetric porous materials with a target pore size gradient. Second, we demonstrate the capability of asymmetric-ALCPF to control both solid and void spaces by generating virtual asymmetric materials with different types of solid matrix geometries and void space pore size gradients. Third, we use these materials to assess the accuracy of PTM results in comparison with the solution from a direct numerical simulation. This work demonstrates that the ALCPF method successfully generates porous microstructures with desired asymmetric geometry with less than 4% error compared to target pore size gradient. Also, PTM estimates permeability with an average error of less than 7% compared to direct numerical simulation results.