Models and data for flood modelling F-X. Le Dimet, C. Mazauric, W. Castaings LMC-IMAG (UJF) BP 53, 38041 Grenoble Cedex 9, France
[email protected] Abstract The prediction of the evolution of a flood requires to gather all the available information. Basically this information is provided by field observations and remote sensing, the mathematical equations governing the flow is another source of information, the last information is the statistics of the past events. The problem of data assimilation is to link these sources in order to retrieve the state of the river at a given date. In the framework of the European IST Project ANFAS, developed in cooperation with China, we are using two mathematical models: CARIMA and FESWMS. Both rely on the shallow water approximations of fluid dynamics, we will present these models and their adaptation to the Loire river pilot site. The basic principle of Data Assimilation, via optimal control techniques will be presented and the application to river hydraulics discussed.
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River modelling
1.1
Overview
The more widely used mathematical models for operational purposes in fluvial hydraulics are 1D and 2D models. Both rely on the unsteady open channel flow equations (Barr´e de Saint-Venant equations).
Quasi 2D models These models do not refer to the unsteady flow equations in two spatial dimensions. The modeling system include two flow regimes: • main channel flow: assumed to be unidirectional (along the longitudinal axis of the valley) is represented by 1D Saint-Venant equations. Each computational point corresponds to a cross section, these points are linked by 1D computational reach. In addition, particular equations are used to simulate flow trough singularities (bridges, gates, ) • floodplain flow: an inundated floodplain is represented by a series of interconnected cells delineated by natural borders such as dikes, roads, hillsides ... The flow
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in the hydraulic links between cells is calculated according to a chosen hydraulic law (Manning/Strickler, weir, orifice ...) Such models due to their relatively low computational cost can be used on very long river reaches ( hundreds of kilometers) but require considerable skill and experience if a reasonable level of accuracy is to be achieved.
2DH models When the width to depth ratio of the water body to be modeled is large, the horizontal distribution of flow quantities are the variables of interest. Thus, the integration of the depth averaged flow equations (2D Saint-Venant equations) provide a good description of the flow. However, specific 1D flow structures are introduced for hydraulic structures were the vertical variation cannot be simulated accurately using the previously cited equations. Current applications are for situations when there is significant flow over a floodplain or in meandering rivers. However, the resulting computational cost is much more important than for the previously cited model type. However, predetermination of the flow directions is not needed and and while one can calibrate the one dimensional model to reproduce physical phenomena, the key components of the flow field will be naturally reproduced by the two-dimensional model.
1.2
CARIMA
Overview CARIMA is a Quasi 2D model, developed by SOGREAH, Grenoble (France).
Governing equations CARIMA is based on the 1D Saint-Venant equations: • Continuity equation: 1 ∂Q ∂Y . + =0 B ∂X ∂t
(1)
∂V γ.β ∂V 2 ∂Y + . +g + gSf = 0 ∂t 2 ∂X ∂X
(2)
• Momentum equation:
Where X is the longitudinal distance, t is the Time, Y (x, t) the water elevation , V (x, t) the Velocity, Q(x, t) the Discharge β the Boussinesq coefficient, γ the local friction loss coefficient Sf the friction loss coefficient. The discretization of the CARIMA equations are based on the finite difference Preissmann method.
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Figure 1: CARIMA adaptation Input data To run CARIMA, we need to build a geometric file with a set of cross sections and hydraulics links in order to define the river and a set of interconnected cells which will allow us to define the floodplain. Finally, hydraulics input (boundaries conditions and initial condition) define in a second file will allows us to run different scenarios.
Output data The output data consist of: • discharge Q(t) and water depth H(t) at the computational points • water depth H(t) in the floodplain basins • Q(t) and/or H(t) for the hydraulic links
CARIMA adaptation to the Loire pilot site A topological discritization from extended similar model (reach of 400 kilometers) was used and adapted to provide bulk flow characteristics for this reach of 50 kilometers using CARIMA model. It includes 130 computational points and 40 floodplain basins (see figure ??). For model application six hydrologic scenarios (from 50 to 500 years flood) were simulated for the area.
1.3
FESWMS
Overview FESWMS-2DH is short for the Finite Element Surface Water Modeling System: 2Dimensional Flow in a Horizontal Plane. This is a 2DH a hydrodynamic modeling code supporting both super and subcritical flow analysis, including area wetting and drying.
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The effects of bed friction and turbulent stresses are included, as are optionally, surface wind stress and the Coriolis force. The flow is assumed to be strictly two dimensional except for the special cases of weir of culvert flow.In fact, the FESWMS model allows users to include weirs, culverts, drop inlets, and bridge piers in a standard 2D finite element model. Moreover, both steady state and transient solutions can be performed with FESWMS.
Governing equations For many practical surface-water flow applications, knowledge of the full threedimensional flow behavior is not needed and it is sufficient to use the depth average flow quantities in two perpendicular horizontal directions. Thus, the depth average velocity components U and V in the x and y coordinate directions are defined by
U=
1 H
Z zs zb
u dz
V =
1 H
Z zs zb
v dz
(3)
where H is the water depth, zb the bed elevation and zs = zb + H the water surface elevation. By integrating the tree-dimensional mass and momemtum transport equations with respect to the vertical coordinate from the bed to the water surface and assuming that vertical velocities and accelerations are negligible one can obtain the depth average surface water equations which are given by µ
¶
∂(HU ) ∂ 1 ∂ ∂zb + βuu HU U + (cosαx cosαz )2 gH 2 + (βuv HU V ) + cosαx gH ∂t ∂x 2 ∂y ∂x · ¸ ∂(Hτxx ) ∂(Hτxy ) 1 τbx − τsx − − = 0 (4) −ΩHV + ρ ∂x ∂y for the x-direction, and µ
¶
∂(HV ) ∂ 1 ∂ ∂zb + βvv HV V + (cosαy cosαz )2 gH 2 + (βuv HV U ) + cosαy gH ∂t ∂y 2 ∂x ∂y · ¸ 1 ∂(Hτyx ) ∂(Hτyy ) −ΩHU + τby − τsy − − = 0 (5) ρ ∂x ∂y for the y-direction. In equations ?? and ??, βuu , βuv , βvu and βvv are the momemtum flux correction coefficients that account for the variation of velocity in the vertical direction; τbx and τby the bed shear stresses, τsx and τsy the surface shear stresses and τxx , τxy , τyx and τyy the shear stresses caused by turbulence; Ω is the Coriolis parameter. The coefficients αx , αy and αz are given by µ
αx = arctan
∂zb ∂x
¶
µ
αy = arctan
∂zb ∂y
The continuity equation is also given by
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¶
³
αx = arccos 1 − cos2 αx − cos2 αy
´
∂(H) ∂(HU ) ∂(HV ) + + =q ∂t ∂x ∂y
(6)
where q is a unit source or a unit sink term. The depth average flow equations (??), (??) and (??) are a coupled system of nonlinear partial differential equations. The method of weighted residuals using Galerkin weighting (i.e weighting functions same as interpolation functions) is applied to the set of equations. For a complete description of the finite element equations see [?] and [?]. Discretization of the computational domain is carried out using two dimensional elements: • six-node triangles (3 vertex nodes and 3 midside nodes) • eight-node ”serendipidity” quadrilateral (4 vertex nodes and 4 midside nodes) • nine-node ”Lagrangian” quadrilateral (4 vertex nodes, 4 midside nodes and one center node) Each node point is associated to 3 degrees of freedom U , V and H. In order to contribute to numerical solution stability, mixed interpolation is used: quadratic interpolation functions are used to interpolate depth average velocities and linear functions are used for flow depth. The finite element solution of the governing equations produces a set of global discretized equations. Because of the nonlinear inertia and bottom friction terms that appear, the coefficient matrix formed during the finite element method will be nonsymmetric. The simultaneous nonlinear system of equations is solved using a strategy that combines full-Newton iteration, quasi-Newton iteration, and the frontal solution scheme. Moreover, a time-integration factor control the degree of implicitness of the integration scheme.
Input data To run FESWMS, data files which define the boundary conditions, the initial conditions, the material properties, the model parameters and the finite element network information are needed.
Output data FESWMS is used to compute water surface elevations and flow velocities at nodes in a finite element mesh representing a body of water such as a river, harbour, or estuary. Therefore, the output of FESWMS at each finite element node consists of: • the depth averaged velocities U and V • the water surface elevation H (from which can be easily extracted the water depth)
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Figure 2: FESWMS adaptation FESWMS adaptation to the Loire pilot site Data was gathered and pre-processed for the design of the 2D model for the area. The finite element mesh shown in figure ?? ( 3000 elements) was used for preliminary computations to investigate model behavior with very coarse topographic data. The design of a model that will contain 30000 finite elements in order to benefit from computational approach, data precision and parallel version of model solver developed by project partners is in progress. This model will be applied to 3 hydrological scenarios (50, 100 and 200 years flood) and hydraulic scenarios for the operation of the spillway which is the main control point of the area.
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Data assimilation
2.1
Motivation
Earth Sciences have some specific features which have to be taken into account when a prediction (in real time or after some virtual perturbation) is carried out. • Each episode is unique. Therefore geosciences are not stricto sensu experimental sciences in the sense that an hypothesis could be (un) validated by duplicating a field experiment. For instance, the validation of a model will be funded on the comparison of the results of this model with data issued from various situation. The invariant parameters of the model have to be clearly identified • The governing equation are non-linear. This fact implies cascades of energy between the different scales of the motion, because a model has to be discretized to get a numerical solution a truncation will be done. The subgrid effect should be represented by a numerical artefacts which will include some artificial ( i.e. not physically measurable) parameters. These parameters should be estimated from observations. • Models are not closed. Most of the time a model requires boundary and/or
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initial condition. The definition of the geographic domain of the model is often artificial. Prescribing boundary conditions is a difficult task and if they are not mathematically natural they should be deduced from the physical observations. These remarks are quite general; they are valid for the study of flood and should be kept in mind. It’s clear that a model without data doesn’t make sense. On the same token data without models (i.e. a framework for their understanding) are not useful for prediction. Data Assimilation is the techniques for coupling Data and Models in order to get the best estimation (in a sense to be defined) of the geophysical fields in the past ( analysis) or in the future ( prediction). During this last decade D.A. has known a huge development especially in meteorology and in oceanography and there is a strong demand for applications to the water cycle.
2.2 2.2.1
Overview Theory
There are two basic approaches of Data assimilation • Deterministic approach based on Optimal Control Theory • Stochastic methods based on the theory of statistical estimation and Kalman filter. The second method requires to handle huge covariance matrices, for the time being they cannot be applied to realistic models. Therefore in the following we will work on the first method. The ingredients are: • A state variable describe the fields (water velocity, water elevation at the grid points). X is this state variable it is of great dimensionality • An observation Xobs , distributed in time and space • Control, variables, U is the initial condition, V is some variable to the model it may be a boundary condition and/or some unknown parameter to be identified. U and V are the input which have to be provided to the model to get a unique solution between time 0 and time T • A model which can be written: dX
= F (X, U )
Xdt(0) = V
(7)
• A cost function which estimates the discrepancy between the solution of the model associated to (U, V ) and the observations. Because the variable X and the observations are not in the same set we will consider C mapping the mathematical variable into a physical variable (e.g. an interpolation in time and space) 1 J (U, V ) = 2
ZT
kC.X(U, V ) − Xobs k2 dt 0
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(8)
The problem is: Determine U ∗ and V ∗ minimizing J, i.e. we seek for the inputs of the model leading to solution of the model closest to the observation. Therefore the closure of the model is stated as a problem of unconstrained optimization for which there are many known efficient algorithms. U ∗ and V ∗ are characterized by cancelling U ∗ and V ∗ the gradient of the cost function, this gradient is also necessary to carry out optimization algorithms. Computing the gradient using the Adjoint Model: We introduce P the ”adjoint variable” which has the same dimensionality as X as the solution of the adjoint model: dP
·
∂F + ∂X dt P (T ) = 0.
¸t
.P = C t (C.X − Xobs )
Then it can be seen that the gradient of J is given by: Ã
∇J =
∇V J ∇U J
!
−P (0) · ¸ = ∂F t − .P ∂U
Therefore the gradient is obtained by a backward integration of the adjoint model. An algorithmic solution is adopted. A descent-type method is carried out , these methods are iterative, they requires several iterations and at each iteration an evaluation of the model and of the gradient.
2.2.2
Implementation
The most challenging task is the writing of the adjoint model. One could drift the continuous direct model and discretize it. However, using this method the precision in the evaluation of the cost function’s gradient is not sufficient for the convergence of the optimization algorithm. Therefore, the determination of the adjoint could be divided in two phases: • The derivation of the direct model which provide the linear tangent model. This operation consists in the derivation of all the direct code instructions. • The transposition of the linear tangent model. The temporal transposition corresponds to the backward integration. The last instruction of the linear tangent code is executed and every single instruction is transposed. The most difficult is the identification of hidden dependencies. Some software such as TAPENADE ( former ODYSEE developed by INRIA) are available for automatic derivation of adjoint, nevertheless the process is long and painful specially when the direct code does not respect some programming standards.
2.3 2.3.1
Application to river modelling Model closure
In classical river modelling, the observations within the computational domain are used only to carry out manual calibration, sensitivity analysis and verifications of the
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model results. Data assimilation provide a global approach for the couple (Models, Data). Optimizing the content of this couple, the model inputs are used to get the best adjustments to the observations. Thus, in ??, the gradient of the objective function could be carried out with respect to: • the model boundary conditions. The prescription of theses values which determine the flow dynamics inside the computational domain is of prime importance. However, the same field data will be used for different river models (1D, 2D etc ...). However, the interpretation of this field data will be different if the governing equations and/or the computational approach differ. • the model initial condition. Before any unsteady computation, a steady state computation has to be carried out in order to retrieve the initial state of the river. For prediction, this initial state could be computed using the adjoint model in combination with the observations. • the input empirical parameters necessary for models closure. In equation [??], [??] and [??] there are some empirical coefficients that theoretically could be evaluated by measured discharges and velocity distribution. The adjoint should permit the optimal control of the model input parameters to get the best adjustment to the observations. For example for CARIMA, the friction loss coefficient Sf is related to the cross section conveyance, which itself is related to the channel roughness. In ?? and ??, τbx and τby directional components of the bed shear stress coefficients are related to a non-dimensional bed-friction coefficient which depend on the Manning roughness coefficients. This channel (or floodplain) roughness is the main parameter used for model calibration. It depends on the channel or floodplain vegetal cover but also on: – the season – the flow conditions, roughness can vary with flow depth – the modelling approach, for example the depth-averaged flow equations directly account for horizontal variations of velocity and the effect of turbulence; thus the roughness based on one-dimensional flow assumptions might be slightly larger than necessary – the spatial resolution, for example the value adopted for roughness coefficient in a 2D model depends on the refinement of the grid. Indeed, a refined grid with only slight topographic changes variations from node to node presents a shape roughness which must not be included in the overall roughness coefficient. In contrast, when a grid is relatively coarse the roughness coefficient includes both skin roughness and small topographic features. Of course in order to show its potential benefits, this prospective investigations should be carried out for test cases for which data is available. Moreover it should help for the identification of the field data that has to be collected to improve flood prediction.
2.3.2
Sensitivity analysis
The use of mathematical models to represent a physical phenomenons require an analysis model uncertainty that could be carried out using sensitivity analysis. Using
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the notations employed in ??, sensitivity analysis can be defined as follows : • X is the state variable of the model and K the vectorial parameter of the model F (X, K) = 0. • G(X, K) = 0 the response function. The model sensitivity is the gradient of the function G in relation to K. To compute the sensitivity, we could introduce the adjoint variable P as the solution of: ·
∂F ∂X
¸T
.P =
∂G ∂X
Thus, the gradient follows : ·
5G =
∂G ∂F − ∂K ∂K
¸T
.P
The sensibility vector is obtained from one run of the adjoint model instead of a large number of integration of the direct model with different input parameters. The price to be paid is the computation of adjoint code.
2.3.3
Models coupling
The hydraulic models considered here provide a different description of the river/floodplain physical system. Moreover, the computational cost is much more important for 2D models. Therefore: • depending on the study purpose spatially distributed results might be needed in some locations of a river reach but not every where. • depending on the site features (extension, topography), the 1D flow hypothesis could be adopted wherever is is possible to reduce computational cost. • depending on the study objectives, one could consider that for example a diked up river bed should be modelled in 1D and the floodplain in 2D Let us consider the following equations are the governing equations for models 1 and 2 : dUi = Fi (Ui ) dt Ui (0i ) = fi (t) dU i (M ) = vi dt where i = {1, 2}, M is the common boundary of each model and 0i is others boundaries. Data assimilation could be used to control the common border with the help of the following cost function : J(v) =
Z T 0
[U1 (M ) − U2 (M )]2 dt
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Conclusion
The mathematical models described in this paper differ from both governing equations and computational approach. Thus, the field described by those models will follow the same rule. Moreover, most of the time data are scarce and not consistent with this field. The possibility to combine models and data using data assimilation was underlined. Research is in progress in order to demonstrate that its implementation could permit to improve flood prediction by providing: • an optimal control for the model inputs (empirical parameters, initial and boundary conditions) • tools for an extensive analysis of model uncertainty • an optimal method for the use of models in combination • guidelines for the type of measurements that should be gathered in the future
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