Assessment of rating curve through entropy-based Manning s equation

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1 Assessment of rating curve through entropy-based Manning s equation DOMENICA MIRAUDA domenica.mirauda@unibas.it School of Engineering Basilicata University viale dell Ateneo Lucano 1, 851 Potenza ITALY MICHELE GRECO School of Engineering Basilicata University vialedell AteneoLucano 1, 851 Potenza ITALY michele.greco@unibas.it Abstract: - The rating curve is the most used methodology for river flow quality and quantity monitoring and control. Nevertheless, it is still difficult to obtain a reliable rating curve because it takes a long time and involves high costs, especially if the measurement cross-section is unstable. Besides, the difficulties faced by operators who carry out the measurements lead to water discharge values being calculated on a statistical basis or with numerical models, which could involve significant errors in rating curves.to overcome these difficulties, the use of a modified form of the Manning s equation, which contains information about the geometric development of the cross-section and the hydraulic behaviour, is suggested here. This formulation, derived from the entropy velocity theory, allows assessing water discharge in an expeditive way, maintaining a sufficient level of reliability, through the use of global parameters which consider the geometric information and the roughness of the river or, more extensively, of the areas of fluvial pertinence.in a first step, the procedure is validated on four river sections in the Grand-Duchy of Luxembourg and in the Basilicata region of Southern Italy. The results seem to demonstrate the validity and potential of the proposed approach. Keywords: entropy, Manning s roughness, rating curve, river, waterdepth, water discharge. 1 Introduction The rating curve of a hydrometric station, which permits the establishment of a relationship between water depth and discharge at a given cross-section, is the most frequently used methodology for continuous river flow measurements. A reliable rating curve is not only difficult to establish, but it also takes time and involves considerable costs. The validity of the relationship should be questioned after each significant change of the river bed geometry. Obtaining a reliable rating curve becomes impossible if the measurement cross-section is unstable [1], [2], [15]. To overcome this problem, the water dischargedepth relation is calculated on a statistical basis or with numerical models, which could involve significant errors in rating curves. Therefore, it is always relevant to develop reliable methods allowing the measurement of water discharges in an expeditive way, through the use of global parameters which consider the geometric information and the whole roughness of the river or, more extensively, of the areas of fluvial pertinence. Recent studies have shown how the entropy theory [3], [4], [7], [12], [17] is able to reconstruct the flow field and acquire the water discharge in an expeditive way. Such theory presents a simple analytical structure based on the evaluation of a single parameter, taking into account the influence of the cross-section geometry and roughness. The parameter can be calculated by the ratio between the mean and maximum flow velocities in the cross-section. Some studies, based on river observations, have underlined a linear dependence between the mean and maximum cross-section ISBN:

2 velocities, with a reduced variability of the ratio between the two velocities when the measurement cross-section changes[1], [6], [9], [13], [16], [17],[2]. This variability seems to be strongly dependent on the local stream morphology.such result allows to address the measurement only to the knowledge of the maximum flow velocity. In order to further reduce the field activities, while performing experiments on cylindrical free surface flows with different geometry, Greco and Mirauda [12] have found a fixed location where the maximum flow velocity occurs. Additionally, analysing the dependence of the entropy parameter on the hydraulic and geometric characteristics of the river cross-section, Moramarco and Singh [18] have proposed the formulation of the Manning s roughness, n, based on the entropy parameter and on the positions in which the velocity is hypothetically zero (y ) and maximum (y max ), respectively. Therefore, in the present paper a modified form of Manning s equation, derived from the entropy velocity theory, was used to obtain the rating curve. In a first step, the methodology was applied to four river sections presenting different geometric and morphological characteristics, and whose data acquisition was processed with different equipment at different water stages. 2 Methodology The ratio between the mean and maximum velocities sampled at a river cross-section [8] can be expressed as: (1) In Eq. (1) the mean velocity can be evaluated using Manning s formula: where u*=(grsf).5 is the shear velocity (g=gravity acceleration); k is the von Karman constant equal to.41; y is the distance at which the velocity is hypothetically equal to zero; α is the dip-correction factor, depending only on the ratio between the relative distance of the maximum velocity location from the river bed, y max, and the water depth, D, along the y axis, where u max is sampled. The location of the maximum velocity, supporting the dip-phenomenon hypothesis, can be obtained by differentiating Eq. (3) and equating du/dy=, which gives: (4) Experimental studies (Greco and Mirauda, 22) have shown that, for channels at different shapes of the cross-section, the velocity maximum is below the free surface around the 2 25% of the maximum depth. Thus, considering y max equal to ¾ of the maximum depth, D, according to Eq. (4), α becomes equal to 1/3. Replacing the value of αin Eq. (3), and after a little algebraic manipulation, the maximum flow velocity can be expressed as: (5) Therefore, inserting Eqs. (2) and (5) in Eq. (1), Φ(M) can be expressed in terms of hydraulic and geometric characteristics of a river: (6) From this latter equation a new formulation of Manning s roughness, n e, based on Φ(M) is derived: (7) (2) where n is Manning s roughness, R the hydraulic radius and Sf the energy slope. To determine the maximum velocity of the crosssection, u max, along the y axis assumed perpendicular to the bottom, the dip-modified logarithmic law for the velocity distribution in a smooth uniform open channel flow, proposed by Yang et al.[21], is considered: (3) Therefore, if Φ(M) is available at a gauged site of the river, then Eq. (7) allows estimating the n value in the cross-section. Replacing Eq. (7) in Eq. (2), the modified form of Manning s equation is obtained: (8) which takes into account the variation of a river s hydraulic and geometric characteristics following the change of the water discharge. ISBN:

3 Application to field data The methodology described above was applied preliminarily to field measurements collected on three ungauged cross sections located along the AlzetteRiver, in the

The choice of these four sections was suggested by their similar bed slope, around.1%, and a regular geometry of their crosssection.

River. It is characterised by a seasonal variation of the flow: high winter levels and low summer levels, with minimum levels during the month of September.

Investigated sections on the Alzette River: (a) Hunsdorf; (b) Lintgen and (c) Mersch. Figure 1. Alzette River basin.

3 3 Application to field data The methodology described above was applied preliminarily to field measurements collected on three ungauged cross sections located along the AlzetteRiver, in the Grand-Duchy of Luxembourg, and on one gauged section located on the SinniRiver, one of the main Rivers in the Basilicata region of southern Italy. The choice of these four sections was suggested by their similar bed slope, around.1%, and a regular geometry of their crosssection. The AlzetteRiver origins in France at around 4 kilometres from the frontier of Luxembourg and is the main tributary of the Sûre River, flowing into the Moselle River, which is tributary of the Rhine River. It is characterised by a seasonal variation of the flow: high winter levels and low summer levels, with minimum levels during the month of September. The three measured sections are located in the northern part of the river, in the valley where the slope presents values of about.1% (Fig. 1). a) b) c) Figure 2. Investigated sections on the Alzette River: (a) Hunsdorf; (b) Lintgen and (c) Mersch. Figure 1. Alzette River basin. The first section, named Hunsdorf, is located in the northern part of the basin, where the slope presents values of about.11% (Fig. 2a). The reach is straight and the bed is characterised by the presence of pebbles. The banks are covered by shrub and cane vegetation. The measurement of the water depth was processed manually with a graduated rod, and the velocity measurements were acquired through a propeller current-meter, stabilised by a heavy weight lowered from the bridges with a mobile trolley system. The second section, named Lintgen, is located in the northern part of the basin, where the slope presents values of about.9% (Fig. 2b). The reach is straight and the bed is characterised by the presence of fine sand and silt sediments. The banks are covered by shrubby and cane vegetation. Even in this case, the measure of the water depth was processed manually with a graduated rod and the velocity measurements were acquired through a current meter. The third section, named Mersch, is also located in the northern part of the basin, where the slope presents values of about.8% (Fig. 2c). The reach is straight and the bed is characterised by the presence of fine sand and silt sediments. The measured section has the banks and the bed covered by reinforced concrete. The velocity measurements were acquired through an Acoustic ISBN:

Current Doppler Profiler (ADCP) Workhorse Rio Grande. The Sinni River sources in the south-west Lucanian Appennines and flows into the Ionian sea after about 94 km (Fig. 3a).

4 Current Doppler Profiler (ADCP) Workhorse Rio Grande. The Sinni River sources in the south-west Lucanian Appennines and flows into the Ionian sea after about 94 km (Fig. 3a). The river is characterised by rainfall with significant floods in autumn and in winter, and low discharges during summer. The analysed measurement section, named Pizzutello, is located in the middle-upper part of the river, where the slope is about.1-.2% (Fig. 3b). a) Table 1. Ranges of the river sections main flow characteristics. The river stretch, including this section, presents a plan pattern tending to a single reach and the banks are characterised by thick vegetation. The measure of the water depth was automatically recorded through an acoustic hydrometer, and the velocities were acquired with a handheld Acoustic Doppler Velocimeter Flow-Tracker. The whole set of data collected on the gauged section presents similar values of water discharge, flow depth and aspect ratio, which allows comparing further analyses. Table 1 reports the ranges of water depth, D, aspect ratio, B/D (B is the channel width) as well as the mean, ū, and maximum flow velocities, umax, and water discharge, Q, of the four investigated river crosssections. 4 Field data analysis Fig. 4 shows how the hypothesis of maximum velocity location at ¾ of the depth seems to be robust for all measured sections. Inspecting Fig. 4 further, the maximum velocity is also below the free surface for values of the aspect ratio, B/D, higher than 6, as described in Ferro and Baiamonte[11]. b) Figure 3. (a) Sinni River basin; (b) Pizzutello gauged section. Section name D (m) Hunsdorf.5 2. Lintgen Mersch Pizzutell o B/D ū (m/s ) u max (m/s ) Q (m 3 /s ) Figure 4. Relation between the distance of the maximum velocity location from the river bed (y max ) and the maximum water depth (D). Eq. (7) computes Manning s roughness once the values of Φ(M) are known and the values of y are calibrated. The values of Φ(M) can be estimated for the investigated river sites, on the basis of the measured pairs (ū; u max ), as reported in Fig. 5. umax was considered as the maximum value in the data set of velocity points sampled during the measurements. The observed values of the parameter Φ(M) are consistent to those reported in literature by ISBN:

5 Moramarco et al.[18] for data collected on Italian and Algerian rivers. The value of y could be calculated by the value of the roughness size on the cross-section bed, according to the relationships of Wilcock[19], Yang et al. [21]and Cheng [3]. Figure 6. Relation between the maximum velocities (u max,v,c ) of all the measurement verticalscalculated using the y value of the y axis and the observed ones (u max,v ). Fig. 7 compares, for the four sites, the observed Manning s n values to those computed by Eq. (7), surmising that all quantities shown on the righthand side of the equation are known. Fig. 7 shows how the Manning s roughness values are reproduced fairly well by Eq. (7) at all fluvial sections. Figure 5. Relation between the mean (ū) and maximum (u max ) velocities for all the fluvial sections. In this case, lacking information on the size of river-bed sediments for all sections, y was evaluated by Eq. (5) knowing the maximum velocity on the y axis and the shear velocity. The energy slope, Sf, was calculated for each investigated section acquiring the water depth upstream and downstream the measured crosssection. The estimated value of y on the y axis was assumed for the whole cross-section, because it slightly changes along the section. In fact, this assumption was verified by comparing the maximum observed velocities for all the measurement verticals with those calculated by Eq. (5), assuming the same value of y for the whole cross-section (Fig. 6). Because in Eq. (5) the maximum depth, D, and the hydraulic radius, R, take into account the value of y, an iterative method was applied to find new values of y, D and R, useful then to determine the value of the maximum observed velocity. Figure 7. The computed Manning s values (n e ) versus the observed ones (n). Once the Manning s roughness, ne, was evaluated, the mean velocity was recalculated according to Eq. (8). With the availability of topographical surveys at measured cross-sections, the water discharges, Qc, were then evaluated and compared with the observed ones Q (Fig. 8). The result shows the perfect correlation between the two values and enforces the use of the proposed Manning s equation (7), derived by the employment of the entropy velocity theory and the assumption of a constant value of the dip velocity. The method leads to improve water discharge assessment by integrating information about hydraulic and geometric characteristics of the flow. ISBN:

6 Figure 8. Comparison between the computed (Q c ) and observed (Q) discharges at Hunsdorf, Lintgen, Mersch and Pizzutello sites. The following Fig. 9 reports the theoretical rating curves in logarithmic scale obtained by the modified Manning s equation and experimental data, considering all velocity measurement points. Such curves are represented only for the Mersch and Pizzutello sites, because for the other two sections there are not enough water discharge measurements to allow the evaluation of the rating curve in an accurate way. 5 Conclusion The use of a rating curve formulation, which takes into account the variables describing the geometric and hydraulic characteristics of a river branch, should allow the improvement of water discharge estimations, particularly for unstable sections. A possible choice for the evaluation of the rating curves is a modified form of Manning s equation, derived from the entropy velocity theory and from the assumption of a constant value of the dip velocity. The presence of the entropy parameter inside Manning s roughness equation leads to consider the evolution of the cross-section geometry, which causes the modification of the rating curve in time. This approach was tested, in a first phase, on a suitable data set of water discharge measures collected on both the Alzette River, in the Grand- Duchy of Luxembourg, and the Sinni River, in the Basilicata region (Southern Italy). The rating curve evaluation, derived for the monitored river sections, underlines a standard error of less than 5%, favouring an expeditive assessment of the flow stage with a sufficient level of reliability. Finally, the proposed approach appears to be suitable enough to reduce the acquisition time and cost during the field activities for river monitoring and control. Figure 9. Observed data and calculated rating curves for: (a) Mersch section; (b) Pizzutello section. Finally, a further test to validate the suitability of the proposed Eq. (8) was derived by the evaluation of the standard error, as suggested by the ISO 11-2 (1998), through the following relationship: (9) where Q is the measured discharge, Qc is the one computed by the rating curve built with Eq. (8), and N is the number of available measures. The observed Se is permanently less than 5%, giving a boost to the use of this methodology. References: [1] Burnelli A., Mirauda D., Moramarco T., Pascale V. 28. Applicability of entropic velocity distributions in natural channel. Proc. of 4th Int. Conf. Fluvial Hydraulics, River Flow, Izmir Cesme, Turkey. [2] Carter R.W Accuracy of current meter measurements. In: Symposium on Hydrometry, vol.ii, IAHS Publ. no. 99, 86-98, Koblenz, Germany. [3] Cheng N.S. 27. Power law index for velocity profiles in open-channel flows. Adv. in Water Resources 3(8): [4] Chiu C.L Entropy and probability concepts in hydraulics. Journal of Hydraulic Engineering 113(5): [5] Chiu C.L Velocity distribution in open channel flow. Journal of Hydraulic Engineering 115(5): [6] Chiu C.L., Hsu S.H. 26. Probabilistic approach to modelling of velocity distributions in fluid flows. Journal of Hydrology 316: [7] Chiu C.L., Murray, D.W Variation of velocity distribution along non uniform open- ISBN:

7 channel flow, Journal of Hydraulic Engineering118(7): [8] Chiu C.L., Said C.A.A Maximum and mean velocities and entropy in open-channel flow. Journal of Hydraulic Engineering 121(1): [9] Chiu C.L., Tung N.C. 22. Maximum velocity and regularities in open-channel flow. Journal of Hydraulic Engineering 128(4): [1] Chow V.T Handbook of Applied Hydrology. McGraw-Hill, New York, USA. [11] Ferro V., Baiamonte G Flow velocity profile in gravel-bed rivers. Journal of Hydraulic Engineering 12(1): 68. [12] Greco M., Mirauda D. 22. Experimental analysis for the entropic parameter evaluation, Proc. of 2nd Int. Conf. New Trends in Water and Environmental Engineering for Safety and Life: Eco-compatible Solutions for Aquatic Environments, Capri, Italy. [13] Greco M., Mirauda D. 24. Expeditive methodology for river water discharge evaluation. Proc. of 2nd Int. Conf. Fluvial Hydraulics, River Flow, Naples, Italy. [14] ISO 11-2/ Measurement of liquid flow in open channel - Part 2: Determination of the stage-discharge relation. [15] Leonard J., Mietton M., Najib H., Gourbesville P. 2. Rating curve modelling with Manning s equation to manage instability and improve extrapolation. Hydrological Sciences-Journal-des Sciences Hydrologiques 45(5): [16] Moramarco T., Ammari A., Burnelli A., Mirauda D., Pascale V. 28. Entropy Theory Application for Flow Monitoring in Natural Channels. Proc. of 4th Biennial Meeting Int. Congress on Environmental Modelling and Software, Barcelona, Spain. [17] Moramarco T., Saltalippi C., Singh V.P. 24. Estimation of mean velocity in natural channels based on Chiu s velocity distribution equation. Journal of Hydrologic Engineering 9(1): [18] Moramarco T., Singh V.P. 21. Formulation of the entropy parameter based on hydraulic and geometric characteristics of river cross sections. Journal of Hydrologic Engineering 15(1): [19] Wilcock P.R Estimating bed-shear velocity from velocity observations. Water Resources Research 32: [2] Xia R Relation between mean and maximum velocities in a natural river. Journal of Hydraulic Engineering 123(8): [21] Yang S.Q., Tan S.K., Lim S.Y. 24. Velocity distribution and dip-phenomenon in smooth uniform open channel flows. Journal of Hydraulic Engineering 13(12): [22] Yang S.Q., Tan S.K., Lim S.Y. 25. Flow resistance and bed form geometry in a wide alluvial channel. Water Resources Research 41(9): W9419. ISBN:

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