Membranes in real foam samples such as polyurethane or metallic foams only account for a very small fraction of material in the overall mass of the porous media. Yet, their role might be of primary importance in the understanding of transport and acoustical properties of these foams. As a long-wavelength wave propagates, the visco-inertial and thermal interactions between the disordered interconnected pores and the surrounding air poses a fundamental physical challenge to the microstructural identification of features which are characteristic of the overall transport phenomena. Part of the solution of this problem lies in the fact that the overall dissipation of the real porous media is expected to be dominated by a few key linkages responsible for the main energy dissipation mechanisms; and in the successful identification of the critical-path responsible for viscous ones. Here, we demonstrate that a complementary part of the solution involves the fluid-structure interaction between the (thermally conducting) air inside the interconnected pores and the membranes that partially close them. Using finite element analysis on a periodic unit-cell local geometry model, experimental estimations of transport parameters, and high resolution imaging of real foam samples, we characterize the closure rate of membranes at the junction between interconnected pores. We find that the presence of non-closed membranes between pores effectively corresponds to the introduction of a second set of critical characteristic sizes, which governs the inertial effects and meanwhile enables a correct description of the thermal ones. For an increasing rate of semi-open membranes, because of the fact that the throat size reduces, then the length Λ which has been identified as a weighted volume-to-surface ratio for the porous medium diminishes, whereas the infinite tortuosity factor that traduces strong cross section changes increases. An increasing membranes closure rate promotes the emergence of a stronger contrast between two distinctive critical characteristic sizes inside one periodic unit-cell, the size of a pore itself and the size of the interconnections between pores, which provides a scaling behavior of real polyurethane foam samples for both viscous and inertial effects. The presence of membranes also favors surface effects known to have a strong influence in transport phenomena such as diffusion controlled reactions. As characterized from the low frequency asymptotic behavior of thermal exchanges between the solid frame and the surrounding air measured in a standing wave tube, the trapping constants of the real foam samples tend to agree well with the one simulated from the previously identified visco-inertial scaling behavior of the unit-cells. A combination of advanced experiments and detailed numerical modeling of fluid-structure interactions at the pore scale reveals the basic microstructural features behind transport phenomena and shows quantitatively how these thin elements are crucial to the correct microstructural description of long-wavelength acoustic waves propagation and dissipation in real foam samples.