MASCARET : a 1-D open-source software for flow

MASCARET : a 1-D open-source software for flow hydrodynamic and water quality in open channel networks N. Goutal, J.-M. Lacombe, F. Zaoui & K. El-Kadi-Abderrezzak LNHE, EDF R&D, Chatou, France

MASCARET modeling framework is a set of numerical codes simulating one-dimensional (1-D) hydroenvironmental problems through a network of open channels. The governing equations underlying MASCARET are the shallow water equations for unsteady flow propagation and the advection-dispersion equation for water quality and contaminant transport. The hydraulic component accounts for floodplains and storage areas, and flow at channel junction can be treated using a 2-D approach. Additionally, flood propagation over dry beds (e.g. dam-break flows) and non-hydrostatic waves can be simulated. The user interface FUDAAMASCARET manages the input data, allocation of parameters, running of simulations and viewing outputs. MASCARET can be easily compiled as a dynamic library, offering special interfaces to be used with three main steps : Initialization, Run and Finalization of the calculation. With these features, MASCARET can be coupled or integrated to other softwares without requiring significant efforts. Since July 2011, MASCARET is worldwide distributed as an open-source code. In this paper, are presented various applications of this tool covering both flow hydrodynamic and water quality. Model-data comparisons show the performance of the modeling framework, and demonstrate the interest of this open-source software for the scientific community.

models are commercial or freeware. MASCARET modeling framework is a set of numerical codes simulating one-dimensional (1-D) hydro-environmental problems through a network of open channels. This modeling package has been developed by Electricit´e de France-Recherche & D´eveloppement (EDF-R&D) in collaboration with Centre d’Etudes Techniques Maritimes et Fluviales (CETMEF) over more than 25 years1 . MASCARET is suitable for a wide range of engineering and environmental applications, from calculating simple backwater profiles to modeling networks of open channels including floodplains and storage areas. Applications include flood risk assessments and mapping, developing management plans, flood alleviation scheme designs, dam-break flows, and water pollution management. Since July 2011 the software package is worldwide distributed as an open-source project for the benefit of students, engineers and researchers. The remainder of the paper is organized as follows. Section 2 details the main feature of the flow component of MASCARET, including the various numerical schemes employed for resolving the water governing

1 INTRODUCTION The study of free surface water flow and water quality in open channels has many important applications, such as flood prevention measures, becoming ever more ambitious, and accidental release of contaminants in rivers. With the advances made in computational techniques over recent decades, numerical models have become proactive tools for investigating flow and water quality in networks of open channels. The selection of one model for each specific problem is usually contingent on the knowledge about the system, the available measurements, and the specific objectives of the study. One-dimensional (1-D) models require the least amount of field data, and the numerical schemes used for solving the water and sediment governing equations are more stable and offer order of magnitude gains in computational efficiency over 2-D and 3-D models (Sobey 2001). Several one-dimensional (1-D) numerical models have been proposed in the literature (e.g., HECRAS (USACE 2002), MIKE1-D (DHI 2001), ISIS (HRW 2001) and every model has its own capabilities and limitations. The development of these models has focused mainly on such aspects as numerical schemes, flow-resistance relations and main channelfloodplains flow exchange. Most of these numerical

1 http://innovation.edf.com/recherche-et-communautescientifique/logiciels/code-mascaret-41197.html

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equations. In Section 3, the water quality and pollutant transport module is described along with simulated physical/chemical processes. Section 4 presents four applications of the model modelling framework, namely: estimation of uncertainty in flood predictions for dyke design, data assimilation in the Durance River (France), flow propagation due to cascade dam break, transport of tritium in the Middle Loire River (France). This is followed by conclusions drawn in section 5.

2-D elements (Rissoan et al. 2002). In this last case, MASCARET implements a simplified 1-D-2-D coupling. The junction, see figure 1, is modelled by a 2-D representation with 12 cells :

2 HYDRODYNAMIC MODELLING The well known shallow water equations can be solved with three different computational kernels. The choice of the specific kernel depends on the case study : steady or unsteady, with subcritical and/or supercritical flows.

The junction hexagon has a geometry similar with the real one. It is built using the direction of reaches and the width of valleys given by the user. This should lead to a good discharge repartition in the different reaches.

• 6 triangles defining an hexagon, models the junction itself and can be defined as exchange cells; • 6 quadrilaterals covering the 1-D domain for the overlap coupling.

2.1 SARAP Kernel This kernel is for steady cases with subcritical, critical and supercritical conditions. The 1-D equations are solved in one step with a finite difference scheme where time derivatives are cancelled. A special treatment is performed for the shock capturing with the solution of a non-linear equation within the supercritical region in order to determine the flow level at the point of the supercritical-subcritical transition. The validity of the shock position is checked with the principle of conservation of momentum (Carlier 1998). SARAP is extremely fast due to the need of only one iteration to solve the problem. It is recommended in order to quickly find correct initial conditions for the two others unsteady kernels, or when a large amount of calculations is necessary for a parametric study (model calibration, optimisation, incertitude study, etc.).

Figure 1: Overlap coupling for the junction

MASCARET kernel is suitable for every kind of computational fluid dynamics. One will only take care of time step restriction due to the Courant-FriedrichsLewy condition if the explicit version of the time discretization is chosen.

2.2 REZO Kernel REZO implements the classical finite difference Preissmann scheme for the unsteady subcritical solution of the shallow water equations (Preissmann 1961). It is implicitly coupled with a code of water storage areas in order to take into account floodplain inundations. This code is extremely robust and fast under subcritical conditions for operational use.

3 WATER QUALITY 3.1 Governing equation and numerical scheme The water quality module of MASCARET has been designed to compute time-dependent concentrations of a series of constituents, which are primarily governed by the processes of advection, dispersion, and chemical/physical reactions. The flow and advection-dispersion equations are solved sequentially at each time step. First the flow routing module of MASCARET is called to provide the time-dependent hydraulic parameters throughout the model domain, and then these variables are passed to the water quality module for the solute transport simulation. The advection and dispersion terms are computed in two separate but consecutive steps (i.e. the well-known split-operator approach). The pure advection equation (i.e. the hyperbolic partial differential equation) can be solved using two approaches:

2.3 MASCARET Kernel The last hydrodynamic computational kernel is based on a well-balanced finite volume Roe scheme (Roe 1981) (Vazquez Cendon 1994) (Goutal and Maurel 2002). It has been designed to perform well with the simulation of dam break waves for EDF needs. It offers special capabilities comparing to other codes like the modelling of some non-hydrostatic waves (Bristeau et al. 2011), or the modelling of junctions with 2

• Method of characteristics, which can be applied either to the conservative or non-conservative form of the advection term;

• the computation code can be easily compiled as a dynamic library for being used outside the GUI; • the algorithm is divided into three main steps: Initialisation, Run and Finalisation (IRF) of the calculation;

• Finite volume method based either on a first or second order upwind scheme. For the second order scheme, a superbee limiter function suppressing numerical oscillations has been included. Because the numerical scheme is explicit, the time step is limited by the CourantFriedrichs-Levy (CFL) condition.

• it has special interfaces written in C or Fortran language that define several entries in the code, one entry per specific action; • it is also instantiable allowing multiple usage of the code in a single run.

In the second step, an implicit finite difference scheme is applied to the pure dispersion equation. This method has the advantage of being unconditionally stable. Regarding the boundary conditions, the Dirichlet condition is used at the upstream end, whilst the Neumann condition (i.e. zero-gradient) is imposed at the downstream end. The longitudinal diffusion coefficient can be either calibrated using field tracer studies or calculated using empirical formulations. The water quality module incorporates dispersion coefficient formulas proposed by many authors, see (El Kadi Abderrezzak et al. 2012).

With all these features, MASCARET is also ready for an easiest coupling or integration with other softwares while requiring no significant efforts. For the considered example, the SARAP kernel is used within the R2 software environment for statistical computing (Bivand et al. 2008). The case study is the Garonne river (France). We want to evaluate the impact of uncertainties on the water level corresponding to a 1000 years return period discharge. The uncertainties taken into account concern :

3.2 Simulated water quality processes The water quality module can be applied to simulate various processes in open channels, such as migration of nuclear power plants releases, water temperature evolution, oxygen demand, growth rate of a population of phytoplankton, and radionuclides decay including interaction with suspended material. The fate and transport of water quality constituents involving a variety of chemical and physical processes (interaction between pollutants) is mathematically expressed in the source terms of the advection-dispersion equation. The following processes have been implemented : micro-pollutants,dissolved oxygen, phytoplankton biomass, water temperature and eutrophication.

• the upstream discharge; • the friction factors for the river main bed and the floodplains; • and on the river’s topography with a stochastic process based on a kriging method (Krige 1951) Figure 2 shows some computed results on the water levels along the river considering two and three kind of uncertainties. These results, on the probability of water levels along the river, are important in order to study the role of dykes for the protection of EDF power facilities against extreme flood. Due to its low computation cost, the SARAP kernel is particularly convenient for such probabilistic studies that can require several thousands of computations.

4 APPLICATIONS Four recent applications are presented here after demonstrating some of the capabilities of MASCARET.

4.2 Data assimilation with REZO The assimilation of water level observations is done in order to improve the flood forecasting with the REZO kernel in real time conditions. The case study is a part of the Durance river (France) concerning hydroelectricity. Main uncertainties of the problem come from hydrological conditions on the Durance upstream flow and lateral inflows of river #1 and #2, see figure 3. With such unknowns, it is not possible to model correctly the water level at the dam as a function of time.

4.1 Uncertainties with SARAP The first application concerns the uncertainty affecting flood hazard assessments. Interest for this problem is increasing within the scientific community and among decision makers (Bernardara et al. 2010). The treatment of uncertainty both involves the proper definition of a deterministic model and of the probabilistic uncertainties on its inputs (De Rocquigny et al. 2008). MASCARET is well adapted for such parametric studies since it offers a modern interoperable interface:

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Figure 4: Water level forecasting with and without data assimilation Figure 2: 90% confidence interval analysis

Table 1: Errors between measurements and computations on the water level at the dam Error (cm) Mean Max

Free run 12.5 21.6

Assim run 2.2 7.8

Starting from an initial estimation of the control vector xb = (a0 , b0 , c0 )T , xa is given by the following formula which is a Best Linear Unbiased Estimate (BLUE): xa = xb + (B −1 + H T R−1 H)−1 H T R−1 d

where B is the background error covariance matrix, R the observation covariance matrix, y0 the observation and d the innovation vector :

Figure 3: Hydropower example on Durance (France) The data assimilation process will use all available measurements (water level at the dam and discharge at the plant in a 24 hours period) to correct the upstream flow. This would lead to a significant improvement in the water level computation in both analysis and forecast modes. The stationary Kalman filter is chosen as the data assimilation technique to adjust the upstream flow Q(t). The flow is corrected through a threeparameters linear transformation (Ricci et al. 2011) : ˆ = a × Q(t − b) + c Q(t)

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d = y0 − H(xb )

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Figure 4 clearly shows the interest of the filtering technique (assimilation) in order to find optimal parameters that will modify the upstream discharge (forecast). Compared with no corrections made on the inflow, the data assimilation greatly improve the forecast of water level at the dam as indicated in table 1. Due to the robustness and fastness of the Preissmann scheme, the Rezo kernel is well suited for operational problems. Once again, the code interoperability makes its integration easy.

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4.3 Dam break simulation with MASCARET In order to illustrate the ability of the finite volume scheme to deal with dam-break shock wave, a case study concerning four dams in a real valley is presented, see figure 5. The first dam upstream collapses instantaneously at the beginning of the simulation.

The control vector xa = (a, b, c)T is calculated through the comparison of simulations and observations. An observation operator H is defined to find an equivalence relation between the model results and the measured values over the chosen time window. 4

The dam #2 collapses when the wave arrives while the two others #3 and #4 downstream are supposed to resist all along the simulation.

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4.4 Tritium transport with TRACER The modelling system is applied for simulating the transport of tritium in the Middle Loire River (France) (Fig. 1), 350km long, over the period 01/01/199931/12/1999. Four main tributary streams feed the river: Vienne, Indre, Cher and Maine. Four Nuclear Power Plants (NPP) are located along the Middle Loire River and one NPP is located along the Vienne River. These NPP generate low-activity radioactive liquid waste, including tritium, which is released into the river in a controlled way. The Middle Loire River is modelled as one continuous reach with the tributaries Vienne, Indre, Cher and Maine as inflows.

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Figure 5: Initial water level conditions

This case includes several reaches where the modelling of 2-D junctions is necessary. The 1-D mesh has a size of about 2650 nodes. The time step of the explicit scheme is variable depending on a given Courant number smaller than 1. With these settings, the results shown in figure 6 can be obtained in approximately 1 mn on a standard workstation.

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Figure 7: Location map showing the study reach

Water level at 31974 s Water level at 15678 s Water level at 5629 s Water level at 1573 s Bottom

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Three hundreds and sixty-eight cross-sections spaced at intervals of about 1 km are used to describe the river geometry. The downstream boundary condition at MontJean sur Loire is a water stage-flow rate curve. For the solute transport module, the tritium discharge recorded at Belleville NPP during the period 01/01/1999 to 12/31/1999 is specified as the upstream boundary. The tritium releases from Dampierre, Saint Laurent and Chinon NPP on the same period are introduced as lateral source terms. The tritium discharge due to release from Civaux NPP is estimated by applying the MASCARET System to the Vienne River between Civaux NPP and Vienne-Loire junction. Monitoring of tritium concentration was performed during 1999 at Anger city. Calibration of the hydraulic module (i.e. Stricklers coefficients) is carried out using measured water level measurements collected along the Middle Loire River at low, medium and high flow discharges. The Strickler coefficient is set at 30m−1/3 .s−1 . The longitudinal

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Figure 6: Water levels at different simulation times

The validity of the MASCARET results is not shown for this particular case. However, several studies have already demonstrated the skill of the code for such problems. See for example (Goutal and Maurel 2002) or (Malleron et al. 2011) for a comparison with analytical solutions, measurements and other codes. 5

dispersion coefficient is computed using the formula proposed by (Seo and Cheong 1998). The first order finite volume method is used. The numerical run is carried out using a time step 10s and the space step is 200m. The courant number is set at 0.5. Figure 8 compares the numerical predictions of the tritium concentration against measurements at Anger city for two months. Generally, the model-data fit is reasonably well. The model reproduces the magnitude and general temporal evolution of the tritium concentration at Angers. The mean relative error is approximately 43%.

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REFERENCES Bernardara, P. et al. (2010). Uncertainty analysis in flood hazard assessment: hydrological and hydraulic calibration. Canadian Journal of Civil Engineering 37(7), 968–979. Bivand, R. S. et al. (2008). Applied spatial data analysis with R. Springer. Bristeau, M.-O. et al. (2011, August). Numerical simulations of a non-hydrostatic shallow water model. Computers & Fluids 47(1), 51–64. Carlier, M. (1998). Hydraulique g´en´erale et appliqu´ee. Eyrolles. De Rocquigny, E. et al. (2008). Uncertainties in industrial practice — A guide to quantitative uncertainty management. Wiley-Blackwell.

Measurements Predictions

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DHI (2001). MIKE 11 Reference Manual. Appendix A Scientific background. Danish Hydraulic Institute.

Concentration (Bq/l)

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El Kadi Abderrezzak, K. et al. (2012). Applicability of longitudinal dispersion coefficient formulas in 1-d numerical modeling of solute transport in open channels. In River Flow 2012, San Jose, Costa Rica.

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Goutal, N. and F. Maurel (2002). A finite volume solver for 1d shallow water equations applied to an actual river. International Journal for Numerical Methods in Fluids 38, 1–19.

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Figure 8: Comparison between calculated and measured tritium concentration at Angers city

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HRW (2001). ISIS 2001-User Manual. HR Wallingford. Krige, D. (1951). A statistical approach to some basic mine valuation problem on the witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa 52(6), 119– 139.

CONCLUSIONS

The open-source MASCARET software is a system dedicated to the solution of 1-D shallow-water equations with transport of passive pollutants. This system benefits from 25 years of research and development at EDF-R&D in collaboration with the CETMEF (technical branch of the Public works french department). As detailed previously, the more appropriate scheme has been integrated for each flow regime: Preissmann scheme for subcritical flow and finite volume scheme for unsteady supercitical flow. Moreover, a strong validation has shown the robustness and relevance for dealing with complex industrial studies. The last developments highlights the capability of the system to evolve towards new applications : uncertainties, data assimilation which are of main importance for real time studies and flood hazard assessments. In a near future, the MASCARET system will offer the capability to deal with flood plains by coupling between 1-D and 2-D models.

Malleron, N. et al. (2011). On the use of a highperformance framework for efficient model coupling in hydroinformatics. Environmental Modelling & Software 26(12), 1747–1758. Preissmann, A. (1961). Propagation des intumescences dans les canaux et rivi`eres. In 1er congr`es de l’Association Franc¸aise de Calcul, Grenoble, France. Ricci, S. et al. (2011). Correction of upstream flow and hydraulic state with data assimilation in the context of flood forecasting. Hydrology and Earth System Sciences 15(11), 3555–3575. Rissoan, C. et al. (2002). 1d hydraulic simulation of a dam break wave on the rhone river. In 5th International Conference on Hydro-Science & Engineering, Warsaw, Poland. 6

Roe, P. (1981, October). Approximate riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 43(2), 357–372. Seo, I. W. and T. S. Cheong (1998). Predicting longitudinal dispersion coefficient in natural stream. Journal of Hydraulic Engineering 124(1), 25–32. Sobey, R. (2001). Evaluation of numerical models of flood and tide propagation in channels. Journal of Hydraulic Engineering 127(10), 805– 824. USACE (2002). HEC-RAS Hydraulic Reference Manual. U.S. Army Corps of Engineers. Vazquez Cendon, M. (1994). Estudio de esquemas descentrados para su aplicaci`on a las leyes de conservaci`on hiperb`olicas con t`erminos fuente. Ph. D. thesis, Universidad Santiago de Compostella.

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