Molecular and energy transport in several porous materials can occur through both the pore space as well as the solid phase. In the case of paper and paperboard structures, moisture can be conducted via diffusion in the air through the pore space under low moisture conditions, but primarily through the fiber phase at high moisture contents. Prediction of the overall diffusivity or conductivity in such cases can become quite complex and the impact of the pore-fiber microstructure on transport is not possible. In the present work, a new method based on random walk inside the pore space allowing for a restricted walk inside the conducting fiber phases was applied to show the impact of this dual conduction on the overall transport of moisture (or analogously, heat). Model verification using periodic array of spheres showed the non-linear relationship between normalized effective diffusivity and porosity of the structure in correspondence with Archie's equation and agreeing reasonably well with literature. The structures of samples of paperboard were first assembled using X Ray Computed Tomographic (XR-CT) reconstruction after which a hybrid random walk algorithm was implemented. The walker was allowed to penetrate the fiber space in proportion to the conductivity ratio and also with an adjusted path length. Paper and fiber structures are strongly anisotropic, with the structure in the thickness dimension (z) being different from the in-plane dimensions (x, y). This is reflected in the anisotropy of the diffusivity tensor. Assuming fibers to be nonconductive, laboratory made paper fiber structures showed a nonlinear relationship between normalized effective diffusivity and porosity with in-plane direction being more conductive than the transverse direction. Since both the pore and fiber spaces are conductive to moisture, the diffusivity tensor has contributions from each space and can be assembled from the diffusivities in each of these phases individually. An important result from the present work is that initially anisotropic porous structures become more isotropic as the fiber phase becomes more conductive. Experimental diffusivity data and model simulations were compared showing the effect of structure on the anisotropic diffusion characteristics of paper. The results are presented in the form of a proposed 'effective diffusional tortuosity' and the intrinsic fiber diffusivity, as fundamental characteristics of cellulose fibers. With the development of new nanofibrous cellulosic materials, it is expected that net effective conduction parameters accounting for multiple phase random walks will be more prominent in controlling material properties. (C) 2018 Elsevier Ltd. All rights reserved.