NUMERICAL ASPECTS IN SIMULATING OVERLAND FLOW EVENTS Pierfranco Costabile1, Carmelina Costanzo1 & Francesco Macchione1 LAMPIT, Dipartimento di Difesa del Suolo , University of Calabria , Italy email:
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ABSTRACT 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.
MATHEMATICAL MODELS The implemented codes are based on the fully conservative shallow water equations: ∂U ∂F ∂G + + =S ∂t ∂x ∂y
(1)
where: r− f hu hv h 2 2 U = hu ; F = hu + gh / 2 ; G = huv ; S = gh ( S0 x − S fx ) hu 2 2 huv gh ( S0 y − S fy ) hv + gh / 2
(2,3,4,5)
in which: t is time; x, y are the horizontal coordinates; h is the water depth; u, v are the depth-averaged flow velocity in x- and y- directions; g is the gravitational acceleration; S0x, S0y are the bed slopes in x- and y- directions; Sfx, Sfy are the friction slopes in x- and y- directions, which can be calculated from Strickler’s formula; r is the rain intensity and f are the infiltration losses. By neglecting the local and convective acceleration in the momentum conservation equations, it is possible to obtain the following diffusive model: ∂U d ∂Fd ∂G d + + = Sd ∂t ∂x ∂y
(6)
with: r− f hu hv h 2 U d = 0 ; Fd = gh / 2 ; G d = 0 ; S d = gh ( S 0 x − S fx ) 0 gh 2 / 2 0 gh ( S 0 y − S fy )
(7,8,9,10)
and ignoring also the depth gradient terms one may obtain the following kinematic model: ∂Uk ∂Fk ∂G k + + = Sk ∂t ∂x ∂y
(11)
where: r− f h hu hv U k = 0 ; Fk = 0 ; G k = 0 ; S k = gh ( S 0 x − S fx ) 0 0 0 gh ( S0 y − S fy )
(12,13,14,15)
NUMERICAL MODEL AND TECHNIQUES FOR PREVENTING INSTABILITY PROBLEMS The systems (1), (6) and (11) have been solved using the Mac Cormack scheme, not reported here for a lack of space, that is of second order of accuracy in both time and space; the TVD algorithm has been also considered in the simulations on real topography. The MacCormack scheme is widely used in the flood propagation analysis and its performances have been analyzed in deep, also by comparisons with other numerical schemes, in previous works of the authors (Costanzo et al. 2002, Macchione & Morelli 2003, Costanzo & Macchione 2005). As it will be shown for the complete model (eq. 1), the MacCormack-TVD scheme may present serious problems of stability or significant errors in terms of water volume conservation. More in general, it is well-known in the literature that small depths over complex topography and wet-dry interfaces may lead to several numerical problems; in overland flow simulations these problems clearly are amplified by the presence of a great number of computational dry cells that become wet because of the rainfall input and subsequently dry out due to high bed slopes. The influence of some numerical techniques will be thus analyzed herein in order to evaluate the improvements induced on the solution in those cases in which the simulations have to be carried out in a complex topography. Three kind of controls have been implemented in the MacCormack-TVD scheme. The first one deals with the interface between dry and wet cells. Let's consider a wet cell next to a dry cell on the right where hL≠0, hR=0 and the surface water level HR>HL; in this case, without any modification, a non physical flux will be necessarily computed across the common interface. So a temporary modification of the right cell bed level has to be introduced to avoid this spurious flux and the wet cell velocity component normal to the wet-dry cell interface is set to zero to ensure the no flow condition (Burguete et al. 2004; Liang & Borthwick 2009). The equation (1) solved with the MacCormack-TVD scheme in which the above described modifications are used will be called Model 1. Another source of errors and instabilities comes from steep bed slopes: in particular, more water than that actually contained in the wet cell may be computed as flowing in the dry cell and, as a consequence, the negative water depth will induce numerical instability; in this case a number of modifications have to introduced in the scheme to prevent the code failure. First of all, a minimum
water depth hs has to be considered and when a lower value is computed the cell is considered dry with the velocity components set to zero and the cell itself is filled up to hs; then, to ensure the mass conservation, water is subtracted from the adjacent cell containing most water and, in the same cell, the variables hu and hv are also modified in a such way to maintain u and v the same as before. This technique implemented within the Model 1 leads to another model that hereafter will be called Model 2 The last modification concerns the spurious effects induced by the friction slope. Singularities may arise due to the bed friction term when the water depth becomes very small as it is often present in overland flow simulations: for small water depths, the bed friction term dominates over other terms in the momentum equation, as the term h4/3 appears in the denominator. The most commonly and simple procedure is the pointwise discretization of the terms independent of the methodology used for the rest of the system. However this criterion in presence of high friction or low water depth leads to numerical instabilities. Therefore an implicit or semi-implicit treatment of the friction source term is fundamental (Brufau et al. 2004, Costanzo & Macchione, 2006, Liang et al. 2007). The model 2, with a semi-implicit treatment of the friction terms will be called Model 3. On the other hand, Burguete et al. (2008) amd Heng et al. (2009) proposed a limitation of the friction term in order to avoid that friction force exceeds the sum of the other terms in momentum equations with consequently change of the velocities sign; anyway this condition have not applied herein. NUMERICAL APPLICATIONS Several numerical tests concerning overland flow phenomenon exist in the literature and some of them have been already used by the authors in another paper for model validation purposes (Costabile et al. 2009). In this work, in particular, three numerical tests have been used to compare the performances of the kinematic, diffusive and complete models proposed herein. Then the attention will be focused on an ideal case of overland flow in a real topography to highlight the numerical problems encountered due to small water depths on irregular topography. Variable in time rainfall intensity over a plane These tests, proposed for example in Govindaraju et al. (1988) and Gottardi & Venutelli (2008), consist in a variable in time and constant in space rainfall intensity over a plane, 22m long, with constant slope and Chézy coefficient χ=1.336 m1/2/s. Two different slopes have been considered: 0.001 and 0.04. The numerical results are shown in figures 1a and 1b respectively. In these tests the numerical runoff computed by simplified models are compared with the solutions obtained by the complete models. The computational domain for both tests has been divided in the cells of dimensions 0.1 x 0.1 m while the Courant number is set to 0.1.
Figure 1. Comparison among the simulated runoff hydrographs at the channel outlet: slope 0.001 (a) and slope 0.04 (b).
It is interesting to observe that when using a slope equal to 0.001 the simulations are quite different and in particular the kinematic approximation gives a poor prediction because, in this case, the depth gradient contribution is not negligible in comparison to the bottom slope (figure 1a). However the solutions of the implemented schemes are very similar when using a slope equal to 0.04 (figure 1b). For both cases the numerical results agree with those presented by other authors. Variable in space constant in time rainfall intensity over a cascade of plane These experiments, carried out by Iwagaki (1955) and used as validation test in Feng & Molz (1997) and Fiedler & Ramirez (2000), consist in a variable in space but constant in time rainfall intensity over a cascade of three planes. Each plane section was 8 m long, with slopes of 0.02, 0.015 and 0.01 in the downstream directions; each section received a constant rainfall input of 389, 230 and 288 cm h-1 respectively. Discharge and water depth hydrographs are available with reference to three rainfall duration t=10 s, t=20 s, t=30 s but only the test relative to the shortest rainfall duration has been used herein; indeed it is the most difficult scenario, from a numerical point of view, because in this experiment a shock wave, which arrives at the downstream end at approximately 25 s, is produced (Fiedler & Ramirez 2000). The computational domain has been obtained using a structured mesh with a cell size equal to 0.1 m; the Manning coefficient was set equal to 0.01 s/m1/3. In the figure 2 the comparison between the numerical results and the experimental data is shown; in particular, the water depth profiles (figure 2b) refer to the instant of time in which the rain ends (10 s). For this test, the numerical simulations gave different predictions of the flood wave at the end of the last plane (figure 2a). In particular the fully dynamic and kinematic model make good predictions of the observed
discharge peak value while the diffusive model provides a significant underestimation of the maximum discharge itself. The prediction of the water depth profiles provided by the simplified models is poor: in the first plane, a systematic underestimation of the water level is simulated; then the numerical results give a sudden rise of the water level, not observed in the experiment, at the beginning of the second plane along which the water depth is clearly overestimated. The hydraulic jump, that occurs at the beginning of the third plane, is simulated very badly. The fully dynamic model works very well as for the flood wave as for the water level profiles showing a good agreement with the experimental data. Once again the numerical results achieved by the implemented codes are similar to those presented in the literature by other authors for the same tests.
Figure 2. Comparison between experimental data and numerical results: (a) flood wave at the channel outlet, (b) longitudinal water depths profile at the end of the rainfall input
Ideal test in real topographies The above discussed tests have shown that the MacCormack scheme is quite suitable to provide accurate solutions without any numerical oscillations or instability problems and ensuring, at the same time, a quite good conservation of the volume. Indeed the idealized topography used can be considered only a drastic simplification of what one may encountered in practical studies characterized by sudden bottom variation and adverse slope. So the suitability of a numerical model has to be evaluated using more complex topographies than planar situations. In the figure 3, two different topographies are shown: the surface elevation depicted in the figure 3b is the same of that represented in the figure 3a except for the presence of two hills put in the domain to make it more complex with presence of both adverse slopes and of sudden bottom variations. The dimensions of the computational domain are about 950 m x 1100 m. Rainfall intensity has been assumed constant in time and space: in particular a low value has been considered (10 mm/h) for dealing with a very difficult conditions from a computational point of view. The domain has been divided according to a structured grid whose cell size is 20 m. The Strickler’s coefficient is constant in the whole domain (10 m1/3/s) and the infiltration rate is set to zero. In all the simulations the water depth threshold hs has been assumed equal to 10-7 m.
Figure 3. Simple (a) and complex (b) topography used in the numerical tests
In the figure 4a the simulated discharge hydrographs, obtained using the simple topography, are shown; indeed, for this case, the Model 1 is quite sufficient to obtain fairly good results without particular oscillations or physical anomalies; therefore the other models (Model 2 and Model 3) do not induce any significant improvements to the solution even if the Model 3 seems to provide the most accurate solution. In particular, there are no particular errors in terms of volume conservation during the whole simulation as it may be observed in the figure 4b in which the comparison among the cumulative rainfall water volume falling onto the domain and the simulated volumes, computed for each instant of time as the outflow volume plus the volume within the basin, is shown.
Figure 4. Simple topography: discharge hydrographs in the comparison section (a) and overall volume over time simulated by the models (b)
The conclusions are very different for the simulations carried out over the complex topography. As it may be noticed in the figure 5b, the results of both the Model 1 and 2 show a loss of volume for the whole rainfall duration and then a significant increase of the simulated volume; in particular, the Model 2 provides some benefits limiting this spurious effect while the Model 1 completely fails in terms of mass conservation. The Model 3 gives the best result ensuring a very good respect of the volume conservation during the period of the rainfall input and a reasonably good behavior during the recession phase of the phenomenon. The differences may be observed also in terms of discharge hydrographs in the comparison section that highlight the increase of the volume simulated by the Model 1 and some significant physical anomalies in the shape of the flood wave; a better simulation is provided by the Model 2 which, however, still suffers of numerical problems while the Model 3 seems to give the most reliable result.
Figure 5. Complex topography: discharge hydrographs in the comparison section (a) and overall volume over time simulated by the models (b)
CONCLUSIONS The analysis of the hydraulic processes associated to the overland flow phenomena starts from the choice of a suitable model able to describe the main features of propagation dynamic. So, in this paper, the performances of three overland flow models, based on the fully dynamic shallow water equations and the relative approximations (kinematic and diffusive wave), have been compared and discussed with reference to three experimental tests. The numerical integration of the systems have been carried out using the MacCormack scheme. The results have highlighted the models predict the same discharge hydrograph at the end of the domain in the case of constant rainfall over a steep plane. In the other two tests the simulations are different. In particular, the diffusive and complete model still continue to provide similar results in the case of constant rainfall over a mild slope while the kinematic wave solution significantly overestimates the flood peak values. On the other hand, the kinematic and the fully dynamic model give similar results, in terms of discharge hydrographs, in the simulation concerning variable in space and constant in time rainfall intensity over a cascade of planes while the diffusive model does not work well providing an underestimation of the peak value; in terms of water depth profiles, the simplified models gives very poor results compared with both the experimental data and the complete model. These considerations underline that the simplified model have to be used carefully in the practical studies; the complete model is thus recommended. However, when applied to complex topography, the MacCormack-TVD scheme shows a number of numerical problems that may result in significant errors in the mass conservation and numerical oscillations that induce the code failure. These problems have been highlighted making two different simulations: the first one deals with a real but simple topography while the second one is the same as before but complicated by the presence of two steep hills. The first test has been carried out without particular problems with a reasonably good conservation of the volume instead the simulation of the second test highlights significant increasing errors that occur as during the rainfall input as after the end of the rainfall. So a number of modifications have been introduced in the model and their relative importance have been analyzed; in particular, the
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