Due to the intricate structure of carbonate rocks, relationships between porosity or saturation and petrophysical transport properties classically used for reservoir estimation and recovery strategies are either very complex or nonexistent. Thus, further understanding of the influence of the rock structure on the petrophysical transport properties becomes relevant. We therefore present a Dual Pore Network approach (D-PNM) applied to mu-CT images of bimodal porous media. The major advantage of this method lies in the fact that it takes into account the real architecture of the connected macropore network as well as the microporosity unresolved by mu-CT imaging. Whereas governing equations are solved in each individual macropore, transport behavior of microporosity is simulated by average quantities. Thus, D-PNM is particularly suited for the investigation of carbonate rocks, characterized by broad pore size distributions. We describe the principles of the image acquisition and network extraction procedure and the governing equations of D-PNM. The model is tested on three carbonate samples, two outcrop, and one reservoir carbonate. Calculated petrophysical transport properties are compared to experimental data and we show that D-PNM correctly reproduces conventional as well as unconventional electrical transport behavior. A major restriction of D-PNM is the requirement of a connected macropore network, that is, especially in the case of carbonates, not always available. Solutions to that are presented.