1. Introduction

Water discharge assessment in open channel still represents a fundamental aspect for hydraulic engineer in several operative and technical fields like water resources management, ecological flow assessment and control, drainage and irrigation system as well as runoff and flood routing model calibration and implementation. Nevertheless, the water discharge evaluation in generic open channel is heavily affected by local fluid dynamics and geometric conditions, which well arise once flow velocity measurements and morphological boundaries are available

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

at the same site. On the other hand, the drainage and irrigation channel present a regular cross section which might provide facilities in water discharge assessment and control, inducing also reduction in time and operative costs. That is, the implementation of operative procedures enabling operative charges simplifying the commitment of field activities, indeed, plays a fundamental role in channel monitoring for natural flow and manmade hydraulic structures. The main idea is related to the definition of expeditive procedures for flow field assessment and water discharge evaluation capable to optimize the surveying resources in time and efforts. Thus, the opportunity to manage with a simple and straightforward velocity law, different from the classical logarithm formulation but capable to provide suitable results is all the more technically fruitful. That is, an operative tool for expeditive velocity distribution assessment basing on simple and immediate parameters.

Therefore, M might represent an intrinsic parameter of the gauged site and this insight led several authors to explore the dependence of M on hydraulic and geometric characteristics of the flow site [3, 7]. In the case of river flows, Greco [9] enlightened a different behavior of Φ(M) depending on the roughness dimension: the velocity ratio is heavily influenced by the magnitude of relative submergence if large or intermediate scale [10]. Finally, the results support and validate a robust and fruitful operative chain to be implemented for expeditive water dis-

Informational Entropy Approach for Rating Curve Assessment in Rough and Smooth Irrigation Ditch

The concept of informational entropy as a measure of uncertainty associated to a probability distribution was formulated for the first time in the field of hydraulics by Shannon [11]. The principle of maximum entropy introduces the least-biased probability distribution of a random variable constrained by defined information system as well as the theorem of the concentration for hypothesis testing, introducing the informational entropy theory [12]. A direct evaluation of uncertainty related to the probability distribution of a continuous random variable expressed in

<sup>H</sup> ¼ � <sup>ð</sup>þ<sup>∞</sup>

where, p(x) is the continuous probability density function of random variable x.

�∞

Using POME, entropy can be maximized through the method of Lagrange multiplier as

in which, m > 0, gi(x) is the ith constraint function and λ<sup>i</sup> is the constrain Lagrange multiplier as

Chiu [1, 2] applied the concept of entropy to open-channel analysis to model velocity and shear stress distribution as well as sediment concentration. In such a way, the velocity distribution in the probability domain allows to obtain the cross-sectional mean velocity and the momentum and energy coefficients disregarding the geometrical shape of cross sections, which is generally

Further, an assumption on the probability distribution in the space domain is needed to relate the entropy-based probability distribution to the spatial distribution. Therefore, defining u by the time-averaged velocity placed on an isovelocity curve with the assigned value ξ, the value of u is almost 0 at ξ0, which corresponds to the channel boundary, while u reaches Umax at ξmax, which generally occurs at or below the water surface, depending on the dip-phenomenon. Thus, the velocity u monotonically increases from ξ<sup>0</sup> to ξmax and for each value of the spatial coordinate

p xð Þ <sup>1</sup> � p xð Þ � �<sup>m</sup>�<sup>1</sup> n odx <sup>þ</sup><sup>X</sup>

p xð Þlog p xð Þdx (1)

http://dx.doi.org/10.5772/intechopen.78975

7

N

i¼1

λ<sup>i</sup> gi

ð Þx (2)

charge assessment in rough and smooth irrigation ditch.

2. Entropy velocity profiles in open channels

terms of entropy, H, is defined as follows

<sup>L</sup> ¼ � <sup>1</sup>

a weight in the maximization of entropy.

complex in natural channels [2, 13].

m � 1

ðþ<sup>∞</sup> �∞

follows:

Recent theoretical and experimental studies endorse the informational content hold into the distributed velocity measurements following an entropy-probabilistic approach. That is, Chiu [1, 2] drew the correlation between the mean flow velocity and maximum flow velocity defining the entropy parameter, M introducing the velocity ratio Φ(M). Considering the important implication that this finding could have for monitoring of high flows in rivers, many authors investigated the reliability of this relationship using field data [3–7]. Overall, they found M as a river site depending and not influenced by the flood intensity both in terms of amount and duration. Thus, M should be considered a specific factor of the gauged cross section as outlined by Moramarco and Singh [7] exploring the dependence of M on the hydraulics and geometries of the river cross sections.

The study was able to explain the constancy of M value on the ground that M is not depending on the dynamic of flood, such as expressed by the energy or water surface slope, Sf and to identify a formula expressing M as a function of the hydraulic radius, Manning's roughness and the location, y0, where the horizontal velocity is hypothetically equal to zero. For the latter, it was preliminarily found that if y0 was assessed by distinguishing low flows from high flows, then a better estimation of M would have been obtained across a gauged river site. However, considering that the y0 location is not of simple assessment and then might have high uncertainty, the assessment of M should be addressed, mainly for ungauged river sites, using hydraulic and geometric variables easy to acquire. Such a thought might be discussed introducing the relative submergence D/d (in which, D = average water depth and d = roughness dimension). That is, the velocity distribution in natural rivers depends on several variables like channel geometry, bed and bank roughness, and the vertical velocity distribution generally increases monotonically from 0 at the channel bed, to the maximum at the water surface and can be assumed 1-D flow dominant. Moreover, whenever the channel cannot be considered "wide", that is the aspect ratio (B/D with B channel width and D water depth) is less than 6, besides the presence of the boundary, the velocity varies even transversely and a twodimension distribution occurs, leading G as the 2D entropy parameter. The maximum velocity places below the water surface inducing dip-phenomenon and the position of maximum velocity is also influenced by the aspect ratio [8], which is of simple assessment once channel cross-section geometry is known. Thus, investigating the influence of bed roughness and cross section geometry on medium and maximum velocity ratio at the global scale assumes a relevant interest in the field of open channel flow.

Therefore, M might represent an intrinsic parameter of the gauged site and this insight led several authors to explore the dependence of M on hydraulic and geometric characteristics of the flow site [3, 7]. In the case of river flows, Greco [9] enlightened a different behavior of Φ(M) depending on the roughness dimension: the velocity ratio is heavily influenced by the magnitude of relative submergence if large or intermediate scale [10]. Finally, the results support and validate a robust and fruitful operative chain to be implemented for expeditive water discharge assessment in rough and smooth irrigation ditch.
