nModelling of flood propagation in a semi-arid environment case of the N'FIS basin - Moroccan Western High Atlas

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nModelling of flood propagation in a semi-arid environment case of the N’FIS basin - Moroccan Western High Atlas Abdelhafid El Alaoui El Fels , Noureddine Alaa , Ali Bachnou PII: DOI: Reference:

S2468-2276(20)30044-2 https://doi.org/10.1016/j.sciaf.2020.e00306 SCIAF 306

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Scientific African

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2 June 2019 9 November 2019 31 January 2020

Please cite this article as: Abdelhafid El Alaoui El Fels , Noureddine Alaa , Ali Bachnou , nModelling of flood propagation in a semi-arid environment case of the N’FIS basin - Moroccan Western High Atlas, Scientific African (2020), doi: https://doi.org/10.1016/j.sciaf.2020.e00306

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Modelling of flood propagation in a semi-arid environment case of the N'FIS basin - Moroccan Western High Atlas Abdelhafid El Alaoui El Fels1. Noureddine Alaa2. Ali Bachnou1 1

Laboratory of Geosciences and Environnement, Department of Earth Sciences, Faculty of Science and Technics Gueliz, (P.B. 549), Cadi Ayyad University, Marrakech, Morocco. 2 Laboratory Applied mathematics and informatics, Department of math and informatics, Faculty of Science and Technics Gueliz, (P.B. 549), Cadi Ayyad University, Marrakech, Morocco.

Abstract The modeling of flood propagation in semi-arid environments can be extremely useful for a better assessment and flood risk planning to reduce the catastrophic impacts of this phenomenon. The main purpose of this work is to make a comparative study between two different models in order to choose the most suitable one for the monitoring of the hydrological status of N'fis watershed and for the optimal management of the dam reservoir of Wirgane during flooding periods. In this work, the modelling of the phenomenon of flood propagation is carried out by two models Hayami and Muskingum. Our analysis of the models first passes through a calibration process that is based on the identification by Genetic Algorithm (GA) the optimal parameters, the second phase is validation by testing the deliverability of the models through the parameters of the previous phase, the last phase consists of evaluating the models by performance criteria. The results obtained show that the parameters optimized by the GA, K (storage constant) = 11580 and x (weighting factor) = 0.14 for Muskingum and C (Celerity) = 3 and D (Diffusion coefficient) = 3200 for Hayami, are found to have a better response with a Nash greater than 0.7 and a correlation coefficient d of Order 0.9. According to the sensitivity test, the parameters K and C remain parameterized with the greatest sensitivity. In general, the comparison of the two models shows that the Hayami model is easy to manipulate to simulate flood routing; this aptitude makes it one of an interesting tool capable of adequately representing the behavior of floods in semi-arid environment Key words: flood propagation, Genetic Algorithms, Hayami, Muskingum-cunge.

1. Introduction Hydrological modelling is a tool that describes in a comprehensible and simple way the behaviour and the mechanism of functioning of a complex real hydrological system [1] and to simulate different scenarios, in order to exploit the information provided to the decision within the framework of the management or the preservation of this system. Flood modelling is one of the major concerns of hydrologists. In the literature, there is a diversity of hydrological models of floods [2-5]. In the literature, there is a diversity of flood hydrological models that have been created and developed according to the desired objectives as well as available input data. Otherwise, in arid or semi-arid regions, a multitude of exploitable models are used, each of which has a restricted domain of application and validity [611]. Unlike the models that deal with the genesis of floods, namely models that are based on the rainfall-runoff mechanism that involves poorly known physical processes such as evapotranspiration and seepage etc., the propagation of flood waves in a fluvial environment obeys laws derived from the mechanics of fluids and whose practical validity is confirmed by real observations. In 1871 [12], Barré de Saint-Venant proposed a system of equations describing the phenomenon. This system, which bears his name, constitutes a fundamental theoretical basis of the study of propagation. Nevertheless, depending on the complexity of this system, some terms of the equations can be neglected through mathematical approximations, leading to a more reduced and simpler form of the wave propagation equation. We then speak of models of wave dynamics, wave scattering and wave kinematics [13]. For the simulation of floods, the latter two models are often used. In the general case, the resolution of these equations relies on finite difference numerical schemes (see a review in [14-16]). The complexity of hydraulic models in describing the phenomenon of propagation has produced new

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generations of models called hydrological models which are centred not on hydraulic concepts but on considerations of storage along rivers [17]. These models therefore use the equation of the continuity and the relations between the storage and the inlet and outlet flows of the section. Despite the weakness of the basic concepts of hydrological models, they are widely used in hydrology, because of the simplicity in its applicability and the low computational demand for their implementation. Among the best-known hydrological models used in the simulation of flood propagation is the Muskingum model [18]. The task is to select the propagation model most suitable for flood forecasting in the N’Fis basin, in order to have a continuous monitoring of the hydrological situation and optimal management of the reservoir of Wirgane during floods. The choice is based on a comparative study between the two models Hayami [19] and Muskingum [18] which is different at the conceptual level of these formulations. This comparison is based on the performance of the model productivities which is controlled by the identification of their optimal parameters in order to arrive at a simulation very close to the observed flow. Deterministic methods are most often used in this kind of problem, but they have the disadvantage of being able to converge towards a local optimum, which leads us to intervene in our study of more robust optimization methods. [20-23]. However, this efficiency is strongly linked to a judicious choice of the parameters of the algorithm.

2. Material and Methods 2.1. Study Area : The N'Fis basin is part of the western Haouz reigned by a semi-arid climate. It is located on the northern front of the Moroccan High Atlas, between latitudes 31 °, 22 N-30 °, 50 S and longitudes 7 °, 55 E-8 °, 40 W (Figure 1) with an area of 1200 Km2. On the morphological side, it is characterized by an elongated shape, a rugged topography, and a branched hydrographic stream network, the elevations vary from 740 meters at the outlet to 4079 meters at the highest point (High Atlas Mountains). On the geological hand, the N'Fis basin is formed by shales characterized by low permeability and impermeable magmatic rocks. Precipitation is characterized by its spatio-temporal irregularity and its occasional high intensity appearance, with an average annual of 500 mm. The high precipitations are located at high altitudes which are generally exposed from the north and northwest ocean currents. In the study area, there are two major seasons: a wet season from October to April, where almost all rainfall events occur (average of 84% annual rainfall) a dry season from May to September (only 16% annual rainfall on average).

2.2. Hydrological data Hydrological data are composed of instantaneous flows measured at the two hydrometric stations Imine El Hamame (downtream) and IN'kouris (Upstream). These data concern only generalized floods and cover a period of 27 years (1973 to 2000) which was before the building of the Ouirgane dam in 2004 in order to regularize the flows. We have formed the samples consisting of the instantaneous flows calculated from the observed water heights and a height-flow relationship (rating curve) established by the Tensift Basin Hydraulic Agency.

2.3. Flood propagation modelling 2.3.1. Muskingum-cunge The Muskingum method of flood routing was developed in 1930s by Maccarthy, it is based on a variable discharge-storage relationship in fluvial systems; it is well known and extensively used in river engineering

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[24]. Although this method behaves better in river systems where the inertia effects and back water influences are low, Chang [25] and Cunge [26] proved that the conventional Muskingum equations are numerically equivalents to Saint-Venant equations that are derived from the principles of mass and momentum conservation laws, where they are assimilated to a convective-diffusion equation. The Muskingum-Cunge method is given by the following equation [27]: 𝑂𝑡+1,𝑚 = 𝐶1,𝑚 𝐼𝑡+1,𝑚 + 𝐶2,𝑚 𝐼𝑡,𝑚 + 𝐶3,𝑚 𝑂𝑡,𝑚 + 𝐶4,𝑚 𝑄𝐿 (1) With 𝐶1,𝑚 , 𝐶2,𝑚 𝑎𝑛𝑑 𝐶3,𝑚 are dimensionless parameters 𝐶1,𝑚 =

∆𝑡−2𝐾𝑚 𝑥𝑚 2𝐾𝑚 (1−𝑥𝑚 )+∆𝑡 ∆𝑡+2𝐾𝑚 𝑥𝑚 𝑚 (1−𝑥𝑚 )+∆𝑡

𝐶2,𝑚 = 2𝐾

2𝐾 (1−𝑥 )−∆𝑡

𝐶3,𝑚 = 2𝐾𝑚 (1−𝑥𝑚 )+∆𝑡 𝑚

𝐶4,𝑚 = 2𝐾

𝑚

∆𝑡

𝑚 (1−𝑥𝑚 )+∆𝑡

(2) (3) (4) (5)

Where 𝐼𝑚 and 𝑂𝑚 represent the simultaneous amounts of storage, inflow and outflow, respectively at a given time t. The km could be assimilated to the transit time of the wave between two river points [28]. In fact, the variable 𝐾𝑚 represents the offset between the gravity centre of the input and output hydrographs [29]. Here, we will use the same units as for ∆𝑡 (hour, days …). 𝑥𝑚 is a dimensionless weighting factor that allows to quantify the respective influences of the inlet and outlet flow rates on the stored volume.

2.3.2. Model Hayami The transport diffusion model or Hayami model [30] is a simplification of complex systems of SaintVenant equations, assuming that the forces of friction, gravity and pressure are dominant in the description of the phenomenon, so the terms of inertia are negligible. After that, it was detailed by (Moussa) in 1996 [13]; which is a hydrodynamic model of flow-flow type whose formulation is based on the fluids mechanics to describe the flood wave propagation phenomena, by considering it as a propagating wave in space with temporal variation. The Model equation is given in the following form: 𝜕𝑄 𝜕𝑡

𝜕𝑄

+ 𝐶 𝜕𝑥 − 𝐷

𝜕𝑄 2 𝜕𝑥

=0

(6)

Where C: The wave Celerity, the movement velocity of the flood wave (m/s). D: Diffusion coefficient, damping of the flood wave (m2/s).

Generally, C and D present a variable character and equation (6) is considered to be nonlinear [31]. Under some considerations and mathematical approximations, we obtain the analytic solution of our simplified Hayami Model, which is expressed as follows 𝑡 𝑄𝑂 = ∫0 𝑄𝐼 (𝑡 − 𝜏)𝐾𝐶,𝐷 (𝜏) 𝑑𝜏 (7)

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𝐾(𝑡) =

−𝐿 𝐶𝑡 − +2) 𝐶𝑡 𝐿 3⁄ 2

𝑒𝑥𝑝(

𝐿 1 2(𝜋𝐷) ⁄2

𝑡

(8)

where 𝑄𝑂 is the output flow, 𝑄𝐼 the input flow and L is the length of river.

2.4. Mechanism of Optimization of Parameters by Genetic Algorithms The hydrological or hydraulic models are characterized by a variable predictive aspect controlled by parameters that are specific to the forecast site. Then the calibration is a "learning" phase of selecting the set of parameters of a model so that it simulates the hydrological behavior of the watershed of the best possible way [32]. Therefore, it is about determining the values of its parameters that allow it to get the best performance by minimizing the error between the observed and simulated data. This minimizing function is defined as follows 𝑇

𝑓(𝑝𝑎𝑟𝑎𝑚é𝑡𝑟𝑒𝑠 𝑑𝑢 𝑚𝑜𝑑é𝑙𝑒𝑠) = ∫ (𝑄𝑠𝑖𝑚(𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑟𝑒𝑠) − 𝑄𝑜𝑏𝑠(𝑡))2 𝑑𝑡 0

It is often that the model functions represent local minima that we must avoid. In this case, the classical optimization methods and the deterministic method are discouraged [33-34]. In order to solve this problem, we have used the optimization method based on some specific techniques derived from genetics and evolution problems. They are called genetic algorithms. These algorithms form a class of biology inspired optimization methods based on techniques derived from the genetic and Darwin’s theory of evolution [35-36]. They incorporate a “survival to the fittest” strategy to intelligently handle the manipulation of a population of individuals. The strategy is implemented at four different levels that an initial random population undergoes namely: selection, crossover, mutation and elitism. Reynés [37] claimed that the intelligence of genetic algorithms is in the selection mechanisms, he defined the genetic algorithms as the probabilistic optimization methods based on the mechanism of natural selection. The originality of these algorithms is in their ability to extract the fittest solution form an initial population that evolves through a succession of generations, until the convergence of all the individuals towards optimum solution. The genetic algorithm mechanism began by the evolution of population of individuals, using objective function based on the pertinence criterion to ensure the quality of adequacy (fitness). The fit individuals will be able to reproduce more frequently than the others and they will have more descendants than their less adapted competitors. The main objective is to find the ideal combination of these elements that gave the maximum “Fitness”. At every iteration (population generation), a new population is created using parts of the best elements of selected individuals from the previous generation, as well as, occasionally innovative parts. These generation of offspring is then mutated under a certain non-uniform mutation probability. In other words, genetic algorithm exploit the distributed information, obtained previously to generate new individuals to explore, with the hope of improving performance. After the operations of selection, crossover and mutation, elitism is introduced when establishing the transition from the current to the next generation. The two generations are gathered (parents and offspring) into one population composed of the best individuals satisfying the pertinence criterion while providing the best fitness score.

Algorithm Initialize the population (randomly generate a population of n individuals 𝑓(𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠)) Calculate the degree of adaptation 𝑓(𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠) of each individual As a non-finished or non-convergence Select 2 individuals to the times for the reproduction

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Apply the genetic operators (crossing, mutation) Keep only the best individuals Calculate the degree of adaptation 𝑓(𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠)of each child Select the survivors among parents and children End as long as Find the best optimum

2.5. The choice of optimal interval of the parameters of the models The settings are inputs from model that characterize a given environment, in other words, they vary depending on the case of the use of a template. The determination of the optimal values of the parameters are based on the identification of the interval of the Games of the latter so that they can be representative of the hydrological behaviour of the watershed in general. With the sensitivity analysis, it is rather to explore the "neighbourhood" of an optimal solution, i.e. to determine the range of variation in which a variable can be capable of presenting the hydrological behaviour of the basin of a mannered correct. The sensitivity of the model is to vary the values parameters in a given interval and then calculate for each torque of the parameters of our Hayami model (𝐶𝑖 , 𝐷𝑗 ) and Muskingum (𝐾𝑖 , 𝑥𝑗 ) the value of the functioncriterion Nash, therefore it is possible to represent surfaces of iso-value of the function-criterion or response surfaces as proposed in the figure 2. It shows that the sensitivity analysis of the function-criterion in relation to variations of the parameters of the model allows to consider not more an optimal value parameters, but an interval in which their combination gives simulations acceptable, it also allows us to draw the parameter which presents a high degree of influence on the response of the mode. The analysis of the figure 2 allows us to extract the best intervals of the parameters that control the good answers of the models served to predict the downstream flow (flow of imine station El hamame); this response is translated by the performance criterion Nash. For the Hayami model the performance of the model increases as a function of the growth of the haste, which shows that this last is the most sensitive parameter, by against we find that the diffusion coefficient has a small influence to the response of the model and makes the determination of a specific interval difficult. The strong Nash are marked for a speed which exceeds 2.5 m/s. The Muskingum performance is strongly controlled by the parameter K, this performance degrades with a rate decreasing in more away from the interval [10000 s; 15000 s]. We can also note that despite the small influence of the parameter 𝑥, the strong Nash are registered for an interval [0; 0.3] of this last.

2.6. Evaluation of the models The quality evaluation of a model is based on numerical criteria; they are different from the "subjective" judgment of the observer. These criterions are based on the use of statistical formulas that generally measure the difference between the observed and simulated data, in order to translate the simulated performance of the model. Below all the formulations of each criterion used: 𝑁𝑎𝑠ℎ = 1 − 𝑅𝐸𝑃 =

𝛴(𝑄𝑜𝑏𝑠 −𝑄𝑠𝑖𝑚 )2 𝛴(𝑄𝑜𝑏𝑠 −𝑄𝑚 )2

𝑄(𝑀𝑎𝑥)𝑜𝑏𝑠 −𝑄(𝑀𝑎𝑥)𝑠𝑖𝑚 𝑄(𝑀𝑎𝑥)𝑜𝑏𝑠

(8) (9)

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With: 𝑄𝑜𝑏𝑠 ∶ 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑓𝑙𝑜𝑤 𝑄𝑠𝑖𝑚 ∶ 𝑠𝑖𝑚𝑢𝑙𝑖𝑡𝑒𝑑 𝑓𝑙𝑜𝑤

3. Results and Discussion 3.1. Calibration of the model Given the problems related to the dependence of the parameters and to the existence of local optima, our Calibration procedure is to set the terminals of plausible values based on the study of the sensitivity and then to use the GA in the adjustment to search for the settings that give the best criterion of Nash at the outlet of the watershed. According to the various tests conducted, the best adjustment obtained for the two models is the adjustment of the flood of 1980. The table 1 and the figure 3 above summarizes the results obtained. According to the analysis of the figure 3 and the table 1 of criteria we find that the models Hayami and Muskingum are able to represent well the hydrological response of the basin N'Fis, with a correlation coefficient and Nash close to 1, the simulated flows have a variance in very low with the observed flows. Also at the level of the simulation of peak observed hydrograph the two models have been well simulated with an error very low. After the optimization of model parameters by genetic algorithms, we make a simple comparison of the parameters based on established relationships by Weinmann and Laurenson in 1979 [39] and Cunge (1969) [26]: 1 𝐷 𝑥= − (10) 2

𝐶𝐿 𝐿

𝐾 = 𝐶 (11) According to the two equations we can convert the parameters of each model assessed as new parameters, in other words, compare the response of the Hayami model by the parameters of the Muskingum and the reverse. The table 2 summarizes the converted settings

The analysis of figure 4 shows that the simulations obtained by the two models by using the parameters converted by equations (10) and (11) are good, the response of the Muskingum model with the converted settings of the Hayami model is almost the same of that of Hayami, but the model parameters Hayami does not respect the conditions of the Muskingum model in its form discretized, despite the good simulation of the latter, this comparison between the parameters of the models optimized for each model, leads us to see that the use of the Hayami model in order to describe the phenomenon of spread and more flexible than the Muskingum model that adjusts to certain conditions in their application , we also noted that the settings of The Muskingum model is eligible after the conversion of the used model Hayami.

3.2. Validation and evaluation of models At the end of the calibration, it is necessary to check the reproducibility of results and the representativeness of the parameters adjusted by the GA. The objective of this phase is the testing of our

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models with the new parameters identified in different events. The models wedged on the flood of 1980 are applied on four events of Flood 18-01-1979; 26-09-1976; 06-01-1978; 15-11-1983, the results of the simulation in the validation phase are presented in Figure 5.In spite of the great distance that separates the two stations of the basin of N'Fis (33 km) models are able to faithfully reproduce the hydrographs of floods with a Nash means of order 0.7, we also note that the hayami model has well simulated the peaks of the hydrographs with minimal error, while the Muskingum model gives remarkable underestimations (see the table 3). Nevertheless, for some floods for example that of 18-01-1979, the simulated flows according to the two models are slightly underestimated compared to the flow rates observed. This result illustrates the impact of the spatial variability of precipitation; the flows of 15-11-1983 flood are underestimated by the models because of the non-integration of the lateral contributions generated by the tributaries of the basin, so we can say that the 15-11-1983 flood is a magnifying flood; Finally, for the 18-01-1979 flood; Finally, the lag in the second peak of the 18-01-1979 flood hydrograph can be explained by the fact that this complex flood is due to two different rainy episodes, the first episode is located only in the upstream of the basin while the second is a rainy episode divided throughout the basin.

4. Conclusion The main objectives of our research are to characterize the ability of models to describe the phenomenon of the spread of floods and to select the model which represents best our river system. This study is based on the use of a method of global optimization of the parameters which avoids the general problems of the deterministic methods, in order to have a good simulation which brings us subsequently to choose the most adequate model in an optimal management of the withholding of the Wirgane dam in flooding period. Our work is based on two different models Hayami and Muskingum. The GA has proved their effectiveness in optimizing the different parameters of our models. The parameters identified by this probabilistic method are able to obtain a good calibration, which is characterized by a perfect simulation of flows of spikes with a minimum error between the flow rates of the hydrographs of simulated flood is observed. The present study has shown us that the parameters related to the speed of propagation of floods are the most sensitive parameters, and also, that the use of the Hayami model is more flexible than the Muskingum model which easily adapts to different conditions. The comparison of the results of the two models obtained at the level of the validation phase show that the Hayami model has an advantage over the Muskingum model. In fact, Hayami model is able to predict the flow of the complex hydrograph and also the simulation of the time to fit the peaks. Overall, it is a quite interesting tool, capable of properly representing a significant range of conditions of water and floods. Finally, Improve the predictive quality of propagation models required a consideration of losses and lateral inflow. The lack of knowledge of these can therefore be a major handicap in flood forecasting. Indeed, the good simulation of the downstream flow resides in the coupling of the propagation models with rainfallrunoff models able to estimate the lateral contributions not previously taken into account in the modeling.

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Figure list

Figure 1 : the location of N’Fis watershed

Figure 2 : Sensitivity of the parameters of the models Hayami (A) and Muskingum (B)

Figure 3 : Calibration of the two models

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Figure 4 : Comparison between the parameters responses of the two models

Figure 5 : Validation of Two Models

Table list Table 1 : The settings optimized by the GA in the phase of stalling and the performance of the flood shimmed MODELS

PARAMETRES

NASH

CORRELATION

REP

MUSKINGUM

K = 11580

x =0.1497

0.9735

0.9867

0.0957

HAYAMI

C=3

D = 3200

0.9333

0.9899

0.0653

Table 2 : Converted parameters of the two models

Optimized parameters Hayami model C=3  D = 3200  Muskingum model K = 11580  x = 0.1497 

Converted parameters Muskingum model Knew = 11000 xnew = 0.467 Hayami model Cnew = 2.84 Dnew = 34670

Table 3 : Performance of the two models (Phase of Validation)

Muskingum - Cung Flood

Hayami

Nash

REP

Cor

Nash

REP

Cor

15/11/1983

0.826

0.175

0.953

0.796

-0.007

0.925

06/01/1978

0.889

0.207

0.946

0.818

0.045

0.955

18/01/1979

0.754

0.188

0.897

0.750

-0.014

0.872

26/09/1976

0.575

0.609

0.806

0.810

0.076

0.919

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