Numerical Aspects in Simulating Overland Flow Events

In this paper a comparative analysis of 2D numerical models for overland flow simulations have been performed; in particular they are based on the shallow water approach according to fully dynamic, diffusive and kinematic wave modeling. The systems have been numerically solved by using the MacCormack second order central scheme. The models have been tested, validated and compared using numerical tests commonly used in the literature. Afterwards the implemented codes have been also applied to simulate overland flow over real topography. In this contest some numerical anomalies appear due to the presence of small water depth over high slope and irregular topography. A careful analysis of these aspects has been made and some numerical techniques have been implemented in order to prevent them. INTRODUCTION Surface runoff is a dynamic part of the response of a watershed to the rainfall; it is known to cause surface erosion and it is quite often associated to a sudden rise of the stream hydrograph. In particular, convective rainstorm events often induce extreme floods, known as flash floods, that are one of the most destructive natural hazards in the Mediterranean region. So the use of modeling approaches seems to be necessary for both predicting the flood-prone areas and, consequently, planning the damage minimization policies. The use of an accurate model is very important to manage the risk associated with potential extreme meteorological events at basin scale. The mathematical modelling of overland flow is very complex because it involves the description of several phenomena such as the surface flow and groundwater flow with seepage at the ground surface (Singh & Bhallamudi 1998). In particular, the hydraulic description of the overland flow is very important for flood risk assessment because it allows the computation of flow depths and velocities. So a strong effort has been made in the literature for modelling these situations leading to a number of numerical models based on different levels of detail according to the simplifications introduced to describe the hydraulic processes. The most powerful approach to deal with these situations is represented by 2D shallow water equations because of their capacity to simulate a number of complex phenomena that occur in real topography such as backwater effects and transcritical flow. The work of Zhang & Cundy (1989) represents one of the first attempt to model overland flow using the fully 2D shallow water equations; the authors used a finite difference scheme to solve them. Esteves et al. (2000) and Fiedler & Ramirez (2000) developed numerical models that couple the surface flow and infiltration process considering the variations in the topographic elevation and in the soil hydraulics parameters; both the models are based on the MacCormack scheme for solving the unsteady flow equations. Indeed a number of numerical problems exist in the use of the 2D unsteady flow modeling for the propagation of a surface runoff in complex topography; these are mainly due to small water depths over high slope, adverse slope and irregular topography. For example Ajayi et al. (2008) proposed a model to simulate hortonian overland flow in which, within a numerical model based on the leapfrog scheme for the integration of 2D shallow water equations, a time filtering is introduced to ensure smooth results in consecutive time steps. Unami et al. (2009) used the 2D complete unsteady flow equations, solved with a finite volume method, to study the runoff processes in Ghanaian inland valleys during flood events; in their model particular attention is paid to achieve stable computation in complex topographies. Heng et al. (2009) proposed a numerical model to describe the overland flow and the associated soil erosion phenomena; the numerical scheme, based on a MUSCL-Hancock method, minimizes the spurious oscillation that may arise from both the numerical imbalance between source terms and flux gradient and the treatment of wet-dry fronts with very shallow flows. The problems of instabilities and convergence due to highly non linear nature of the governing equations have been somewhat limited, in the past, the use of the complete shallow water equations. Moreover, in recent years there is a tendency to assume that the uncertainties over the topographic detail and roughness coefficients are predominant over the mathematical representation of the wave propagation dynamics; as a consequence recent researches in hydraulics examine reduced complexity approaches (see for example Hunter et al. 2007, Moussa & Bocquillon 2009). So in the literature the use of simplified approaches as the kinematic and diffusive wave models is common in order to simulate the overland flow processes (Di Giammarco et al. 1996; Feng & Molz 1997; Howes et al. 2006, Kazezyilmaz-Alhan & Medina 2007; Gottardi & Venutelli 2008). Several authors have studied the conditions in which those approximations are completely justified (Woolhiser & Liggett 1967; Ponce et al. 1978; Moussa & Bocquillon 1996; Moramarco et al. 2002). A comprehensive review of the applicability criteria may be found in Tsai (2003) in which the backwater effects have been also included in the analysis. In this paper, the attention is focused on two aspects. First of all, a comparison of complete and simplified 2D approaches based on the shallow water equations have been performed using literature tests. Then an analysis of the numerical problems that arise in the propagation of small depths over complex topographies are presented together with a number of numerical techniques to prevent them.