Modeling Overland Flow Mauricio Santillana and Clint Dawson

1. Modeling Overland Flow Mauricio Santillana and Clint Dawson OVERLAND FLOW EQUATION The main scaling assumption in overland flow modeling is the fact that frictional forces are dominant, since the water depth can be extremely shallow (in the order of centimeter or millimeters). For such scenario, the acceleration terms in the 2-D SW momentum equations are neglected leading to a relationship between the horizontal velocities and the surface water slope. The 2-D SW continuity equation combined with the free surface boundary condition become a doubly nonlinear degenerate parabolic equation given by: ABSTRACT The objectives of this investigation are to provide a detailed analysis of the equations of overland flow using Manning’s formula and Chézy’s formula and to develop a two dimensional mathematical model capable of simulating the shallow water flow, characteristic of overland flow, using Finite Element Techniques. The latter will be computationally implemented and verified. t = 30 seconds Rain duration = t t = 20 seconds MOTIVATION This investigation is part of a project whose objectives are to develop, analyze, and implement numerical algorithms which honor the scales of ground and surface water flows, and which accurately model the coupling between these flow regimes. The goal of this work is to thoroughly understand an important surface water flow model used to simulate overland flow. Understanding the assumptions and limitations of this model from a physical and a mathematical perspective will lead to the application of appropriate coupling strategies. Overland flow is a term that is used mostly to describe the shallow movement of water across land surfaces both when rainfall has exceeded the infiltration rate of the ground’s surface (Hurton Overland Flow), and when the entire soil column becomes completely saturated and water exfiltrates at the surface (Dunne Overland Flow). The main assumption in overland flow is that the fluid motion is dominated by gravity and balanced by the boundary shear stress. Bodies of water such as wetlands, see Figure 1 (a), where water flows through vegetated surfaces, have been successfully modeled using the differential equation governing overland flow, often referred to as the diffusive wave approximation of the Shallow Water (SW) equations. The diffusive wave approximation of the Shallow Water Equations Depth = h = H - z (meters) t = 10 seconds X coordinate (meters) Figure 2. Comparison of the numerically calculated depth profiles at the cessation of rainfall with Iwagaki’s experimental results on a three plane cascade. Solid lines are numerical results and dashed lines experimental results. where and See Figure 1. (b) Water Surface THEORETICAL ANALYSIS A detailed study of existence, some regularity results, a comparison result, uniqueness and nonnegativity of weak solutions of this equation is presented for the zero Dirichlet initial/boundary value problem in Alonso, Santillana and Dawson (2007). The properties of the Galerkin method as a means to approximate the solution of this equation, such as stability and a priori error estimates, are obtained in Santillana and Dawson (2007). h(x,y) H(x,y) Land Surface 8 m. 8 m. 8 m. z(x,y) 24 m. y NUMERICAL EXPERIMENTS In order to verify the performance of this model to simulate real life experiments, we decided to set and Such a choice in parameters corresponds to Manning´s friction formula. The main reason why these parameters are a good starting point is the existence of experimental data in order to validate results. A 1-D Model was implemented numerically using both, a semi-implicit in time continuous Galerkin and a discontinuous Galerkin formulation in order to reproduce the results of a set of laboratory experiments performed by Iwagaki (1955). These experiments were designed to produce unsteady flows in a channel 24m long. The channel was divided into three sections of equal length and different slopes (θ = 0.02,0.015, 0.01). See Figure 3. During experiments, three different rainfall intensities (0.108, 0.064 and 0.80 cm/s) were simultaneously applied to each section. The profiles of the numerical examples are shown in Figure 2. The numerical results matched reasonably well with the experimental results. Both formulations showed similar performance. Datum Figure 3. Iwagaki’s laboratory set up. Unsteady water flow generated by rainfall with different durations (10, 20, and 30 seconds). x (b) (a) • REFERENCES • R. Alonso, M. Santillana, and C. Dawson, Analysis of the diffusive wave approximation of the Shallow Water equations, SIAM Journal on Mathematical Analysis, in review, 2007 • V. Aizinger and C. Dawson, A discontinuous Galerkin method for two-dimensional flow and transport in shallow water, Advances in Water Resources, 25, pp. 67-84, 2002. • Daugherty, R., Franzini, J. and Finnemore, J., Fluid Mechanics with Engineering Applications, McGraw-Hill, 1985. • Feng, Ke, Molz, F.J., A 2-D diffusion based, wetland flow model, Journal of Hydrology 196, 230-250, 1997. • Iwagaki, Y., Fundamental Studies of runoff analysis by characteristics. Bulletin 10, Disaster Prevention Res. Inst., Kyoto University, Kyoto, Japan, 25pp, 1955. • M. Santillana and C. Dawson. Analysis of the continuous Galerkin formulation to solve • the diffusive wave approximation of the shallow water equations, in preparation, 2007 • Turner, A.K., Chanmeesri, N., Shallow Flow of Water Through Non-Submerged Vegetation, Agricultural Water Management 8, 375 – 385, 1984. • Vreugdenhil, C.B., Numerical Methods for Shallow-Water Flow, Kluwer Academic Publishers, 1994. • Zhang, W. and Cundy, T.W., Modeling of two dimensional flow. Water Resources Res. 25, 2019-2035,1989. Figure 1. (a) Diagram of ground and surface water. Source: www.agr.gc.ca (b) Land surface elevation (Bathymetry) is given by z, water surface elevation by H, and depth by h. SURFACE WATER MODELS Models for surface water flows are derived from the incompressible, three dimensional Navier-Stokes equations, which consist of momentum equations for the three velocity components , and a continuity equation Depending on the physics of the flow, scaling arguments are used in order to obtain effective equations for the problem in hand. In Shallow Water Theory, the main scaling assumptions are that the vertical length scale as well as the vertical velocity are small relative to the horizontal scales. These reduces the third momentum equation into the hydrostatic pressure approximation and leaves us with two effective momentum equations in the horizontal direction. Upon vertical integration, we can obtain the 2-D shallow water momentum equations. A third equation comes from combining the depth averaged continuity equation with the free surface boundary condition. FUTURE WORK A 2-D Local Discontinuous Galerkin formulation is currently being implemented. The performance of such model still needs to be tested. The next steps in the investigation are two: 1) The first one is to couple the 2-D LDG overland flow model with a 2-D / 3-D SWE model. Such model has been already implemented by Vadym Aizinger and Clint Dawson as an improvement to UTBEST ( University of Texas Bays and Estuaries Simulator ). 2) The second one is to couple such models with a two phase flow model solved using a Discontinuous Galerkin formulation developed by Shuyu Sun and Mary Wheeler.