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EROSIONAL NARROWING OF A CHANNEL RAPIDLY INCISING INTO A RESERVIOR DEPOSIT IN RESPONSE TO SUDDEN DAM REMOVAL Alessandro Cantelli, Miguel Wong, Chris Paola and Gary Parker St. Anthony Falls Laboratory University of Minnesota Mississippi River at 3rd Ave SE Minneapolis, MN USA Some Preliminary Results

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CONSIDER THE CASE OF THE SUDDEN REMOVAL, BY DESIGN OR ACCIDENT, OF A DAM FILLED WITH SEDIMENT Before removal

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REMOVAL OF THE DAM CAUSES A CHANNEL TO INCISE INTO THE DEPOSIT After removal

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AS THE CHANNEL INCISES, IT ALSO REMOVES SIDEWALL MATERIAL Can we describe the morphodynamics of this process?

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A SEEMINGLY UNRELATED PROBLEM: DEGRADATION OF THE LITTLE WEKIVA RIVER, FLORIDA The Little Wekiva River drains a now heavily-urbanized area in the northern suburbs of Orlando, Florida. The stream was straightened, and the floodplain filled in in the period 1940 – Urbanization of the basin a) reduced infiltration, increasing the severity of floods, and b) cut off most of the supply of sediment to the stream. As a result, the stream degraded severely and produced substantial sidewall erosion.

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MORPHODYNAMICS OF DEGRADATION IN THE LITTLE WEKIVA RIVER Two of us (Cui and Parker, consulting) developed a morphodynamic model of the evolution of the Little Wekiva River as a tool for designing bank protection and grade control structures, which have since been installed.

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EXNER EQUATION OF SEDIMENT CONTINUITY INCLUDING SIDEWALL EROSION B b = channel bottom width, here assumed constant b = bed elevation t = elevation of top of bank Q b = volume bedload transport rate S s = sidewall slope (constant) p = porosity of the bed deposit s = streamwise distance t = time B s = width of sidewall zone = volume rate of input per unit length of sediment from sidewalls > 0 for a degrading channel, i.e. b / t < 0

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EXNER EQUATION OF SEDIMENT CONTINUITY INCLUDING SIDEWALL EROSION contd. Reduce to obtain the relation: or That is, when sidewall erosion accompanies degradation, the sidewall erosion suppresses (but does not stop) degradation and augments the downstream rate of increase of bed material load.

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MODELING OF SEDIMENT PULSES IN MOUNTAIN RIVERS DUE TO LANDSLIDES AND DEBRIS FLOWS Landslide into the Navarro River, California, USA, 1995 Test of model (Cui, Parker, Lisle, Pizzuto) developed for EPA

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ADAPTATION TO THE PROBLEM OF CHANNEL INCISION SUBSEQUENT TO DAM REMOVAL: THE DREAM MODELS (Cui, Parker and others) Saeltzer Dam, California before its removal in 2001.

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THE DREAM MODELS Specify an initial top width B bt and a minimum bottom width B bm. If B b > B bm, the channel degrades and narrows without eroding its banks. If B b = B bm the channel degrades and erodes its sidewalls without further narrowing. But B bm must be user-specified.

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SUMMARY OF THE DREAM FORMULATION But how does the process of incision really work? Experiments of Cantelli, Paola and Parker follow. (Tesi di Laurea, Cantelli)

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NOTE THE TRANSIENT PHENOMENON OF EROSIONAL NARROWING! (Experiments of Cantelli, Paola, Parker)

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EVOLUTION OF CENTERLINE PROFILE UPSTREAM (x 9 m) OF THE DAM Upstream degradationDownstream aggradation Former dam location

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CHANNEL WIDTH EVOLUTION UPSTREAM OF THE DAM The dam is at x = 9 m downstream of sediment feed point. Note the pattern of rapid channel narrowing and degradation, followed by slow channel widening and degradation. The pattern is strongest near the dam.

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REGIMES OF EROSIONAL NARROWING AND EROSIONAL WIDENING The dam is at x = 9 m downstream of sediment feed point. The cross-section is at x = 8.2 m downstream of the sediment feed point, or 0.8 m upstream of the dam.

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SUMMARY OF THE PROCESS OF INCISION INTO A RESERVOIR DEPOSIT

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CAN WE DESCRIBE THE MORPHODYNAMICS OF RAPID EROSIONAL NARROWING, FOLLOWED BY SLOW EROSIONAL WIDENING?

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The earthflow is caused by the dumping of large amounts of waste rock from the Porgera Gold Mine, Papua New Guinea. PART OF THE ANSWER COMES FROM ANOTHER SEEMINGLY UNRELATED SOURCE: AN EARTHFLOW IN PAPUA NEW GUINEA

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THE EARTHFLOW CONSTRICTS THE KAIYA RIVER AGAINST A VALLEY WALL The Kaiya River must somehow eat all the sediment delivered to it by the earthflow. Kaiya River earthflow

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THE DELTA OF THE UPSTREAM KAIYA RIVER DAMMED BY THE EARTHFLOW The delta captures all of the load from upstream, so downstream the Kaiya River eats only earthflow sediment earthflow

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THE EARTHFLOW ELONGATES ALONG THE KAIYA RIVER, SO MAXIMIZING DIGESTION OF ITS SEDIMENT A downstream constriction (temporarily?) limits the propagation of the earthflow.

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THE VIEW FROM THE AIR Kaiya River The earthflow encroaches on the river, reducing width, increasing bed shear stress and increasing the ability of the river to eat sediment!

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THE BASIS FOR THE SEDIMENT DIGESTER MODEL (Consulting work of Parker) The earthflow narrows the channel, so increasing the sidewall shear stress and the ability of the river flow to erode away the delivered material. The earthflow elongates parallel to the channel until it is of sufficient length to be digested completely by the river. This is a case of depositional narrowing!!!

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GEOMETRY H = flow depth n = transverse coordinate n b = B b = position of bank toe B w = width of wetted bank n w = B b + B w = position of top of wetted bank S s = slope of sidewall (const.) b = elevation of bed = volume sediment input per unit streamwise width from earthflow The river flow is into the page. The channel cross-section is assumed to be trapezoidal. H/B b << 1. Streamwise shear stress on the bed region = bsb = constant in n Streamwise shear stress on the submerged bank region = bss = bsb = constant in n, < 1. The flow is approximated using the normal flow assumption.

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EXNER EQUATION OF SEDIMENT BALANCE ON THE BED REGION Local form of Exner: where q bs and q bn are the streamwise and transverse volume bedload transport rates per unit width. Integrate on bed region with q bs = q bss, q bn = 0; / t(sediment in bed region) differential steamwise transport transverse input from wetted bank region

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EXNER EQUATION OF SEDIMENT BALANCE ON THE WETTED BANK REGION Integrate local form of Exner on wetted bank region with region with: q bs = q bss for n b < n < n b + B w q bn = - at n = n t where q denotes the volume rate of supply of sediment per unit length from the earthflow Geometric relation: Result: / t(sediment in wetted bank region) differential steamwise transport transverse output to bed region transverse input from earthflow

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EQUATION FOR EVOLUTION OF BOTTOM WIDTH Eliminate b / t between Note that there are two evolution equations for two quantities, channel bottom elevation b and channel bottom width B b. To close the relations we need to have forms for q bsb, q bss and. The parameter is specified by the motion of the earthflow. and to obtain

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FLOW HYDRAULICS Flow momentum balance: where S = streamwise slope and B w = H/S s, Flow mass balance Manning-Strickler resistance relation Here k s = roughness height, D = grain size, n k = o(1) constant. Reduce under the condition H/B s << 1 to get:

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BEDLOAD TRANSPORT CLOSURE RELATIONS Shields number on bed region: where R = ( s / - 1) Shields number on bank region: Streamwise volume bedload transport rate per unit width on bed and bank regions is q bsb and q bss, respectively: where s = 11.2 and c * denotes a critical Shields stress, (Parker, 1979 fit to relation of Einstein, 1950). Transverse volume bedload transport rate per unit width on the sidewall region is q bns, where n is an order- one constant and from Parker and Andrews (1986),

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SUMMARY OF THE SEDIMENT DIGESTER Equation for evolution of bed elevation Equation for evolution of bottom width Hydraulic relations Sediment transport relations As the channel narrows the Shields number increases Higher local streamwise and transverse sediment transport rates counteract channel narrowing A higher Shields number gives higher local streamwise and transverse sediment transport rates. The earthflow encroaches on the channel

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EQUILIBRIUM CHANNEL Transports sediment without changing slope or eroding the banks (flow turned off below: Pitlick and Marr)

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EQUILIBRIUM CHANNEL SOLUTION As long as < 1, the formulation allows for an equilibrium channel without widening or narrowing as a special case (without input from an earthflow). Choose bed shear stress so that bank shear stress = critical value Streamwise sediment transport on wetted bank region = 0 Transverse sediment transport on wetted bank region = 0 Total bedload transport rate Three equations; if any two of Q w, S, H, Q b and B b are specified, the other three can be computed!!

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ADAPTATION OF THE SEDIMENT DIGESTER FOR EROSIONAL NARROWING As the channel incises, it leaves exposed sidewalls below a top surface t. Sidewall sediment is eroded freely into the channel, without the external forcing of the sediment digester. B b now denotes channel bottom half-width B s denotes the sidewall width of one side from channel bottom to top surface. The channel is assumed to be symmetric, as illustrated below.

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INTEGRAL SEDIMENT BALANCE FOR THE BED AND SIDEWALL REGIONS On the bed region, integrate Exner from n = 0 to n = n b = B b to get On the sidewall region, integrate Exner from n = n b to n = n t under the conditions that streamwise sediment transport vanishes over any region not covered with water, and transverse sediment transport vanishes at n = n t

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INTEGRATION FOR SIDEWALL REGION Upon integration it is found that or reducing with sediment balance for the bed region,

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INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION For the minute neglect the indicated terms: The equation can then be rewritten in the form: As the channel degrades i.e. b / t < 0, sidewall material is delivered to the channel. Erosional narrowing, i.e. B b / t < 0 suppresses the delivery of sidewall material to the channel.

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INTEGRAL SEDIMENT BALANCE: SIDEWALL REGION contd.

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INTERPRETATION OF TERMS IN RELATION FOR EVOLUTION OF HALF-WIDTH This term always causes widening whenever it is nonzero. Auxiliary streamwise terms This term causes narrowing whenever sediment transport is increasing in the streamwise direction. But this is exactly what we expect immediately upstream of a dam just after removal: downward concave long profile!

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REDUCTION FOR CRITICAL CONDITION FOR INCEPTION OF EROSIONAL NARROWING (on the plane and in the train) Narrows if slope increases downstream WidensEither way Where N S and N B are order-one parameters, At point of width minimum B b / s = 0

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REDUCTION FOR CRITICAL CONDITION FOR INCEPTION OF EROSIONAL NARROWING contd (on the plane and in the train) Where N s and N b are order-one parameters, After some reduction, where M is another order-one parameter. That is, erosional narrowing can be expected if the long profile of the river is sufficiently downward concave, precisely the condition to be expected immediately after dam removal!

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I HOPE THAT MY TALK WAS NOT TOO CONTROVERSIAL

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THANK YOU FOR LISTENING

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