Numerical Investigation on 1D and 2D Embankment Dams Failure Due to Overtopping Flow

Transcription

1 Internatonal Journal of Hydraulc Engneerng 2016, 5(1): 9-18 DOI: /.he Numercal Investgaton on 1D and 2D Embankment Dams Falure Due to Overtoppng Flow Omd Saber 1,*, Gerald Zenz 2 1 PhD Student, Insttute of Hydraulc Engneerng and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA 2 Prof, Insttute of Hydraulc Engneerng and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA Abstract Ths artcle provdes three numercal nvestgatons on the overtoppng falure of embankment dams whch are modelled wth non-cohesve fll materal. The frst nvestgaton s based on a one-dmensonal approach n order to calculate the outflow hydrograph durng dam overtoppng falure. The wrtten program calculates the flow parameters by solvng the one-dmensonal Sant-Venant equaton. Furthermore, the bed evoluton s calculated by solvng the Exner equaton wth the fnte dfference scheme. Here, the model accuracy s ncreased by dvdng bed slopes nto three categores: zero slope (0.0-1%), mld slope (1%-20%) and steep slope (>20%). At each tme step based on the real bed slope the bed load transport s modelled. In the second nvestgaton the two dmensonal open-source TELEMAC software has been modfed n order to model embankment dam falure. Ths modfcaton s done by calbratng slope correcton formula n the sedment part of ths software. Fnally, the nfluence of dfferent geometrcal parameters on the outflow hydrograph of dam falure s numercally modelled. Our modellng results show that the downstream slope has sgnfcant nfluence on the outflow hydrograph n comparson to the other geometrcal parameters. Keywords Embankment dam, Overtoppng falure, Numercal modellng, Steep slopes 1. Introducton Dam falure s a catastrophc event and can cause loss of lfe and property damage to the structures and propertes below the dam. Accordng to the Internatonal Commsson on Large Dams (ICOLD [7]) a large number of dams worldwde have been faled as embankment dams n comparson to the other types of the dams. Wthn ths type of dam, overtoppng falure s one of the most common causes of falures n comparson to the other types of falures lke ppng and slope falure. Therefore, n hydraulc engneerng, water resource management and rsk management felds, overtoppng falure s categorzed as more dangerous falure n comparson to other types of falures. In ths paper, dam falure due to overtoppng s reassessed. Generally, dam falure parameters can be calculated ether by emprcal or numercal methods. In emprcal methods, dam falure parameters can be predcted by fndng relatonshps between prevous dam falures parameters by usng regresson analyss. The man dsadvantage of ths method s gnorng many ste specfc parameters lke gran sze, roughness, densty, coheson, velocty, stress, etc. * Correspondng author: omdsaber@gmal.com (Omd Saber) Publshed onlne at Copyrght 2016 Scentfc & Academc Publshng. All Rghts Reserved Therefore, usng these smplfcatons, can lead to errors and the necessty to nclude more approprate falure parameters. In the numercal methods dam falure parameters are calculated by solvng the governng equatons of flud lke Naver Stokes, shallow water or Sant-Venant. The man advantage of ths method s consderng more reasonable parameters based on mechancal relatonshps lke coheson, frcton, etc. Ths means f approprate parameters are avalable, the numercal method s more accurate n comparson to the emprcal method. Numercal modellng heren s performed n two steps: frst to solve the hydrodynamc part, and second the eroson part. In the hydrodynamc part, flow parameters lke water depth and velocty of water n downstream of the dam s determned by usng one-dmensonal Sant-Venant equaton. In the eroson part by usng the pror calculated flow parameters, a bed evoluton durng falure process s calculated wth the sedment contnuty equaton. However, the fnal results of these two steps show that ths s not accurate. The man problem arses from the formulaton of the bed load equatons. Avalable bed load equatons are formulated for unform flow wth mld slopes. Whle n the embankment dam falure, flow regme s unsteady and the bed slope changes from steep to mld rapdly. Ths phenomena s nvestgated n ths study and as a soluton, a one-dmensonal program s wrtten n Fortran 90. Moreover, n the eroson part three dfferent equatons are used. The orgnal form of Meyer-Peter & Muller [13] equaton s used

2 10 Omd Saber et al.: Numercal Investgaton on 1D and 2D Embankment Dams Falure Due to Overtoppng Flow for modellng zero slopes, the SMART [18] equaton s used for modellng slopes up to 20% and for modellng slopes more than 20% the modfed form of SMART [18] equaton s calbrated and used n ths program. The second nvestgaton n ths paper s done on the sedment part of the open-source TELEMAC software (SISYPHE software) to model embankment dam falure. Ths modfcaton s done by calbratng slope correcton formula n the SISYPHE software. The accuracy of the 1-D program and the TELEMAC-2D software are valdated and compared to each other by usng L. Schmocker [17] small scale dam falure tests. In the last part of ths paper the nfluences of geometrcal parameters on the breach outflow hydrograph s numercally nvestgated and compared to each other. The fnal results show that the downstream slope of the embankment dams have more nfluence on the breach outflow hydrograph n comparson to the other geometrcal parameters of the embankment dams. 2. One-Dmensonal Dam Falure Modellng Dam falure parameters as falure tme and peak outflow dscharge are determned by usng ether emprcal or numercal methods. Emprcal methods use the data from the falure dam hstores to predct dam falure parameters. However, n numercal methods by solvng governng equatons of flow and consderng a wder range of scenaros and parameters results based on physcal processes are ganed. Furthermore, the numercal approach s normally performed n two steps: Hydrodynamc part Eroson part In the followng of ths paper these two steps are explaned n more detal Hydrodynamc Part The hydrodynamc part n the numercal approach for dam falure modellng concerns on one-dmensonal flow parameters lke water depth and ts velocty. These parameters can be determned by solvng the Sant-Venant equaton. Ths one-dmensonal equaton s vald under four man assumptons: 1. The flow velocty and water depth change only n the drecton of flow. 2. Vertcal acceleraton s neglected and hydrostatc pressure s consdered along the channel. 3. The densty of flud s constant along the channel and flud s ncompressble. 4. Slope of the channel s small. The conservaton form of ths equaton can be descrbed as the followng equatons (Cunge [3]): U = A Q,F = U + F = S (1) t x Q Q 2 A +gi 1, and S = 0 ga (S 0 S f ) Here, I 1 s the hydrostatc pressure. However, ths complex form can be smplfed f a rectangular channel wth constant wdth along the channel are assumed. Ths means that I 1 can be wrtten as equaton (2): I 1 = h2 B = A2 (2) 2 2B There are dfferent schemes to solve ths equaton lke MACCORMACK Scheme, FTSC Scheme, Lax Scheme and Leap-Frog scheme. In ths study the revsed MACCORMACK scheme (Garca Navarro [4]) s selected for solvng the Sant-Venant equaton. The man advantages of ths method n comparson to the other methods are: Fgure 1. The MACCORMACK Scheme

3 Internatonal Journal of Hydraulc Engneerng 2016, 5(1): The MACCORMACK scheme has two steps, predctor step and corrector step whch s capable of capturng the dscontnutes n the flow. 2. The MACCORMACK scheme has hgher accuracy because of usng two dfferental equatons both n space and tme. 3. In the MACCORMACK scheme the prmary results whch are determned durng the predctor part, are used durng the corrector part as shown n the Fgure 1. In general, n the MACCORMACK scheme, the Sant-Venant equaton s solved n two steps: - Predctor step - Corrector step In the predctor step the forward fnte dfference method normally s used to calculate the prmary flow parameters as shown n the equatons (3) and (4): A = A t (Q x +1 Q ) (3) Q = Q t x g t A Q 2 +1 A +1 Q 2 + g I A 1+1 I 1 S 0 S f Here, S 0 and S f are bed slope and frcton slope and are defned as the followng equatons: S 0 = (Z Z +1 ) x, S f = (n 2 Q Q + (4) ) (A R ) (5) In the corrector step the backward fnte dfference method s used to calculate the secondary flow parameters as shown n the equatons (6) and (7): A = A t (Q x Q 1 ) (6) Q = Q t x g t A S 0 Q 2 A Q g I A 1 I S f Here, S 0 and S f are bed slope and frcton slope and are defned as the followng equatons: S 0 = (Z 1 Z ) x, S f = (n 2 Q Q (7) ) (A R ) (8) Fnally, the value for each hydraulc parameter s determned by averagng between the results from these two steps (predctor and corrector): Q +1 = 1 2 (Q + Q ), A +1 = 1 2 (A + A ) (9) The above methodology, descrbes a procedure to solve the Sant-Venant equaton, the stablty of ths calculaton should be controlled by usng the Courant-Fredrch-Lewy (CFL) formula. In general, calculaton get unstable when the CFL number become larger than 1.0. Therefore the value of the CFL number must be less than 1.0 or 1.0 to have stable calculaton. The CFL number s expressed as equaton (10): CFL n = t x U + C 1.0 (10) C = (gh ) (11) Here, C (m/s) s the wave velocty, U (m/s) s the flow velocty, A (m 2 ) s the cross secton area and Z (m) s the bed elevaton. Now, the only unknown parameters remanng n the Equaton (9) are the mannng number and the boundary condton values. In the followng sectons these parameters are determned and explaned n more detal Boundary Condton In the MACCORMACK method lke other explct methods, all computatonal nodes can be calculated except two nodes, the frst node (n=1) and the last node (n=n). These nodes represent n total four unknown parameters whch are related to the heght of water and water velocty at ponts 1 and N. In order to calculate these unknown parameters a new method has been used n ths paper. Ths method was suggested by Garca Navarro [4]. In ths method unknown parameters are estmated by usng characterstcs and lnear nterpolaton approaches Roughness Computaton In open channel, the total flow resstance results from the nteracton between dfferent elements whch are located on the bed of the channel. Among these elements, some of them have more nfluence on flow resstance lke gran sze, vegetaton, bed slope, bed alment and obstructon n the channel. In fact, roughness coeffcent shows the effect of these parameters n stream flow. The mportance of calculatng roughness coeffcent s related to defne exact value for velocty n the mannng equaton. In ths nvestgaton the followng equatons are used for calculatng roughness coeffcent: The Ghan [5] equaton, for mld slope (S ) n = 4e10 8 ( H d 50 ) 2 5e10 5 ( H d 50 ) (12) The Jarrett [8] equaton for steep slope (S 0 > 0.01) n = 0.39S f 0.38 R 0.16 (13) Here, R (m) s the hydraulc radus, H (m) s the water depth and d 50 (m) s the mean dameter Valdate Hydraulc Part In ths part, n order to verfy the accuracy and performance of the MACCORMACK scheme, the Goutal and Maurel [6] test s employed. Ths test was carred out n a frctonless rectangular channel wth 25m length and 1.0m wdth. In the mddle of ths channel one bump has been consdered. The bed topography for ths bump s llustrated n the Equatons (14) and (15). Z = (x 10) < x < 12.0 m (14) Z = 0.0 x < 8.0 m and x > 12.0 m (15)

4 12 Omd Saber et al.: Numercal Investgaton on 1D and 2D Embankment Dams Falure Due to Overtoppng Flow In ths part, the model accuracy s demonstrated by usng the transcrtcal Flow over bump (supercrtcal wth hydraulc ump) and the fnal results are compared aganst the analytcal results. Fgure 2. Geometry of bump test Transcrtcal Flow over bump In ths test the downstream water level s kept constant at 0.33 m and nflow dscharge per unt wdth s kept constant at 0.18 (m 3 /s)(1/m). The fnal results are shown n the fgure 3. z +1 = 1 α z + (α(z +1 ( t 2 1 p x) q b +1 + z 1 )) 2 q b 1 (19) Here, p s the porosty and α s the stablty parameter and can be determned, based on Vreugdenhl [20] equaton: α = μ (20) μ = V((Q s Q) 1 Fr 2 )( t x) (21) Now, the only unknown parameter remanng n the equaton (19) s the bed load parameter (q b (m 3 /s)( 1 )). In m order to fnd ths parameter many dfferent expermental equatons wth dfferent accuracy are avalable. Generally, all avalable bed load equatons are formulated for unform flow wth mld slopes. Whle n the embankment dam falure, flow regme s unsteady and the bed slope changes from steep to mld rapdly. Consequently, usng these equatons for modellng embankment dam falures may show excessve errors n the fnal results. In ths study to overcome ths defcency the bed slope parameter n the SMART [18] equaton s calbrated. Furthermore, n order to reduce errors durng calculaton dfferent bed load equatons are selected and appled for slopes less than 20%. The appled equatons for dfferent slopes are dscussed brefly n the followng: For zero slopes, orgnal form of the Meyer-Peter & Muller [13] equaton s used: q b = 8 θ θ c 1.5 [g s rel 1 d 3 ] 0.5 (22) For slopes up to 20% slopes orgnal form of the SMART [18] equaton s used: q b = 4 g s rel 1 d [ d 90 d S (V ghs )θ 0.5 θ θ c ] (23) Fgure 3. Supercrtcal wth hydraulc ump Fgure 3, shows the MACCORMACK scheme has acceptable accuracy for modellng hydraulc ump n the open channels Eroson Part The second step n the numercal modellng of the embankment dam falure s calculatng the surface eroson. In ths step the bed elevaton s defned by solvng the one-dmensonal sedment contnuty equaton. Ths equaton s expressed as the followng equaton. 1 p ( z t) + ( q b x) = 0.0 (16) Equaton (16) s known as the Exner equaton. Ths equaton can be dscretzed by usng the Modfed Lax scheme as shown n the followng equatons: z t = 1 Δt z +1 1 α z + (α(z +1 q b x = (q b +1 + z 1 )) 2 (17) q b 1 ) 2 x (18) By substtutng equatons 17 and 18 nto equaton 16, the bed elevaton for tme step (+1) can be determned as: To model beds wth slopes more than 20%, the calbrated form of the SMART [18] equaton s selected: q b = 4 g s rel 1 d [ d 90 d S 0 ω (V ghs )θ 0.5 θ θ c ] (24) Here, S 0 s the bed slope, s rel s the relatve densty, θ s the sheld parameter, θ c s the crtcal sheld parameter and ω s the calbrated parameter. In ths paper to calbrate ω parameter, the Chnnarasr [1] dam falure tests are selected. The fnal value for ω parameter s shown n the Table 1. Table 1. Calbrated ω parameter (for slope more than 20%) Slope (%) 1V:5H - 1V:4H V:4H - 1V:3H More than 1V:2H TELEMAC MASCARETE The second nvestgaton n ths paper s related to updatng the sedment part of the TELEMAC MASCARETE software. Ths system has been developed by the Department ω

5 Internatonal Journal of Hydraulc Engneerng 2016, 5(1): Laboratore Natonal d Hydraulque (LNH) at the Electronc de France Drecton des Etudes et Recherché (EDF-DER).The software system s owned by EDF-R&D and ths software avalable as the open-source software whch s desgned for modellng free surface flud, flood wave propagaton, ground water flow and sedment transport n the open channels and rvers. The TELEMAC MASCARETE system s able to solve the Shallow Water Equaton for two dmensonal modellng (TELEMAC-2D) and the Naver Stokes Equaton for three dmensonal modellng (TELEMAC-3D). For modellng eroson and sedment transport n the rvers the SISYPHE software s used n TELEMAC system. The SISYPHE software can smulate sedment transport and bed elevaton n the complex morphology same as coastal, rvers, lakes and estuares wth the dfferent dscharge rate, dfferent sedment gran sze and dfferent sedment transport equatons. The SISYPHE software can be easly coupled wth the TELEMAC-2D and TELEMAC-3D software. In ths couplng, at each tme step TELEMAC-2D or 3D send the calculated hydrodynamc parameters lke heght of the water (H) and water velocty (U, V) to the SISYPHE software. The SISYPHE software model bed elevaton by solvng the two-dmensonal sedment contnuty equaton whch s called the Exner equaton as shown n the Equaton 25 (TASSI [21]). 1 p z t + Q b = 0 (25) Here, Q b (m 3 /s) s the bed load transport, z (m) s the bed elevaton and p s the bed porosty. Furthermore, SISYPHE software use the Meyer-Peter-Müller, Ensten-Brown and Hunzker equatons to determne bed load transport (Q b ). To mprove the accuracy of the fnal results, the SISYPHE software uses two correcton formula whch are called Koch and Flokstra [9] and Soulsby [10] equatons. In ths paper, n order to model sedment transport over the steep slopes, the β Tel parameter n the Koch and Flokstra [9] formula s calbrated by usng the Chnnarasr [1] dam falure tests as shown n the equatons (26) and (27): Koch and Flokstra formula (1981): M 1 = 1 β Tel ( z f s) (26) Q b = Q b0 M 1 (27) Here, M 1 s the correcton formula, β Tel s an emprcal parameter, Q b0 (m 3 /s) s the bed load transport and Q b (m 3 /s) s the corrected bed load transport. The calbrated value for β Tel parameter s shows n the Table 2. Table 2. Calbrated β Tel parameter Slope (%) β Tel 1V:5H - 1V:4H 1.3 1V:4H - 1V:3H More than 1V:2H Valdate 1-D Program and TELEMAC MASCARETE Software In order to verfy the accuracy and performance of the 1-D program and TELEMAC-2D software, two expermental tests are selected from the Schmocker [17] dam falure tests. The Schmocker tests are carred out n a glass-sde flume wth 8.0m length, 0.2m wdth and 0.7m heght. The small scale dke s nstalled at 4.0m dstance from the channel ntake wth 0.2m heght and 0.1m crest wdth. The upstream and downstream slopes of ths dam are fxed at 1V:2H. Furthermore, these dam falure tests have been done based on two dfferent dam materals: - Homogeneous sand wth mean sedment dameter 2.0mm. - Homogenous sand wth mean sedment dameter 0.31mm. Fgure 4. Comparson between measurement, 1-D and TELEMAC-2D results for d=2.0 mm

6 14 Omd Saber et al.: Numercal Investgaton on 1D and 2D Embankment Dams Falure Due to Overtoppng Flow Fgure 5. Comparson between measurement, 1-D and TELEMAC-2D results for d=0.31 mm In both materals the sedment s non-cohesve wth densty of 2650 kg/m 3. The nflow dscharge also was kept constant at 6.0 l/s durng falure process.the results for 1-D program and TELEMAC-2D software are shown n the Fgures 4 and 5. These fgures confrm that the appled calbraton approach n both software have a good accuracy for modellng slopes more than 20%. Dam wth ntal breach (frst scenaro) Dam wthout ntal breach (second scenaro) 5. Influence of Intal Breach One of the maor dfferences between one and two dmensonal numercal modellng n embankment dam falure s consderaton of the ntal breach. In order to fnd the nfluence of the mentoned ssue on the fnal results, two dfferent scenaros have been made on the Morrs [14] dam falure test. - Frst scenaro, ntal breach s consdered n the center of the crest. - Second scenaro, no ntal breach s consdered on the dam crest (Fgure 6). These two scenaros are modelled numercally wth the TELEMAC-2D software and 1-D program and the fnal results are compared aganst each other IMPACT Proect (Test No.2) The IMPACT proect (Morrs [14]) s one of the well-known large scale dam-break proects whch has been done n Norway n Ths embankment dam was bult manly from non-cohesve sol wth D 50 =4.75mm. The man purpose of ths test was to have a better understandng of the falure mechansm n the homogeneous non-cohesve embankment dams whch are faled by the overtoppng flow. Table 3 shows the more detals about ths test. Fgure 6. Influence of ntal breach on the outflow hydrograph H(m) Table 3. Lst of geometrcal parameters D 50 (m) Porosty (%) Upstream slope Downstream slope V:1.7H 1V:1.7H 5.2. Compare Results of 1-D Program and TELEMAC-2D Software In ths part, the modellng results of the one-dmensonal program and the TELEMAC-2D software, are compared aganst to the measurement results, these comparsons are shown n the Fgure 7. Fgure 7 shows the outflow hydrograph wthout ntal breach gves hgher value n peak outflow dscharge and lower value n falure tme compared wth the outflow hydrograph wth ntal breach. 6. Influence of Upstream Slope, Downstream Slope and Crest Wdth In ths part, the focus manly s on the nfluence of dam s geometrcal parameters lke upstream slope, downstream slope and crest wdth on the outflow hydrograph calculaton durng dam falure process. In order to defne whch geometrcal parameters have a sgnfcant nfluence on the

7 Internatonal Journal of Hydraulc Engneerng 2016, 5(1): breach outflow hydrograph the Chnnarasr [1] dam falure test s chosen. In the followng, ths dam falure test s explaned n more detal Chnnarasr Dam Falure Tests (Test No.1) Ths expermental test was performed n a flume wth dmensons 35m 1m 1m. A small scale dam s consdered n the mddle of ths flume wth 0.8m n heght and 0.3m crest wdth. The upstream slope of ths dam s fxed at 1V:3H and downstream slope of ths dam s fxed at 1V:2H. For all tests, the ntal water level n the reservor s kept constant at 0.83m and downstream water level s kept constant at 0.03m. The sol porosty s 0.35 and sol densty s kg/m 3. Ths dam falure test s numercally modelled by modfyng shape parameters as expressed n Table 4 by usng TELEMAC-2D software. Heren, the fnal results are compared aganst each other wthn the gven dataset and are shown n the Fgures 8, 9 and 10. No. Upstream slope Table 4. Lst of geometrcal parameters Downstream slope Crest wdth (cm) Heght of the dam (cm) 1 1V:2H 1V:2H - 5H V:2H - 5H 1V:2H V:2H - 5H 1V:2H - 5H Fgure 7. Influence of ntal breach on outflow hydrograph n 1-D and 2-D software Fgure 8. Influence of dfferent downstream slope on outflow hydrograph

8 16 Omd Saber et al.: Numercal Investgaton on 1D and 2D Embankment Dams Falure Due to Overtoppng Flow Fgure 9. Influence of dfferent upstream slope on outflow hydrograph Fgure 10. Influence of dfferent crest wdth on outflow hydrograph Fgures 8, 9 and 10 show, the downstream slope of the embankment dam has more nfluence on the outflow hydrograph n comparson to the other geometrcal parameters. 7. Dscusson Our senstvty analyss n the prevous part (secton 6) reveals that the downstream slope n the embankment dams has more nfluence on the breach outflow hydrograph n comparson to the other geometrcal parameters lke upstream slope and crest wdth. The man reason s related to the hgh nfluence of the downstream slope on water velocty and bottom shear stress durng falure process as beng llustrated n the Fgure 11. From the above fgure, t can be concluded by decreasng downstream slope, water velocty and thus the eroson rate of the dam's materal are decreased. Ths means that the rsk of dam falure s reduced sgnfcantly.

9 Internatonal Journal of Hydraulc Engneerng 2016, 5(1): The results of ths nvestgaton can be valdated wth other work studes. The most mportant one s the Mnmum energy loss (MEL) method whch s based on the work of Mackay [23]. In the followng MEL method s explaned n more detal. Energy lne Water surface Bottom S 0 Fgure 11. Water velocty and energy lne n downstream slope 7.1. Mnmum Energy Loss (Mackay [23]) The Mnmum energy loss (MEL) for the frst tme was desgned and ntroduced by Mackay [23] and s one of the unusual overtoppng protecton desgns. It was frst bult on the Chnchlla wer n Australa and t s stll n use wthout any damage. The Mnmum energy loss (MEL) s workng based on the reduce energy of large flood durng passng on the embankment dams to prevent any eroson and damage by decreasng downstream slope as shown n the Fgures 12 and 13. H1 Z Fgure 12. Sde vew of Mnmum energy loss (MEL) Fgure 13. Plan vew of Mnmum energy loss (MEL) Fgure 12 reveals that ths method s n agreement wth our concluson whch was mentoned n the prevous secton (secton 6). Q = B max g( 2 3 H 1 Z ) 1.5 (28) B = B max ( H des Z H des Z 1 ) 1.5 (29) Here: (H 1 -z) (m) s the upstream head above spllway crest B max (m) s the crest wdth (Fgure 1-7) τ V 2 /2g y Bottom shear stress Z1 H des [m] s the desgn upstream head Z 1 (m) s the any elevaton above the toe B (m) s the wdth of the channel at Z 1 elevaton 8. Conclusons In ths paper a one-dmensonal program s ntroduced and developed n Fortran 90 to model non-cohesve embankment dam whch fal by overtoppng flow. Addtonally the two-dmensonal open-source TELEMAC software has been appled and modfed n order to model embankment dam falure. Ths modfcaton s done by calbratng the slope correcton formulas n the sedment part of the TELEMAC software. Durng valdaton of these two software, some nvestgatons have been done on geometrcal parameters to fnd out the nfluence of these parameters on the breach flow hydrograph of embankment dam falure. In the followng the most mportant results are lsted: - Computatonal stablty n 1-D program s dependng on the Courant number. Best result can be reach f the Courant number s less than In 1-D program two basc assumptons are consdered. The frst assumpton s that whole length of the crest s eroded at same tme. In the second assumpton, rectangular shape wth constant wdth s consdered to model the reservor of a dam. - Change of the downstream slope n the embankment dam falure has more nfluence on the outflow hydrograph n comparson to the other geometrcal parameters. Notaton The followng symbols are used n ths paper A Wetted cross secton [m 2 ] B Wdth of channel [m] C Wave celerty [m/s] C n Courant number [-] Fr Froude number [-] g Acceleraton of gravty [m/s 2 ] I 1 Hydrostatc pressure [N/m 2 ] I 2 Force because of change n wdth [kg m/s 2 ] p Sol porosty [%] Q Dscharge [m 3 /s] Q 0b Dscharge [m 3 /s] Q b Modfed dscharge [m 3 /s] q b Unt dscharge [m 2 /s] R Hydraulc radus [m] S 0 Bed slope [%] S f Energy lne slope [%] s rel Relatve densty [-] Z Bed elevaton [m] t Tme step [-] x Cell space [-] α Stablty parameter [-]

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