Proceedings, ASCE World Water and Environmental Resources 2004 Congress, Salt Lake City, June 27-July 1, 10 p.

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1 Modeling Channel-Floodlain Co - evolution in Sand-Bed Streams J. W. Lauer and G. Parker 2 Graduate Student, Saint Anthony Falls Laboratory, Deartment o Civil Engineering, University o Minnesota, 2 3 rd Avenue SE, Minneaolis, MN 5544; laue0050@umn.edu 2 Proessor, Saint Anthony Falls Laboratory, Deartment o Civil Engineering, University o Minnesota, 2 3 rd Avenue SE, Minneaolis, MN 5544; arke002@umn.edu Abstract The imortance o including loodlain eects in one-dimensional river channel hydraulic comutations is well known. However, in morhodynamic modeling, these are oten artially ignored by using a single eective discharge to drive bed elevation changes. This neglects changes in channel caacity (and thus eective discharge) caused by deosition on or erosion rom either the channel bed or the loodlain. This aer resents a model or reach-averaged channel bed and bank to elevation evolution that seciically accounts or changes in channel deth over time. The model considers two grain sizes: one or sand, which interacts rimarily with the bed, and one or mud, which interacts only with the loodlain. The model also describes the evolution o the roortion sand and mud in the loodlain deosits. Sediment transort and loodlain deosition are driven by a simle gradually varied low solution. Erosion rom the loodlain is reresented as a net loss associated with channel migration. Because overbank deosition is strongly aected by low, eective loodlain deosition and in-channel sediment transort are obtained by integrating results rom an entire low duration curve. In the absence o bed elevation changes, the channel and loodlain co-evolve toward a stable bankull geometry where overbank deosition just equals loodlain erosion. Introduction It is generally acceted that alluvial rivers are resonsible or building their loodlains. To accomlish this, rivers must both deosit material on and erode material rom their loodlains. Since loodlain deosits are usually much iner than the river bed sediment, correctly redicting the interaction between a river and its loodlain requires accounting or both bed material and ine sediment (Narinesingh et. al., 999).

2 Most river bed aggradation/degradation models consider only sediment in the bed material size range. Furthermore, while many models allow loodlain eects to be included in the low solution, the loodlain surace is usually not allowed to change over time as a unction o the ine material sulied to the system. This aer resents a model or loodlain evolution that allows a river bed and its loodlain to evolve together by seciically accounting or both bed material and iner grain size ractions. The model is based on relatively simle reach-averaged sub-models or both overbank deosition and loodlain erosion, and a simle -D model or low and sediment transort within the river channel. The model is alied to an examle long term rediction o bed and loodlain evolution. Formulation The model is intended to reresent reach-averaged behavior over relatively large length and time scales. Consequently, it ignores all lateral and most vertical structure within the loodlain itsel and simliies the channel by assuming a wide rectangular section. The model enorces conservation o sediment mass in each o our comletely-mixed sediment reservoirs at each cross section; a) the active layer o the bed (Hirano, 97), b) a basal loodlain layer, c) an uer loodlain layer, and d) the water column (Figure ). The model also allows material to be moved to or rom a assive, vertically unmixed substrate as the channel aggrades or degrades, resectively. B H B c Uer Floodlain Channel Water Column Uer Floodlain Iui F i H c L Floodlain Base Bed Active Layer Floodlain Base F bi F ai L a Substrate Ili si (z) Iai η Datum Figure. Concetual cross sections showing sediment reservoirs and deining variables. 2

3 Geometric variables used in the ormulation are as ollows: η elevation o the base o the bed s active layer, B c channel width, B loodlain width, L a thickness o the bed s active layer, L thickness o the uer loodlain channel bankull deth, H deth o low on the loodlain, and H c deth o low in the channel. Grain size ractions in size class i are reresented in the uer loodlain by F i, in the loodlain base by F bi, and in the active layer by F ai. Several local grain size ractions are also required; Iui raction moving across the uer interace o the loodlain base layer, Ili raction moving across the lower interace o the loodlain base layer, and Iai raction moving across the interace between the channel bed active layer and the substrate. These variables deend on the direction the boundaries are moving. Finally, si (z) raction in class i at any level z in the substrate. The model is simliied or the resent alication to two grain sizes: one or sand, which interacts rimarily with the bed but can be resent anywhere, and one or mud, which can be resent anywhere excet in the active layer o the bed. While sand can be susended in the water column, the sand concentration is always assumed to be in equilibrium with the entrainment rate rom the bed (Garcia and Parker, 99). Figure 2 shows the sediment lux athways o the model. Accounting or the lux along each athway results in a descrition o the evolution o the channel bed and loodlain elevation and the raction sand and mud in each reservoir. From Ustream Water Column From Ustream Bed Active Layer Water Column Uer Floodlain Substrate Bed Active Layer Floodlain Base To Downstream Water Column Fine material (mud) Bed material (sand) To Downstream Bed Active Layer Figure 2. Sediment low athways considered by the model at a given cross section. The low o bed material (sand) is denoted by a dashed line while low o material iner than that on the bed (< 62.5 µm; mud) is denoted by a dotted line Note that the concentration o bed material susended in the water column is ully seciied by the low, so any overbank deosition o sand on the uer loodlain is eectively taken directly rom the bed s active layer, and any sediment o bed material size eroded rom the loodlain is eectively moved directly to the bed s active layer. 3

4 The undamental well-mixed layer is the active layer o the channel bed, which by deinition contains only the bed material size. As the channel migrates, the relatively coarse material on the bed is abandoned in the loodlain. It is not realistic to assume that this material immediately becomes mixed across the entire loodlain thickness, since it enters loodlain at a relatively low elevation. Instead, our model orces it to enter the loodlain base layer, which is deined geometrically based on the thickness o the bed s active layer. Conservation o Mass For Bed Region The conservation o mass equation or the channel bed, which is here assumed to include both the well mixed active layer and a substrate with a otentially arbitrary vertical grain size structure, can be stated as ollows or each grain size range: Rate o change o sediment volume below the bed surace Transer rate rom the uer loolain to the channel bed through loodlain erosion Transer rate to the uer loodlain rom the bed through overbank deosition + net streamwise exchange rate o bedload and susended bed material load with adjacent nodes along the channel roile + net exchange rate with the loodlain base layer as the channel migrates. η + La xc BcFi z dz qei xc qobi xc + Bcqti B x cqti c xc x ( ) _ 0 ( λ ) + ( λ ) cla ( Fbi Fai ) xc Here λ orosity o bed and substrate (assumed constant), x c down-channel coordinate, q ei transer er unit channel length o material in size class i rom loodlain to channel due to bank erosion, q obi transer er unit channel length o material in size class i rom channel to loodlain due to overbank deosition, c seciied lateral stream migration rate, and the index i ranges rom (mud) to 2 (sand). Using the relationshi valley channel length x length x c Σ c () channel sinuosity, (2) taking the limit x 0, alying Leibnitz rule and assuming constant channel width B c, the ollowing conservation equation results: Iai η La + F ai Fai + L a ( λ ) q ei q B c obi q ti cl + x Σ Bc For i 2, F a, and the equation reduces to a orm that describes the time evolution o the change in elevation o the bottom o the channel bed active layer. η F b 2 cl Bc a Ia2 + Ia 2 q e2 q ob2 q t 2 a cl ( F bi L F ( λ ) B ( ) Σ x B c λ c a a ai ) (3) (4) 4

5 Conservation o Mass or Floodlain Base Mass conservation in the basal loodlain layer or each grain size is as ollows: Rate o change o sediment volume in the loodlain base layer Net exchange rate with the loodlain as the uer boundary o the basal layer moves + Net exchange rate with the substrate as the lower boundary o the basal layer moves Net exchange rate with the active layer o the channel bed as the channel migrates. η + La ( ) Fbi B xdz ( ) B ( + La ) xc Iui ( ) xc Ili λ λ η η η (5) + c xc La ( Fai Fbi ) Taking the limit as x 0, assuming no time variation in B, setting i 2, and making the assumtion that the basal layer is well mixed, the ollowing orm results or the time evolution o the raction sand in the loodlain base layer: Fb 2 η Iu2 Il 2 cσ L a Iu2 La cσ F (6) b 2 La B La La B Conservation o Mass or Floodlain For the uer loodlain layer, mass conservation becomes: Rate o change o sediment volume in the loodlain layer Net exchange rate o sediment with the channel net exchange rate with the loodlain base as the uer boundary o the loodlain base layer moves. ( λ ) xb F i L qobi xc qei xc ( λ ) ( η + La ) xb iui. (7) Summing over all grainsizes, n q obt q obi, q et q ei, (8) i ( λ ) n i assuming no time variation in B, and taking the limit as x 0 yields: dl Σ dη dla ( qobt qet ). (9) dt B dt dt This result, which describes the evolution o loodlain thickness, can be substituted back into 7 to give, or i 2: F 2 F 2 L Iu2 η La + + Σ B ( λ ) L ( q F q ) which describes the grain size evolution o the uer loodlain deosit. ob2 2 obt, (0) 5

6 Equations 4, 6, 9 and 0 reresent a system o our equations and our unknowns (η, F b2, L and F 2 ). However, it requires the indeendent seciication several additional terms, seciically the loodlain erosion terms q e and q e2, the overbank deosition terms q ob and q ob2, the streamwise rate o change o total bed material load q t / x, the thickness o the active layer L a, and the time rate o change o the active layer thickness L a /. The other terms, c, Σ, B c, and B can be arbitrarily seciied based on observed data. Erosion Model Erosive transer o material rom loodlain to channel can occur by several rocesses ranging rom direct erosive striing o the loodlain surace (Nanson and Croke, 992) to several kinds o losses associated with river bank migration. One o the migration-related rocesses is the tendency o river systems to migrate until an avulsion or cuto occurs. The average volume o the abandoned channel divided by the average time between cutos reresents a loss rate o loodlain material. Another migration-related loss rocess is the tendency or channels to build oint bars that are somewhat lower in elevation than the cut bank on the oosite side o the channel. I the to o the oint bar reresents the loodlain surace that is being regenerated, then there is a net loss o material since less is redeosited that is eroded. Only the third erosive rocess is included in the resent version o the model. For a given system, we assume a net elevation dierence, η, between the cut bank and the to o the oosite oint bar. Assuming the oint bar comosition is not signiicantly dierent than the cut bank, ( λ )( η) qei F i c. () Assuming that c and η are constant in time results in a model that describes the erosion rate rom the loodlain as constant in time but allows the raction sand and mud eroded to vary as the loodlain grain size distribution evolves. Deosition Model Narinesingh et. al. (999) and Parker et al. (996) rovide similar deosition models that characterize reach-average deosition as a unction o low on the loodlain and susended sediment concentration in the channel. Both are based on advective transort o susended sediment onto a loodlain which is treated as an array o stream tubes that act as individual settling basins. The orm o Parker, (996) is used here since it rovides several coeicients that allow the model to be calibrated. 2 C 0iQ αv si B q obi Fl ex (2) B Q Here, F l and α are dimensionless coeicients that must be obtained by calibration, C 0i average sediment concentration in the water column above the loodlain level (obtained using the Rouse roile or sand, Rouse, 939), v si settling velocity o size class i in quiescent water, and Q volumetric low rate o water on the loodlain. 6

7 Channel Hydraulics and Sediment Transort The deosition model requires a descrition o -D low on the channel loodlain and the concentration o susended sand in the channel above the level o the loodlain. The morhodynamic model also requires a total load sediment transort model, which is used to comute q t / x. A simle -D gradually varied low model is used to artition low between the channel and loodlain. The hydraulic model is concetually straightorward, using the standard orm o the energy equation to comute a gradually varied low water surace elevation roile (Sturm 200). The hydraulic model accounts or orm drag associated with dunes using the riction model o Wright and Parker, (in ress), which also includes the eects o density stratiication on channel bed riction. Manning s equation with a riction coeicient o 0. is used or the loodlain. The susended sediment transort model o Wright and Parker, (in ress) is used to redict susended sediment transort rates, while the model o Ashida and Michiue, (972) is used or bed load. The active layer thickness, L a, is seciied as an arbitrary raction (0%) o bankull deth L. Because overbank deosition is driven by loodlain inundation while in-channel transort is rimarily a unction o low below the bankull level, it is necessary to drive the model using a reresentative set o low rates taken rom a low duration curve. The time rate o change o any o the our sediment storage variables (η, L, F 2, and F b2 ) comuted or any given low in the low duration curve are multilied by the raction o time reresented by the given low and then summed to get the average change rate used to ste orward in time; φ j φ φ j ( Q j ) ; Pj (3) t t j In the above equations, φ is one o the our sediment storage variables, Q j is the low rate or the jth bin in the low duration curve, P j is the raction o the time that low Q j occurs, is the number o bins, φ j / is the comuted value o the time rate o change o one o the sediment storage variables at low Q j, and φ/ is the value o the rate o change o a sediment storage variable used to ste orward in time. Conservation o Mass or Water Column The overbank deosition model requires susended sediment concentrations or both sand and mud size classes. For sand, the concentration is seciied by the channel hydraulics. For mud, the concentration is indeendent o hydraulics, which is why susended mud is oten called washload. For relatively short reaches, it may be aroriate to assume that the washload concentration is simly seciied by a rating curve constructed rom observed data. However, over longer reaches, we would exect washload concentrations to vary in the downstream direction, articularly where loodlain erosion is not in equilibrium with overbank deosition. Since we are rimarily interested in long term, large scale eects, we have balanced the conservation o mass equation or a whole time ste by summing over all lows and 7

8 not considering individual bins o the low duration curve. The equation or washload concentration C at a given low rate j is: where C q, j x ei j H B c, j P q Model Alication j c ei j q e q ob + F b, ; qobi Pjq j η ( ) cl B obi j λ a c Ia (4),. and Ia η 0, 0 η ( η), < 0 The model is alied to the Minnesota River, a tributary o the Mississii with a drainage area o aroximately 43,000 km 2 at the conluence. The model is driven by a set o 20 low rates that characterize the uer hal o the observed low duration curve. Wash load concentrations at the ustream end are taken rom a susended sediment rating curve rom which susended sand concentration has been removed using the Wright and Parker (2003) susended load transort relationshi. The downstream end o the model has a ixed bed elevation and a normal deth water surace boundary condition. The ustream sediment eed in the sand size range is adjusted so that it is in equilibrium with the comuted caacity o the reach. The coeicient η required by the erosion model was obtained rom cross sections taken at several locations along the river. Migration rates were obtained rom sequential aerial hotograhy. The coeicients or the deosition model were then calibrated base on the observed bankull L o 4.6 m so that the resulting deosition rate was exactly equal to the erosion rate redicted by the erosion model and so that the roortion o sand and mud deosited on the loodlain is reasonable. We assumed a sand raction in the loodlain o 30%. This calibration assumes that loodlain erosion and deosition are currently in equilibrium along the modeled river reach. The loodlain elevation was then reduced by m along a 5-km reach o the valley, locally utting loodlain and channel erosion and deosition out o balance. Results Figure 3 shows several longitudinal roiles o the bed, loodlain, and one ercent exceedance water surace taken at various oints in time or this run (at 30, 300, and 3000 years). 30 years into the run, the bed in the zone o loodlain excavation has begun to aggrade due to the reduction in shear stress on the bed. During this time, the loodlain in the excavation area actually exeriences a reduction in overbank deosition as sand that would have been deosited on the loodlain is instead deosited on the bed. During this initial time eriod, the ustream bed degrades in resonse to the additional energy sloe caused by the reduced water levels in the excavated area. Eventually, however, deosition on the loodlain causes both the bed and loodlain to recover something close to the initial re-excavation roile. s (5) 8

9 4 2 % Exceedence Water Suraces Elevation (m) Initial 30 years 300 years 3000 years Floodlain Down Valley Coordinate (m) Bed Figure 3. Evolution history o bed, loodlain, and one ercent exceedence water surace elevation roiles. The model extends ustream an additional 5000 m and downstream an additional 2000 m. Discussion The relatively raid bed elevation changes redicted by the model during the irst 30 years o simulation are driven entirely by the additional conveyance that becomes available on the loodlain ater the excavation occurs. While existing channel bed aggradation and degradation models are able to redict this, they are only able to do so i enough o the lood low distribution is used to account or loodlain low. The eventual return to the initial roile, while erhas not occurring in engineering time (and not quite comlete even ater 3,000 years o simulation), cannot be redicted using standard aggradation and degradation models that do not account or overbank deosition. The eedback that allows the model to return to a stable state is based on the tendency or the more requent looding in the low area to cause deosition, eventually increasing the loodlain s elevation. However, as the loodlain is built, it loods less requently, eventually so inrequently that erosion and deosition come into equilibrium. A similar story can be told or a loodlain that is in some sense too high. As long as erosion rom the loodlain is not a strong unction o loodlain elevation, a high loodlain that loods inrequently should exerience more erosion than deosition, which would tend to decrease its elevation until erosion comes into balance with deosition. 9

10 Since loodlain elevation - bed elevation bankull deth, our model can be thought o as a model or channel bankull cross-sectional geometry. Our model diers in a undamental way rom the usual descrition o channel ormation in which bankull discharge is considered the channel-orming discharge. By deinition, any low, including bankull, that is contained entirely within the banks can not, at least through deosition, build a channel. While the bankull discharge rovides a useul, observable arameter uon which to base alluvial channel geometry models, it can not by itsel be resonsible or a channel s ormation. Rather, the range o lows near and above bankull must lay an imortant role. Our model, though concetually rather simle, catures the essence o the eedback between erosion and deosition we think is driven by this range o lows and ultimately controls a channel s deth. Acknowledgements This material is based uon work unded by the National Science Foundation under agreement numbers CTS and EAR , as well as the STC Program under agreement number EAR Reerences Ashida, K. and Michiue, M. (972) Study on hydraulic resistance and bedload transort rate in alluvial streams. Transactions, Jaan Society o Civil Engineering, 206: (in Jaanese). García, M., and Parker,G. (99) Entrainment o bed sediment into susension. Journal o Hydraulic Engineering, 7(4), Hirano, M. (97) On riverbed variation with armoring. Proceedings, Jaan Society o Civil Engineering, 95: (in Jaanese). Nanson, G.C., and Croke, J.C. (992) A genetic classiication o loodlains. Geomorhology, 4, Narinesingh, P., Klaassen, G.J., and Ludikhuize, D. (999) Floodlain sedimentation along extended river reaches. Journal o Hydraulic Research, 37(6), Parker, G., Cui, Y., Imran, J., and Dietrich, W. (996) Flooding in the lower Ok Tedi, Paua New Guinea due to the disosal o mine tailings and its amelioration. Proceedings International Seminar on Recent Trends o Floods and their Preventive Measures, Hokkaido River Disaster Prevention Research Center, Saoro, Jaan, Rouse, H. (939) Exeriments on the mechanics o sediment susension. Proceedings 5th International Congress on Alied Mechanics, Cambridge, Mass, Sturm, T.W. (200). Oen Channel Hydraulics, McGraw-Hill, New York. Wolmann, M. and Leoold, L. (957) River loodlains: some observations on their ormation. USGS roessional aer 282-C. U.S. Geological Survey. Washington, D.C. Wright, S.A. and Parker, G. In Press. Flow resistance and susended load in sandbed rivers: simliied stratiication model. Journal o Hydraulic Engineering. 0