J. Hydrol. Hydromech., 60, 2012, 2, 125–142 DOI: 10.2478/v10098-012-0011-6

HYDRODYNAMIC INVESTIGATION AND NUMERICAL SIMULATION OF INTERMITTENT AND EPHEMERAL FLOWS IN SEMI-ARID REGIONS: WADI MEKERRA,

KHALED KORICHI1,2), ABDELKRIM HAZZAB1)

1)Laboratory of Modeling and Computational Methods, University of Saida, BP 138 Ennasr Saïda 20002, Algeria; Mailto: [email protected] 2)Faculty of Engineering, Department of Civil Engineering, University of Djillali Liabes, Sidi Bel Abbes, BP 89, Sidi Bel Abbes 22000, Algeria.

Semi-arid regions are characterized by important infrequent rainfall. They often occur in early autumn and give rise to devastating floods. Flooding problems at Wadi Mekerra, located in the Sidi Bel Abbes town (Northwest Algeria), was traditionally the main concern of researchers and government officials. In this work, the magnitude of raging flood wave in the studied catchment and the principal causes are discussed. After this, we present the main hydromorphometric features and the results of numerical simulations of flood wave. This simulation is done by using finite volume shock capturing schemes. It concerns applying the first order Godunov scheme and the second order Monotonic Central scheme. The Manning roughness coefficient was used as a calibration parameter. The comparison of numerical results with observed data confirms more stability and accuracy of applied numerical schemes in rising limb phase than in the falling limb phase. These results provide information on flood forecasting and monitoring of changes in the magni- tude of the flow in Wadi Mekerra.

KEY WORDS: Semi-arid, Temporary Flows, Wadi Mekerra, Flood Wave, Saint-Venant Equations, Finite Volume, Shock Capturing.

Khaled Korichi, Abdelkrim Hazzab: HYDRODYNAMICKÝ VÝSKUM A NUMERICKÁ SIMULÁCIA OBČASNÝCH A KRÁTKODOBÝCH PRIETOKOV V SEMIARIDNÝCH OBLASTIACH: WADI MEKERRA, ALŽÍR. J. Hydrol. Hydromech., 60, 2012, 2; 114 lit., 13 obr., 7 tab.

Semiaridné oblasti sú charakterizované významnými občasnými dažďami. Vyskytujú sa spravidla v jese- ni a spôsobujú devastačné povodne. Povodne vo Wadi Mekerra v meste Sidi Bel Abbes (severozápadné Alžírsko) sú často problémom pre výskumníkov a vládnych úradníkov. Práca analyzuje devastačné povodne v tomto povodí. Okrem toho sa uvádzajú najdôležitejšie hydromorfomertrické charakteristiky a výsledky simulácií povodňovej vlny. Simulácia je aplikáciou Godunovovej schémy prvého rádu a Monotonickej cen- trálnej schémy druhého rádu. Ako kalibračný parameter bol použitý Manningov súčiniteľ drsnosti. Po- rovnanie simulovaných a pozorovaných údajov potvrdzuje väčšiu stabilitu a presnosť počas rastúcej vetvy prietokovej vlny, ako je tomu v klesajúcej fáze. Tieto výsledky sú informáciou o možnostiach predpovede a monitoringu povodní vo Wadi Mekerra.

KĽÚČOVÉ SLOVÁ: semi-aridný, občasné povodne, Wadi Mekerra, povodňová vlna, rovnice Saint- Venanta, konečný objem, pohltenie nárazu.

Introduction damage (Hansson et al., 2008 ; Jonkman et al., 2008). Almost all natural phenomena can produce disas- The inundations risk is considered as the product ters; however flooding is the most important in of the occurrence probability of floods and conse- terms of lost materials and human lives (Mudd, quences associated with these events (Raaijmakers 2006; Huang et al., 2010). Prediction and control of et al., 2008). By possessing a diagram of the situa- damage caused by these disasters require special tion, it would be possible not only to manage risk, identification of vulnerable areas and determination take the most appropriate steps to eliminate loss in of factors amplifying the magnitude of generated life and minimize property loss, but also have di-

125 K. Korichi, A. Hazzab verse opportunities to provide the appropriate di- 2007; Chaponnière et al., 2007; (Pilgrim et al., mensioning of hydraulic structures (Breton and 1988; Hughes, 1995; Sharma and Murthy, 1998; Marche, 2001). Wheater et al., 2006). Other approaches propose the The estimated impact of floods and the risk as- coupling of hydrological, hydraulic models and sessment that follows are issues for which the liter- statistical analysis. This is especially the combina- ature provides different analysis approaches tion of surface and groundwater hydrology (Rozos (Gregory et al., 1993; Haase et al., 2003 ; Olfert et et al., 2004; Bouri et al., 2007; Kingumbi et al., Schanze, 2008). The used tools for the floods 2007), or between the stochastic and dynamic ap- treatment are flow modelling, analysis of flood proaches (Hreiche et al., 2007; Lajili-Ghezal, frequency and evaluation of potential annual cost of 2007). damage (Boyle et al., 1998). Recent works have been performed to calibrate Semi-arid regions are characterized by infrequent and validate various models for Wadis whose flow rainfall but sometimes very important (Colombani regime is intermittent or even ephemeral (cf. Tab. et al., 1984 ; Pedro et al., 2006). These cloudbursts 1). One cites innovative studies distributed in such often give rise to flooding whose consequences are semi-arid zones. It is about of hydrological model sometimes unforeseen (Hansson et al., 2008). developed for Australian drylands (Costelloe et al., However, streams in semi arid areas are character- 2005), or hydro-biological model for the same type ized by irregular flow and a strong hydrologic fluc- of zone in Brazil (Pedro et al., (2006). in the Mid- tuation (Arab et al., 2004). During the year, inter- dle East, Alexandrov et al., (2007) quantify the mittent flows dominate in most Wadis (Argyroudi sediment transport in the Negev watershed. Modern et al., 2009). A mismatch between lotic and lentic techniques involve the use of Digital Elevation conditions, is thus noticed (Morais et al., 2004). Models (DEM) for flood forecasting by parameter- It should be noted that this type of region where izing algorithms for two-dimensional hydrodynam- most rivers are ephemeral, do not have, to date a ic flood simulation (Sanders, 2007). global treatment approach. This is justified by In Algeria, we can refer to the studies of many reasons. (a) The lack of data that represents a Dechemi et al., (1994) in the Tafna watershed and handicap to study of changes in hydrological bal- the works of Achit and Ouillon (2007) for the Wadi ance (Sivapalan et al., 2003). (b) The occasional Abd. change of climate and anthropic conditions, which By tracking all these methods, numerical model- leads to an imbalance in the hydrologic fluxes and ling of transient and complexes free surface flows physical characteristics of the basin (Puigde- can be explored as an interesting approach. Horritt fabregas and Mendizabal, 1998; Xoplaki et al., and Bates (2002) present an example of a two- 2004; López-Moreno et al., 2007). (c) The high dimensional numerical model of flood wave propa- variability of ephemeral events and the hydrology gation. Mathematically, floods waves can be gov- of dryland zones. This implies a strong nonlinearity erned by the Saint-Venant model which is a nonlin- of explored models (Nouh, 2006; Hreiche et al., ear hyperbolic system of partial differential equa- 2007; Kingumbi et al., 2007; Lajili-Ghezal, 2007; tions. Nonlinear problems represent shock and rare- McIntyre et al., 2007; Nasri, 2007). (d) Spatial het- faction waves. To overcome the handicap in the erogeneity generates a gradual change in the hy- treatment of non-linearity, several solving strategies draulic capacity. So flood risks are confirmed (Leb- have been developed (Liggett and Cunge, 1975; di et al., 2006). Cunge et al., 1980; Guinot, 2010). Historical hydrology (Barriendos and Rodrigo, Among all numerical approaches, the shock cap- 2006; Brázdil and Kundzewicz, 2006) represents the turing finite volume methods have several ad- best tool that can respond to treatment of these is- vantages, since they combine between the simplici- sues. The effort is focused on the methodology. ty of the finite difference method and the flexibility Thus, robust methods are designed to ensure a of the finite element method (Godunov, 1959; Lax compromise between the applicability and accura- and Wenderoff, 1960; Van Leer, 1977; Leveque, cy, especially in rainfall-runoff modelling field, 2004 and Toro, 2006). which represents a major preoccupation in arid and However, in the treatment of real problems (eg: semi-arid catchments (Camarasa and Tilford, 2002; channel with complex geometry), some schemes Benkaci and Dechemi, 2004; Nasri et al., 2004; still need to overcome numerous difficulties (Wang Cudennec et al., 2005; Hreiche et al., 2007; Moussa et al., 2000). For instance, the treatment of the et al., 2007; McIntyre et al., 2007; Ouachani et al.,

126 Hydrodynamic investigation and numerical simulation

T a b l e 1. Targeted papers relevant to flood wave in Wadis and rivers.

Reference River basin or region Type Main contribution Cherif et al. (2009) Wadi Mekerra, Algeria Ephemeral Statistical Study Dechemi et al. (1994) Tafna Wadi, Algeria Intermittent Statistical study Horritt and Bates Severn river, UK – 2D Numerical model (2002) Nasri et al. (2004) El Gouazine and Dekekira, Tuni- Intermittent Flood impact sia Costelloe et al. (2005) Lake Eyre, Australia Both Hydrological models Pedro et al. (2006) Taperoá, Brasil Intermittent Hydro-biological model Alexandrov et al. (2007) Eshtemoa, Negev Ephemeral Sediment tranport Achit and Ouillon Wadi Adb, Algeria Intermittent Sediment transport (2007) Bracken et al. (2008) Nogalte and, Torrealvilla, Spain Both Rainfall and flood Sanders (2007) Santa Clara River – DEM model source term is one of the difficulties that may result order scheme. These schemes have proved their in significant numerical error in such cases. This ability to obtain robust, stable and monotone solu- has been a problem that many researchers are seek- tions (Ying et al., 2004; George, 2006; Begnudelli, ing to solve (Leveque, 1998; Garcia-Navarro and 2007). Such approach is based on solving the Saint- Vasquez-Cendon 2000; Zhou et al., 2001) and more Venant equations by numerical treatment of Rie- recently (Roger et al., 2003; Audusse et al. 2004; mann problem. The simulation results are promis- Benkhaldoun et al. 2007; Kesserwani and Liang ing and confirm the stability and accuracy of these 2010; Liang and Wang 2010; Lee and Wright numerical schemes. 2010). The remainder of the paper is organized as fol- Sidi Bel Abbes town makes part of semi-arid re- lows. “Study area” presents a global description gions. It is faced to floods phenomena and inunda- and General information about the study site, flow tions that occur in a catastrophic manner (Stucky, regime and the chronology of floods generated at 2005). These floods are frequent, recurrent and Wadi Mekerra. “Mathematical model and methods” present a major constraint on economic and social presents governing equations of Saint-Venant mod- development (DGPC, 2008). The management of el including numerical methods in which Godunov this phenomenon presents a particular concern for scheme and Monotonic Central scheme are briefly public authorities. The objective is to conceive a presented. “Numerical tests” presents a series of hydraulic management project that responds to the Dam failure and flood wave tests in order to verify city protection requirements against floods while and validate the model. We end by a Summary and preserving the benefits arising from the plain's im- Conclusion. mersion through an important contribution of natu- ral fertilizers elements and the aquifers filling. Study area: Wadi Mekerra Having regard to the importance of these accen- tuated flooding, several preventive works have been General information proposed. Certain concern the study of the flooding phenomenon in Mekerra catchment (Borsali et al., The Wadi Mekerra is the main stream in the wa- 2005). Currently, the works of SAFEGE group tershed which bears the same name. This latter is constitute a promising start in the implementation part of the great Macta basin. It occupies a great of an alert system which provides flood and also for part of Sidi Bel Abbes town (Northwest of Algeria). the delimitation of risk zones. It is located between the latitudes 34°31'N and In this paper, we present numerical simulation of 35°21'N and longitudes 1°16'O and 0°58'O (cf. Fig. a flood wave in October 1995, at Wadi Mekerra. 1). The Wadi has its source in the high valleys of This contribution aims to identify potential risk the steppe at an altitude of around 1250 m. To the areas, the wave magnitude and the time required to right of Sidi Bel Abbes city, the Wadi drains an alert threatened agglomerations. The numerical area of 3000 km2 and develops a talweg of 115 km simulation is achieved by applying finite volume in length. The catchment is elongated and north- methods. It is about of shock capturing first-order south oriented. Its relief is mountainous in the hills. Godunov scheme and Monotonic Central second The absolute heights of the watershed crest are

127 K. Korichi, A. Hazzab

1000 to 1100 m in the north, 1200 m in the oust, station. Tab. 2 summarizes the main hydromor- 1200 to 1260 m in the south and 870 to 1460 m at phometric characteristics of each sub-basin. the east of Mekerra basin. 48% of the catchment The Mekerra catchment is divided into three hy- area has an elevation above 1000 m, wich explains drological areas that manage the hydrological re- the torrential streams character. The average slope sponse of the Wadi. It is about of the floods produc- of Wadi is 5.5%. The Mekerra watershed is divided tion zone at the upstream, discharges transfer zone into three sub-basins (Hacaiba, in middle and expansion zone in downstream. and Sidi Bel Abbes city). Each one has a gauging

Fig. 1. Location of the Mekerra watershed and its major hydrometric stations.

T a b l e 2. Main hydromorphometric features of the Mekerra sub-basins.

Haçaiba Sidi Ali Benyoub Sidi Bel Abbes

Symbol Unit Station Station Station Features (1946) (1949) (1918) Surface S [km2] 957 1890 3000 Compactness index KC [km] 1.15 1.29 1.43 H [m] 1440 1715 1714 Characteristic height max H min [m] 925 635 437 Roche Slope index Ip [%] 0.099 0.0936 0.0913 Wadi's length L [km] 54.00 92.00 115.00 Drainage density Dd [–] 0.06 0.050 0.02 Torentiallity CT [–] 0.20 31.25 6.86 Time of concentration Tc [h] 6.369 10.560 13.448

Vegetation and geology (ANAT, 2000). It is thus noted that the vegetation cover in the basin is developed in an irregular man- Across the Mekerra watershed, only 25% of sub- ner. At the downstream of Ras El Ma city appear soil is impermeable. The northern part is much lands covered with Alfa, which towards the north more permeable than the southern part. Tab. 3 re- give way to brushland. At the upstream, there are sumes the land cover in the Mekerra watershed cereals, vines and citrus occupying Sidi Ali

128 Hydrodynamic investigation and numerical simulation

Benyoub zone. Forests cover over 20% of the basin agglomeration. During wet years the amount of area (DGPC, 2000). rainfall may exceed 800 mm (cf. Fig. 3). In dry The influence study of the vegetation on the lay- years, it decreases to 110 mm. The average interan- out and the stream aspect has been described by nual rainfall is around 400 mm. Filliol (Filliol, 1954). Depending to rates of banks integral fixation or bed partial fixation by vegeta- tion, it results in lateral erosion processes or depth erosion. There's also the factor of flow velocity influence. According Martinez-Mena et al., (1999) and Rachman et al., (2003), vegetation cover can reduce the rainfall kinetic energy at 50% and the runoff power to 75%.

T a b l e 3. Mekerra basin landcover.

Surfaces Landcover % surfaces [km2] Forest and maquis 500.3 27.61 Fig. 2. Variability of mean monthly precipitation and water Annual cultures 649.1 35.82 discharge in Wadi Mekerra (Sidi Bel Abbes gauge Station, Alfa 194.1 10.71 1942–2001). Agglomerations 21.16 1.17 Soil conservation works 7.56 0.42 Unproductive land 50.80 2.8 Route 103.9 5.73

Three main sets of soil dominate the Mekerra catchment. It is about of calcic brown soil, calcare- ous soil and alluvial soil. The first two types belong to the magnesium-calcic soil class. These soils are deeper, having a structure generally well developed granular or lumpy type in the surface horizon and polyhedral in the depth (Bouasria et al., 2010). On the agricultural front, their complex adsorbent is saturated with calcium and magnesium, the pH is above 6.8. The alluvial soils belong to the poorly evolved class. These soils are encountered on the Wadi terraces and characterized by a poorly differ- entiated and low-structured profile. This explains that these soils are good for agriculture (Askri et al., 1993).

Climate, rainfall and runoff

Situated in the semi-arid climatic region, the Fig. 3. Variability of annual precipitation (top) and mean annu- Mekerra catchment is characterized by hot dry al water discharge (down) in Wadi Mekerra (Sidi Bel Abbes summers spanning from April to August and a rela- gauge Station, 1942–2001). tively mild winter and wet from October to March. The average annual temperature is around 15°C and Fig. 3 shows in particular a strong flow fluctua- average interannual day of frost is about 35 days. tions during the period (1942 to 2001) in which the Prevailing winds are from northwest and west. The Wadi Mekerra has passed through two long dry interannual average speed is about 20 m s-1. Rain- periods. The first lasts for nine (9) years (1952 to fall is irregular, characterized by intense autumnal 1962) where the mean annual discharge was 0.221 rains causing major floods (cf. Fig. 2) whose effects m3 s-1. The second period has lasted seventeen (17) are felt between Boukhanifis and Sidi Bel Abbes years from 1968 to 1985 with an average discharge

129 K. Korichi, A. Hazzab of about 0.141m3 s-1. It is interesting to note that the Tab. 4 sums up the main damage, victims, in- lack of pluviometric inflows observed from the jured and homeless, caused by the floods in Wadi mid-seventies, confirms the drought in North Africa Mekerra due by thunderstorms and heavy rains over the past four decades (Tardy and Probst, during twenty five years (1978–2000). One can 1992). This lack is manifested in a dramatic de- add, agricultural land inundation, undermining of crease in water supplies drained by Maghrebi the Wadi banks and vegetation degradation. Wadis. Compared with normal flows, a decrease of Fig. 4 shows the major floods in Wadi Mekerra 67% is observed in the North-West of Algeria presented by peak discharges between (1942– (Meddi et al., 1998). 2001). We observe in particular that the magnitude of floods has increased remarkably during the dec- Floods of Wadi Mekerra ade (1990–2001). Taking into account observations of 34577 instantaneous discharges for the period Overview (1942–2001), spread over the three gauging stations (Hacaiba, Sidi Ali Benyoub and Sidi Bel Abbes) a Inudation's problem generated by cyclical floods statistical study showed that the class of dominants of Wadi Mekerra has always constituted one of the discharges is that less than 10 m3 s-1. While the high main concerns of the managers of Sidi Bel Abbes discharges (more than 200 m3 s-1) represents only city. Further to observations made during the vari- 0.01% (Stucky, 2005) (cf. Tab. 5). ous floods recorded, we can relate it to several The daily flow duration curve (cf. Fig. 5), shows causes, namely: (1) Intense and irregular cloudburst a very pronounced concavity reflecting the irregu- at high Mekerra can reach 200 mm h-1. (2) Mor- larity of the flow regime. A special flow discharges phology of Mekerra watershed, particularly elon- are shown in Tab. 6. gate, where time of concentration is very low (Tc = = 9 hour). (3) Narrowing of the Wadi section at Flow regime some stretches and hydraulic structures. (4) Al- most-total obstruction of the Wadi's sections by The flow regime of the Wadi Mekerra was inves- sediments carried and deposited by anterior floods. tigated based on mean daily discharges recorded at (5) The downstream affluents have very low slopes the station of Sidi Bel Abbes for 59 years (1942– if not negative. (6) Unplanned urbanization in the –2001). The average number of days without flow banks of the Wadi (DGPC, 2000). This factor af- reaches 328 days per year (90% of the year). This fects seriously the magnitude of floods. Indeed, for indicates the intermittent nature of the Wadi. In the semi-arid basin (Wadi Medjerda in Tunisia) referring to the literature for the flow regimes char- Lebdi et al., (2006); Zahar and Albergel (2006) acterization of Wadis in this type of region, we note have shown that people's sensitization affects the the existence of several classifications. Some of vulnerability of flood risks. them cover the world, others are continental

T a b l e 4. Flood statistics in Sidi Bel Abbes City (1978–2000). Inundated area Number of sinister Location Date Cause [ha] Dead Homeless families 160 Ain El Berd 02/10/78 Thunderstorm – 08 24 Ben Badis 06/03/80 Heavy precipitation – 08 10 Boukhanifis 06/03/80 Heavy precipitation – 04 Not estimated Sidi Bel Abbes 04/10/86 Thunderstorm 01 200 Not estimated Sidi Bel Abbes, 30/04/90 Thunderstorm 02 130 Sidi Bel Abbes, Boukhanifis, Sidi Not estimated Lahcen, Hassi Zahana, , 29/09/94 Thunderstorm 01 22 Not estimated , Ain El Berd 16/10/94 Thunderstorm – 70 Not estimated Sidi Bel Abbes, Sfisef, Ain El Berd 05/12/95 Thunderstorm – 03 Not estimated Hassi Zahan 13/06/96 Thunderstorm – 02 Not estimated 17/08/97 Thunderstorm 01 34 Not estimated Sidi Bel Abbes 27/09/97 Thunderstorm 01 - Not estimated Sidi Bel Abbes 13/12/97 Thunderstorm 01 05 Not estimated Ras El Ma 27/07/00 Thunderstorm – 100

130 Hydrodynamic investigation and numerical simulation

250 20/12/1942 10/10/1943 08/09/1944 14/11/1945 03/02/1946 14/05/1947 02/10/1948 01/05/1949 29/12/1950 04/01/1951 26/01/1952 26/09/1953 14/04/1954 18/06/1955 08/04/1956 21/10/1957 14/11/1958 02/12/1959 03/08/1960 27/09/1961 13/02/1962 11/12/1968 06/10/1969 30/09/1970 30/04/1971 24/09/1975 02/10/1976 22/10/1977 29/01/1978 15/09/1979 07/03/1980 13/02/1981 03/11/1982 24/06/1983 11/11/1984 29/11/1985 04/10/1986 04/10/1987 17/05/1988 18/09/1989 16/10/1990 15/03/1991 22/09/1992 13/11/1993 27/09/1994 22/10/1995 05/02/1996 16/09/1997 03/02/1998 27/09/1999 24/10/2000 10/10/2001 --

200 /s) 3

150

100

Peack discherage (m discherage Peack 50

0 Date

Fig. 4. Peak annual discharges recorded at Wadi Mekerra (1942–2001).

T a b l e 5. Distribution in [%] of floods in Wadi Mekerra (1942–2001).

Classes of discharges [m3 s1] > 200 200–100 100–50 50–10 <10 Frequency 0.01 0.5 1.44 5.29 92.77 [%]

T a b l e 6. Special discharge recorded at Wadi Mekerra (1942–2001).

Duration Max Low >10 >30 >90 >180 >270 >355 [days/year] flow flow Discharge 1.473 1.357 0.869 0.318 0.142 0.065 0.003 0.022 [m3 s-1]

Fig. 5. Daily flow duration curve for Wadi Mekerra (1942–2001).

(Smakhtin, 2001). These classifications are based sonal-intermittent. Oueslati et al. (2010) note that on several hydrological parameters. Poff and Ward the stations in eastern and southern parts of the (1989) have developed a conceptual model based Mediterranean basin are more intermittent and pre- on a hierarchical classification of the main compo- dictable in compared to those located in west and nents of the flow regime, namely (intermittency and north. In fact, identified works for the case of Alge- flooding frequency). In tropical regions of Austra- ria's catchments, confirm this finding. As such, we lia, Molière et al., (2009) has developed a classifi- cite the studies of Terfous et al., (2001) in Wadi cation analysis based on discharge variability, the Mouilah; Wadi Mina in eastern Algeria (Achite and flood regime model and the intermittency indexes. Meddi, 2005); Wadi Abd (Achite and Ouillon, This analysis indicates that rivers could be classi- 2007) in western Algeria and Wadi Haddad in cen- fied as permanent, seasonal dry-seasonal and sea- tral Algeria (Touaïbia et al., 2001).

131 K. Korichi, A. Hazzab

In our case, the data exploration confirms the descent during the recession. This regularization is ephemeral character of Wadi Mekerra throughout justified by the spreading fields in the downstream. the studied period except between 1997 to 1999 In the recession phase, the water in spreading ex- where he was intermittent. Further, we find that the tents and the water stored in the alluvium (banks flow regime of Wadi Mekerra hardly reflects the and bed) return by overflow into the streams. On intensity of the heavy rains. Because, a high daily the contrary, water stored in the alluvial part of the discharge does not necessarily result in a high mean Wadi basin assists to the prolongation of the flood annual discharge. This was well observed in 1997 recession and contributes toward the extension of (cf. Tab. 7) where the total volume flowed during low flow during the initial phase of the dry period. the flood was 2,64.106 m3 corresponding to a max- The recharge is relatively slow, which favors the imum instantaneous discharge reaching 170 m3 s-1. filling of the groundwater table and the settling of While in 1999 the total volume flowed during the natural fertilizers elements. Peak discharges are flood was 2,63.106 m3 corresponding to a maximum larger in high Mekerra (Sidi Ali Benyoub) that in instantaneous discharge which was about 90 m3 s-1. its lower part (Sidi Bel Abbes City), where there Floods are more frequent and severe in autumn has been a flat spotting (lamination or natural regu- than in spring. They are characterized by a rapid lation) of the flood wave (Sadeg, 2003). rising in flood phase and a slower and regularized

T a b l e 7. Main total volumes flowed at Wadi Mekerra.

Maximum instantaneous Mean total volume [%] flood volume discharge Total volume flowed flowed in year relative to mean Flood date during the flood annual volume Gross Specific [106 m3] [106 m3] [m3 s-1] [l/s/km2] 29/12/1950 110 60.71 6.52 22.30 29.24 14/04/1954 100 55.19 5.65 9.14 61.82 04/10/1986 104 57.40 4.82 12.89 37.39 15/03/1991 135 74.50 9.29 11.88 78.20 27/09/1994 215 118.65 4.78 30.94 15.45 22/10/1995 154 84.99 7.99 17.41 45.89 05/02/1996 151 83.33 8.64 20.94 41.26 16/09/1997 170 93.82 2.64 42.79 6.17 27/09/1999 89.88 49.60 2.63 13.67 19.24 24/10/2000 161.8 89.29 6.85 20.45 33.50 10/10/2001 111.17 61.35 5.19 23.12 22.45

Mathematical model and methods ⎡Q ⎤ (2) ⎡ A⎤ ⎢ ⎥ U = , F(U ) = 2 and Governing equations ⎢Q⎥ ⎢Q ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ A ⎦ One-dimensional unsteady flows in a natural riv- ⎡0 ⎤ er (Wadi) with irregular cross-section may be de- ⎢ 2 2 ⎥ S(U ) = ⎢ ∂z n Q ⎥ scribed by the Saint-Venant equations (Saint- ⎢−gA − g ⎥ ∂x AR4/3 Venant, 1871) which is derived from the mass and ⎣ h ⎦ momentum conservation principles based on hydro- where A – cross-sectional area [m2], Q – discharge static pressure distribution of incompressible fluid. 3 -1 -2 [m s ], g – gravitational acceleration [m s ]; Z – The conservative matrix form of these equations is water level [m] (cf. Fig. 6), n – Manning coefficient written as: -1/3 [s m ], R = A/P – hydraulic radius and P – wetted ∂U ∂F (1) perimeter of the channel [m]. The above form of + = S(U ) the Saint-Venant equations is commonly used in ∂t ∂x engineering practice. where U, F(U) and S(U) are respectively, the vec- tors of conserved variables, fluxes and sources term, defined ad follows:

132 Hydrodynamic investigation and numerical simulation

Numerical methods

Among all the numerical techniques, the finite volume method is well adapted to treat conserva- tion, hyperbolic and nonlinear problems such as Saint-Venant equations. This method is based on the discretization of the integral form, by subdivid- ing the domain in a number of finite volumes n (cells). The approximate conserved variable Ui represents the average value over each cell. It is defined at the cell centres as

Fig. 6. Definition sketch of cross-sectional geometry. 1 U n Ux,t dx (3) i ≈ ∫ n Δx C () Shallow water equations represent a nonlinear i hyperbolic system of PDEs, where the Jacobian n n matrix And the fluxes F and F are respec- i −1/2 i +1/2 tively calculated at the cells interfaces x and x ∂F(U ) ⎛ 01⎞ i-1/2 i+1/2 J = = ⎜ ⎟ (cf. Fig. 7). Integrating Eq. (1) over the i-th cell ⎜ 2 ⎟ ∂U ⎝ −u +gh 2u ⎠ with length of Δxi yields: is diagonalisable with real and distinct eigenvalues n +1 n Δt ⎡ n n ⎤ 1 2 U = U − F − F + ΔtS . (4) λ = u − gh and λ = u + gh , u is the average flow i i ⎢ i +1/2 i −1/2⎥ i Δx ⎣ ⎦ 2 -1 i velocity [m s ]. The nonlinearity in the Saint- Venant system presents shock and rarefaction waves.

Fig. 7. Definition sketch of cell centered grid.

There are several approaches to evaluate the flux ⎪⎧Ul if x < 0 at the interfaces that construct various conservative Ux(),0 = ⎨ (5) U if x > 0 numerical methods (Ying et al., 2003). We are in- ⎩⎪ r terested in this work to shock capturing schemes based on solving Riemann problem. The solution of the Riemann problem is the key ingredient of Godunov type scheme, the solution of the Riemann problem is computed by what is called a Riemann Solver, it is considered as a tool for the computation of the fluxes at the interfaces between the computational cells. Riemann problem repre- sents (cf. Fig. 8) an initial value problem and de- Fig. 8. Definition sketch of Riemann problem. fined as follows:

133 K. Korichi, A. Hazzab

In the literature, there are several types of Rie- average values in the neighboring cells. This allows mann solver; one can cite exact Riemann solver, the gradients to be located more accurately, thus but also approximate Riemann solvers that are easy leading to more accurate estimates of the fluxes. to implement such as HHL solver (Harten et al., Most second-order schemes use a linear recon- 1983) and Roe Riemann solver (Roe, 1981). Roe’s struction of the conserved variable (cf. Fig. 9b) as: solver, used in this study, is probably the most widely used Riemann solver to date. The basic idea n n n U = U + x − x a (6) is to convert hyperbolic system of conservation law i i ()i i to an equivalent linear one that is easier to solve by n The slope ai of the profile within cell i is calculat- replacing the exact Jacobian in each interval by a ed as the average slope between cells i – 1 and i + 1. constant Jacobian as follows: By limiting the slope of the reconstructed profile,

spurious oscillations are eliminated from the solu- tion. In fact, the contributions of the gradient of the ⎡ 01⎤ variable are limited using a so-called limiter, for Aˆ ⎢ ⎥ i 1/2 = 2 which many formulations have been proposed in − ⎢ −uˆ + gh 2uˆ ⎥ ⎣ ⎦ the literature. In this study, we are interested in Monotonic where Central (MC) scheme. This choice is justified by numerous previous numerical tests (Korichi, 2006). MC limiter function is given by (Guinot, 2010): h u + h u 1 i −1 i −1 i i h = h + h , uˆ = 2 ()i −1 i h + h Φ(θ) = max 0,min 1+θ / 2,2,2θ , (7) i −1 i ()()()

Godunov's approach indicated by the Eq. (4) is the where Φ is the limiting function and θ represent the basis of choc capturing methods (Godunov, 1959) monotony indicator given by: where the conserved variable is approximated by a ⎧ n n piecewise constant function (cf. Fig. 9a). In Godu- Ui −Ui−1 nov’s scheme, the left- and right-states of the Rie- ⎪ if CFL ≥ 0 ⎪U n −U n mann problem are taken from the average values of n ⎪ i+1 i θi 1/2 = ⎨ the variable over the cell on the left- and right-hand + n n ⎪Ui+2 −Ui+1 side of the interface respectively. ⎪ if CFL ≤ 0 U n −U n Higher-order Godunov-type schemes use a re- ⎩⎪ i+1 i construction procedure to estimate the variations of the conserved variable in the calculation cells. The variable in a given cell is reconstructed using the

Fig. 9. Finite volume discretization (a) Piecewise constant (Godunov) (b) Piecewise linear (Monotonic Central).

134 Hydrodynamic investigation and numerical simulation

Fractional step method is used for the treatment ra (October 1995) by applying the selected scheme. of source term. The boundary conditions are treated For calibration, we vary Manning coefficient to by checking the flow regime. Which is verified by a observe the influence of roughness on the numeri- simple calculation of the so-called Froude number cal results. The flood hydrograph is used to validate Fr = u/(gh)1/2. It is defined as the ratio of flow ve- the results. locity u to celerity c of the waves in still water. When the flow is subcritical at a given boundary, Dam failure test only one condition is required. When the flow is supercritical entering the domain, two boundary The first test is an examination of a dam break conditions are needed. When the flow is supercriti- wave on wet bottom in a rectangular, smooth and cal leaving the domain, no boundary condition is horizontal channel. One admits an initial time step required. Such approach is sufficient for finite vol- Δt = 0.1s for a simulation period of 60s. The do- ume method (Sanders, 2001). main length is 2000m divided into 2000 pieces. The rupture point is located at x = 1000m. The upstream

Numerical tests water depth is initially hl = 10m, while the initial water depth at downstream hr = 5m. The initial The primary objective of these tests is to verify discharge is null. the reliability of Godunov scheme and compare it At the end of the simulation (t = 60s) we observe with a MC scheme. The comparative study includes that the Godunov scheme diffuse near discontinui- the examination of unsteady flow. It is a dam break ties, while the MC scheme has more stability and test on a wet bottom in a smooth rectangular chan- precision (cf. Fig. 10). nel. These two schemes are checked using an ana- lytical solution. The other application is a one- dimensional flood wave simulation in Wadi Meker-

(a) (b)

             !                % 

$  !

#

        " !

! 

 !  !   !  !       

Fig. 10. Comparison of different schemes for dam break test (a) Depth, (b) Discharge.

Simulation of flood wave at Wadi Mekerra the flood wave dated on 11–12 October 1995. The in October 1995 channel length is about 44 km from Sidi Ali Benyoub station to Sidi Bel Abbes station. Fig. (11) Used data shows a simplified longitudinal profile of Wadi Mekerra between these two stations where the Based on the results of the first test, the selected stream is treated as a prismatic rectangular channel scheme (Monotonic Central) is applied to simulate of width b = 18.6m. The boundary conditions are

135 K. Korichi, A. Hazzab represented by a flood hydrograph observed at Sidi established by Crausse (Crausse, 1951), Chow Ali Benyoub station while fixing the water depth in (Chow, 1959) and Graf (Graf, 1984) which ranges downstream at 0.6m. in general from 0.012 and 0.15. For earthen and To understand the influence of frictional forces natural channels the Manning coefficient is be- on the flood wave shape, three Manning coeffi- tween 0.02 < n <0.05. The simulated results are cients considered in this study are (n = 0.02, 0.03 compared with recorded measurements at the and 0.04 s m-1/3); this choice is justified by the vari- downstream station. ability in the Wadi bed and the basin Landcover. Complete lists of the Manning coefficient, n were

Fig. 11. Simulated longitudinal profile of Wadi Mekerra between Sidi Ali Benyoub and Sidi Bel Abbes stations.

The flow is initially uniform with a discharge Q0 Observations extracted from the results (cf. Fig. = 1.02 m3 s-1; while the maximum discharge rec- 13) show that the flood strength decreases accord- orded during this flood is 155 m3 s-1 through total ing the increase in the Manning coefficients. It is time t = 48 hours. The studied domain is divided direct consequence of the frictional forces that into 100 cells. To keep the Courant number close to damp the impact of a high velocity gradient. Tab. 8 one, the time step is adapted as: summarizes the main results. One notes that for Manning coefficient n = 0.02, the maximum dis-

charge is largely overestimated with an error of CFL Δt = ( u + gh) 0.07% and an advancement of 15 minutes. For Δx Manning coefficient n = 0.03 and 0.04, the maxi- mum flow rates are respectively 121.3 and 116.28 Analysis and comments m3 s-1 but the rise time of flood wave is retarded. It reached 2.64 hours for the second case.

T a b l e 8. Comparison between the simulated and recorded results for different coefficient of Manning.

Simulated Recorded at n = 0.02 n = 0.03 n = 0.04 Sidi Bel Abbes station Flood rising time [h] 5.76 7.68 8.64 6 Difference in flood rising time [h] –0.24 +1.68 +2.64 – 3 -1 Qmax [m s ] 124.4 121.3 116.28 116 Error (Q) [%] 7 4 3 –

136 Hydrodynamic investigation and numerical simulation

The roughness influence is clearly established. ships (Warmink et al., 2007). Other models consid- However, the spatial distribution of roughness in er the temporal evolution of these dunes (Paarlberg the study area represents a major handicap for the et al., 2006). Einstein and Barbarossa (1952), con- numerical simulation of floods. The channel flow sider that the total roughness height of the main resistance is largely determined by the presence of channel (ktotal) can be divided into a contribution of dunes that usually develop in the beds. This re- grains (kgrain) and dunes (kdunes). sistance varies depending on the nature and topog- According to our results, the time required for raphy of the Wadi, if it is minor bed (main channel) the wave to reach Sidi Bel Abbes station is estimat- or major bed (floodplain). Udo et al., (2007) shows ed at t = 6 hours. We can divide the recession phase the relationship between roughness, expressed by in two stages, the first covers 12 hours (6h to 18h) the Chézy coefficient and discharge (cf. Fig. 12). in which the simulated discharge are larger than They thus prove that the resistance in the main recorded discharge. This is due in part to the water channel is much greater than that in the floodplain. lamination by the spreading field formation Relationship (roughness, flow) is more linear in (swamps), on the other hand, exclusion of affluents floodplains (Perumal et al., 2004). influence to the main thalweg. The second phase In transient flows cases, flood modelling seems lasts from t = 18 hours until the end of the simula- being complicated. In some hydraulic models, the tion. It is thus observed that the simulation results roughness coefficient is directly related to the dunes and recorded hydrograph are clearly coincided. sizes that are dimensioned by empirical relation-

Fig. 12. Calibraion Chezy coefficient of main channel land the floodplain as a function of discharge (Udo et al., 2007).

160      140            120         

) 100  

 80 (

60

40    

20

0

-20 0 1020304050    

Fig. 13. Comparison between measured and simulated hydrographs for different Manning coefficients.

137 K. Korichi, A. Hazzab

Summary and conclusion sediment transport generated in flood period. The silty-clay fraction increases with the width of Wadi Mekerra, located in a semi-arid zone, is Wadis and influences the service life of hillside part of the ephemeral Wadis with temporal flow. In dams located in the upstream (Albergel et al., 2004; early autumn, it generates devastating floods that De Araujo et al., 2006; Remini and Wassila, 2006; cause human and material damage. This is due to Ben Mammou and Louati, 2007). hydrological, geomorphological and human factors Acknowledgements. The authors warmly thank too. These floods threaten twenty five agglomera- anonymous reviewers that their comments and sug- tions. Since the 2006 disaster to date more than 120 gestions were highly appreciated. billion dinars ($ 14 million) have been mobilized to cope with inundations. A substantial budget has List of symbols been devoted to the concreting works on Wadi A – cross section of the channel [m2], Mekerra channel. It is about 13 kilometers connect- ∅ – flux limiter function [–], ing Sidi Lahcen city to the north of the Sidi Bel θ – limiter variable [–], Abbes city adding Wadi banks development. Other λ – eigenvalues of the Jacobian matrix [–], efficient techniques; the hillside dams of various CFL – Courrant Number [–], sizes have been built to store water, contributing to F – vector of flux terms [–], cope with the high variability and reduce flood risk. Fr – Froud number [–], g – gravitational acceleration [m s-2], Methodologically, the flood forecasting and h – water depth [m], monitoring of changes in the flow magnitude are J – Jacobian matrix [–], realized by numerical simulation of the flood wave. k – Dune height [mm], -1/3 This simulation is performed using the simplified n – Manning coefficient [s m ], P – wetted perimeter [m], one-dimensional Saint-Venant model. Finite vol- 3 -1 Q – discharge [m s ], ume shock capturing schemes (Godunov and Mon- R – hydraulic radius [m], otonic Central) are applied. Source term is treated S – vector of source terms [–], by using fractional step method. Numerical results t – time [s], U – vector of conservative variables [–], analysis for the dam break test confirms that slope 2 -1 u – flow velocity [m s ], n limiter Monotonic Central scheme is more advanta- Ui – approximation of the conservative variable [–], geous for the accuracy and stability. However, the x – longitudinal coordinate [–], first order Godunov scheme present more diffusion z – elevation [m]. at discontinuities. The MC scheme is thus applied to simulate the flood wave caused by Wadi Meker- REFERENCES ra dated on October 11th, 1995. The variation of ACHITE M. and MEDDI M., 2005: Variabilité spatio- Manning coefficient (n = 0.02, 0.03 and 0.04) con- temporelle des apports liquide et solide en zone semi-aride. firms that maintenance activities in the bed can Cas du bassin versant de l'Oued Mina (nord-ouest algérien). amortize the impact of the flood wave. Rev. Sci. Eau, 18, 37–56. This promising work could be extended to the ACHITE M. and OUILLON S., 2007: Suspended sediment transport in a semiarid water-shed, Wadi Abd, Algeria application of a two-dimensional model that takes (1973–1995). J. Hydrol, 343, 187–202. into account both the storage coefficient influence ALBERGEL J., NASRI S., LAMACHÈRE J. M., 2004: HY- (inflow-outflow) and also the spatial distribution of DROMED – Programme de recherche sur les lacs colli- Manning coefficient. For more accuracy, applica- naires dans les zones semi-arides du pourtour méditerra- néen. Rev. Sci. Eau, 17, 133–151. tion of well-balanced schemes rather than fractional ALEXANDROV Y., LARONNE B., REID I., 2006: Intra- step method is envisaged. Wetting and drying event and inter-seasonal behaviour of suspended sediment treatment is recommended too. An adaptive refined in flash floods of the semi-arid northern Negev, Israel. Geo- mesh applied to a proper digital elevation model morphology, 85, 85–97. would be a certain advantage (Begnudelli et al., ANAT, Agence Nationale pour la Conservation de la Nature, 2000: Programme national sur la conservation de la nature, 2008 ; Liang and Borthwick, 2008; Berger et al., 40 p. 2010). New procedures in processing satellite im- ANTEVS, ERNST, 1952:"Arroyo-Cutting and Filling." J. ages allow the calibration of the roughness module Geol., Vol. 60, No. 4, pp. 375–385. in the study area Schumann et al., (2007). ARAB A., LEK S., LOUNACI A. and PARK Y. S., 2004: Spatial and temporal patterns of benthic invertebrate com- Other critical parameter has an interest in devel- munities in an intermittent river (North Africa). Ann Lim- opment of an appropriate model. It is about of the nol. – Int. J. Lim, 40, 317–327.

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