Corato G.2, Moramarco T.2, Tucciarelli T.1 1 2 Department of Hydraulic Engineering and Environmental Applications , University of Palermo, Italy, Viale delle Scienze, 90128, Palermo, Italy Research Institute for Geo-Hydrological Protection, CNR, Via Madonna Alta 126, 06128 Perugia, Italy Four possible gauged station configurations for discharge monitoring I. Hydrometric river site with rating curve known II. Hydrometric river site with unknown rating curve 400 2.2 350 2 1.8 1.6 1.4 1.2 Level observations 1 2/3/05 0.00 3/3/05 0.00 4/3/05 0.00 5/3/05 0.00 6/3/05 0.00 7/3/05 0.00 Jones Formula (Henderson , 1966), Fenton (Fenton, 1999), Marchi (1976) 300 250 200 150 100 VPMS model (old) VPMS model (new) Time (h) 0 10 Level observations IV. Equipped river reach with rating curve known at one of ends (Rainfall-runoff modeling, Rating Curve Model) Negligible lateral flows Level observations Rating Curve Significant lateral flows Level observations Observed 50 0 III. Equipped River reach with level observations only (Dyrac (Dottori et al., 2009), MAST (Aricò et al., 2009), VPMS (Perumal et al., 2007; 2010) Inflow Feb. 1999 Discharge (m3/s) Stage (m) (m) Stage 450 2.4 20 30 40 50 60 Level observations III. Equipped River reach with level observations only Negligible lateral flows Level observations • In the context of the third configuration, the hydraulic DORA model (Tucciarelli et al. 2000), based on the diffusive hypothesis, can be applied. • The model starting from observed stage hydrographs at channel ends, allows there of estimating discharge hydrographs by using a calibration procedure of Manning parameter based on the wave speed of the flood computed through observed stages (Aricò et al. 2009) • However the model application, besides the need of topographical data of river sections, was found depending on the Manning’s roughness calibration procedure that affected the model performances Purposes To address the minimum channel length, L, so that the effects of the downstream boundary condition on the computation of the upstream discharge hydrograph is negligible To propose a new procedure for Manning’s calibration also for a real-time context by exploiting instantaneous flow velocity measurements carried out by radar sensors and using the entropic velocity model Using hydraulic modelling to optimize Configuration III Diffusive form of Saint Venant Equation: H H 1 R 2 / 3 A x =Q t T x H n x Boundary conditions Upstream q(0,t) = qu(t) Flow driven h(0,t) = hu(t) Water level driven Downstream h =0 x x =L or 2h 2 x x =L =0 Problem 1: Wath L? In Configuration III each possible downstream b.c. is an approximation of the physical one We need a reach long enough to avoid a strong estimation error of the discharge in the initial section It’s possible to get a rough estimation of the required length? Syntetic Test Hypothesis 1. Large rectangular channel, with constant bed slope 2. Linear variation of water depth in the upstream section Numerical model H 5/3 H h x =Q t x n H x and dh h(0, t ) = h0 t dt x =0 2 h =0 dt 2 x =L Root hydraulic head gradient at upstream section: Qualitative behaviour Dimensionless model The previous problem can be solved numerically once for ever using dimensionless variables, for the most severe case of initially dry conditions Dimensionless variables =h L = t 1/3 nL =x L Dimensionless equations 5/3 = 0 and d (0, ) = 0 d 2 h =0 d 2 =1 with 0' = d d = =0 dh n dt x =0 L2/3 and = -i =0 Error computation The solution is function of L. The reference solution is computed for L= E = maxt dh - x Hx =0 n, , i, L - limL dt 0.006 0.005 dh - x Hx =0 n, , i, L dt i=10-2 i=10-3 i=10-4 E 0.004 0.003 0.002 0.001 0.000 0.E+00 2.E-07 4.E-07 6.E-07 d/d 8.E-07 1.E-06 1.E-06 A priori estimation of L Relative Error Ed = max qmax - q qmax = Hmax - x =0 Hmax Hmax E i x =0 Procedure 1. given Ed, compute E from the above equation 2. from the previous graph, the corresponding value: 0' = dh n dt x =0 L2/3 L Problem 2: Wath n? In Configuration III the calibration of n is carried out using the stage hydrograph of the downstrem section We need a long enough reach to estimate the wave celerity In present method n can be estimated using a single mean velocity measurement Calibration Manning coefficent was determined minimizing the follow objective function: Err (n) = qcomp (tcal , n) - qobs (tcal ) qobs (tcal ) where qcomp(tcal,n) is the computed discharge at the instant tcal in which measurement is carried out, while qobs is the observed discharge. Problem 2: Estimatimation of calibration discharge Entropic Method To develop a practical and simple method for estimating discharge during high floods, Moramarco et al. (JHE, 2004) derived from the entropy formulation proposed by Chiu an equation applicable to each sampled vertical: umax v y y M u(y) = ln 1 e - 1 exp 1 M D h D h Gauged site: M estimated through the recorded pairs of (um, umax) u m = (M)u max um eM 1 (M) = = M u max e - 1 M If the measurement is carried out in the upper part of the flow area, umaxv is sampled for each vertical. Anyway, to drastically reduce the sampling period it is possible to consider only the upper portion where umax typically occurs and assuming that the behaviour of the maximum velocity in the cross-sectional flow area can be represented through a parabolic or elliptical curve. Umax i (ms-1) Problem 2: Estimatimation of calibration discharge Entropic Method a) b) 2 Umax i (ms-1) umax v ( x) = umax x 1 - xS c) measured d) elliptical parabolic Gauged Section: M.te Molino (Tiber River) – 28/11/05 ore 11:30 h=8.2 m riva DX riva SX a) c) b) d) Study Area: Upper Tiber Basin Pierantonio (1805 km2) Event December 1996 April 1997 November 1997 February 1999 December 2000 November 2005 qpM [m3/s] tph [h] hpM [m] Duration [h] 380.53 429.44 308.17 427.93 565.89 779.03 22.5 32.5 18.5 21.5 74 30.5 4.74 5.07 4.22 5.06 5.92 7.1 49.5 74.5 45 59.5 100 64 Ponte Nuovo (4135 km2) Event qpM [m3/s] tpq [h] hpM [m] Duration [h] November 2005 1073.2 32.75 8.52 70 December 2005 804.23 82.16 7.33 115 Test case 1: Pierantonio L estimation Mean bed slope i = 1.6x10-3 Typical Manning = 0.046 sm-1/3 dh = 2.67 x10-4 m/s (observed during Nov. 05) dt x =0 Ed = 0.05 '0 8 x10-7 E = Ed i = 0.0015 (from diagram) 3/2 dh dt x =0 L= ' 0 200 m Test case 1: Pierantonio L= 200 m L = 20000 m Event Qmax Error [%] December 1996 April 1997 November 1997 February 1999 December 2000 November 2005 2.96 -3.59 -3.51 -2.26 -4.69 -4.66 Test case 1: Pierantonio Discharge Estimation Results Cal. Time [h] 12 15 Man [sm-1/3] 0.051 0.050 Qmax err [%] 2.39 3.70 18 0.051 0.90 Cal. Time [h] 10.5 20.5 Man [sm-1/3] 0.051 0.058 Qmax err [%] 20.60 6.92 25.5 0.061 1.66 Test case 2: Ponte Nuovo L estimation Mean bed slope i = 0.85x10-3 Typical Tiber Manning = 0.042 sm-1/3 dh = 4.89 x10-4 m/s (observed during Nov. 05) dt x =0 Ed = 0.05 '0 3.6 x10-7 E = Ed i = 0.0015 (from diagram) 3/2 dh dt x =0 L= ' 0 400 m Test case 2: Ponte Nuovo Discharge Estimation Results Cal. Time [h] Man [sm-1/3] Qmax err [%] 15 0.04 11.7 22 0.044 1.85 24 0.045 -1.36 Cal. Time [h] 10.5 130.5 Man [sm-1/3] 0.045 0.043 Qmax err [%] -2.8 2.8 Conclusions The effect of downstream boundary condition over the upstream stage hydrograph computation has shown that short channel lengths are enough to achieve good performance of the diffusive hydraulic model The coupling of the hydraulic model with the entropic velocity model turned out of great support for an accurate calibration of Manning’s coefficient The developed algorithm can be conveniently adopted for the rating curve assessment at ungauged sites where the standard techniques for velocity measurements fail, in particular during high floods Based on the proposed procedure, discharge hydrographs can be assessed in real-time for whatever flood condition.