On integral and differential porosity models for urban flooding simulation

A novel differential porosity model for urban flooding, namely the Binary Single Porosity model (BSP), is proposed in the present paper. The BSP model, which is derived from the Single Porosity (SP) model by constraining the porosity to attain only the values zero inside the buildings and one in the voids among the buildings, is local and independent on the existence of a Representative Elementary Volume (REV). The BSP model satisfies the Galilean invariance, while the corresponding wave speeds are identical to those of the Shallow water Equations model, and its integral formulation coincides with the original integral model by Sanders et al. (2008). The structure of the BSP model implies that the solution of the SP Riemann problem is the numerical building block for the construction of the corresponding Finite Volume schemes. This observation prompts a further study of the SP model and its solutions, demonstrating that the exact SP Riemann problem solution has the potential to take into account the transient energy losses due to wave reflections through the urban fabric in BSP models. Nonetheless, a further comparison with the two-dimensional SWE results demonstrates that additional stationary energy dissipations must be accurately taken into account through porosity reductions in the case of supercritical flow. The numerical experiments show that available approximate SP Riemann solvers may cause a systematic underestimation of the energy dissipation through the urban fabric and an overestimation of the flood celerity. The improvement of SP Riemann solvers could limit the resort to additional drag and momentum dissipation terms that are frequently added in numerical models. Finally, the investigation of the differential porosity models where different definitions are used for storage and conveyance porosity shows that these models suffer from a fundamental lack of physical congruence, implying that they cannot be used for the analysis of flood wave propagation through the urban fabric