For hydraulic routing through coarse rockfill dams, there is still debate on whether the inertia terms might be neglected as a result of the drag force generated by the rock materials. In this study, a one-dimensional unsteady model for flow-through rockfill dams is built. For this purpose, inertia terms of Saint–Venant equations are disregarded. A semi-implicit scheme adopted for linearizing the nonlinear friction term within the time integration satisfies the Courant–Friedrich–Lewy stability criterion. The most challenging issue in the modeling of flows through rockfill dams is the appropriate definition of boundary conditions at the dam's exit zone. In addition to the analysis of different exit boundary conditions proposed in the literature, a Neumann-type boundary condition suitable for the non-inertia wave equation is also employed to estimate the exit boundary condition. This procedure is basically in appreciation of the nonlinear behavior of the water surface closer to the exit boundary. Due to the existence of the sloping edges in the trapezoidal-shaped dam, an effective length is considered for the solution domain. Finally, the model is compared with observed data and a dynamic wave model. A very good match is observed, which builds confidence in the presented modeling approach.

2 3To predict the phreatic surface and conduct flood routing of rockfill dams, the flow hydraulics and rating curve need to be simulated. In this research, one-dimensional flow hydraulics through a trapezoidal rockfill dam was modelled via the standard step method. Regarding the non-Darcy flow, the relationship between the hydraulic gradient and the bulk velocity in power form was adopted. Using the exit height as the boundary condition significantly improves the prediction of the phreatic surface. For this purpose, a modified Pavlovsky equation was introduced. The advantage of the proposed method is that, unlike in previous studies, the downstream water level is used to estimate the exit height instead of the upstream water level, which makes the model more grounded on steady gradually varying flow. The proposed model has the capability of predicting the reservoir head-outflow relationship and routed hydrographs with high accuracy comparable with that of two-dimensional models.

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Two meshless (or mesh-free) methods that are widely employed in hydrodynamic modeling of problems with steep gradients and large deformation are smoothed particle hydrodynamics (SPH) and moving particle simulation (MPS) that are classified into two categories of incompressible and weakly compressible fluids (Gotoh & Khayyer, 2016). Although SPH was initially introduced for astrophysical applications (Gingold & Monaghan, 1977), in the scope of hydrodynamics, SPH and MPS were developed by Monaghan (1992) and Koshizuka and Oka (1996), respectively, for solving the free-surface incompressible flow using Navier-Stokes equations. Gotoh et al. ( 2001) presented a sub-particle scale closure model for closing large eddy simulation (LES) to simulate turbulence fluctuations in the MPS method. The theory of a weakly compressible flow was adopted by Monaghan (1994) in SPH and Shakibaeinia and Jin (2010) in MPS using a thermodynamic equation of state for calculating the pressure instead of resorting to the Poisson equation.The main difference between SPH and MPS arises from different differential operators employed in the spatial integration of the partial differential equations. In SPH, the differential operators are obtained from the differentiation of a weighted average function, the so-called kernel, without directly applying differencing schemes for flow variables. The superposition of the kernels then computes flow variables' derivatives (Liu & Liu, 2010). In contrast, MPS is the extension of the finite volume method to a mesh-free approach in which the spatial discretization is obtained based on the differentiation of flow variables. The weighted average of physical quantities derivatives of neighboring particles is applied to the target particle. However, unlike the finite volume method in which the velocity divergence is used for pressure calculation, the particle number density is adopted in MPS. The particle number density indicating the number of particles surrounding a particle is a key variable in MPS that is straightforward to fulfill flow incompressibility (Koshizuka et al., 2018).