A Multi-disciplinary Modelling Approach for Discharge Reconstruction in Irrigation Canals: The Canale Emiliano Romagnolo (Northern Italy) Case Study

Abstract

Agriculture is the biggest consumer of water in the world, and therefore, in order to mitigate the effects of climate change, and consequently water scarcity, it is important to reduce irrigation water losses and to improve the poor collection of hydraulic status data. Therefore, efficiency has to be increased, and the regulation and control flow should be implemented. Hydraulic modelling represents a strategic tool for the reconstruction of the missing hydraulic data. This paper proposes a methodology for the unmeasured offtake and flowing discharge estimation along the open-canal Canale Emiliano Romagnolo (CER), which is one of the major irrigation infrastructures in Northern Italy. The “multi-disciplinary approach” that was adopted refers to agronomic and hydraulic aspects. The tools that were used are the IRRINET management Decisional Support System (DSS) and the SIC² (Simulation and Integration of Control for Canals) hydraulic software. Firstly, the methodology was developed and tested on a Pilot Segment (PS), characterized by a simple geometry and a quite significant historical hydraulic data availability. Then, it was applied on an Extended Segment (ES) of a more complex geometry and hydraulic functioning. Moreover, the available hydraulic data are scarce. The combination of these aspects represents a crucial issue in the irrigation networks in general.

1. Introduction

Counting on the intensive exploitation of the water resources, many works of the last decades have addressed agricultural water management practices towards the productivity strengthening and the defeating poverty [1,2,3]. Nowadays, the water scarcity, combined with the rising food demand, has involved a gradual switch of the objectives [1,3] to the following: Resource preservation (quantitatively, qualitatively, and ecologically) in relation to agricultural production (crop irrigation, animal rearing, and on-farm operations) [4,5,6], rural realities economy improvement [7,8], and facing climate change [9].

The sustainable development resulted from these key components is promoted by the Water Framework Directive (WFD/2000/60CE) [10] and policies that are closely related to the EU2020 program [11,12,13]. At the regional scale, the water management practices for irrigation are identified as a primary challenge because of their socio-economic implications [13]. They consist in the improvement of the irrigation consumption knowledge at the field scale and the increase in the efficiency and the discharge regulation at conveyance system scale [14].

Despite the evolution of irrigation infrastructures tends to be focused mainly on pressurized systems, many districts are often fed by dense canal networks that have remained basically unchanged since they were constructed decades ago. They are characterized by significant water losses and irrecoverable outflow at their end [15,16,17,18]. The irrigation systems performances can be improved through hardware (physical/structural) changes, such as the canal lining or the installation of sophisticated control structures [19,20], or through software (operational) techniques, such as appropriate delivery rules and an effective communication between water supply agencies and water users [21].

A common flaw in irrigation delivery systems that are characterized by open canals and by many users is the absence of a proper information system that ensures and collects measured and monitored data about hydraulic status [22,23,24,25]. When considering that the total water consumption for irrigation is projected to increase by 10% by 2050 [26], it will represent a central issue in the near future [27]. Generally, the only known quantities are measured water levels at specific locations, often with limited precision and possible failures [19].

Hydraulic modelling emerges as a strategic tool for: 1) the reconstruction of unmeasured data, such as discharges or water levels at other locations, unknown perturbations (inflows and outflows) [28,29], and hydraulic variables (friction coefficients and hydraulic device discharge coefficients) [19,30]. 2) the visualization and control of the flow at several structures [15,31].

In parallel, irrigation Decisional Support Systems (DSS) can characterize the crops that are served by a specific irrigation delivery system, and also, can indirectly monitor their hydraulic status. In the last few decades, DSS underwent many changes [32,33] ranging from the prevention of extreme events (droughts and floods) and pollution [32] to the irrigation scheduling [34,35,36,37,38,39]. The latter is based on the integration of several models, processes, and factors (i.e., meteorological and soil conditions and types of crops) [40,41].

This study presents a tool for the reconstruction of unmeasured discharges along a specific irrigation delivery canal. The combination of hydraulic modelling and irrigation DSS can solve the problem that was created by the poor hydraulic data collection. The multi-disciplinary approach that is proposed in this paper reflects the merging of hydraulic engineering and agronomy aspects. It was developed on one of the most important irrigation canals in Northern Italy: The Canale Emiliano Romagnolo (CER) [42]. The methodology was developed on a simple geometry 7 km long Pilot Segment (PS) and over a more complex 22 km long Extended Segment (ES).

2. Materials and Methods

2.1. Description of the CER

The CER starts in Salvatonica di Bondeno (Ferrara, Italy) on the right bank of the Po River and it provides the irrigation supply for an area of about 3000 km². That area represents the 93% of the irrigated and the 22% of the agricultural land in the Emilia Romagna Region. The agricultural land covers the 60% of the regional territory [43], where different cultures are irrigated, among which extensive crops, vegetables, and orchards [44]. To convey and to distribute water, the CER hydraulic system uses seven pumping stations (the main one on the Po River) and 165 km of canal networks (Figure 1).

The main reach is 133 km long and its first 104 km are characterized by 60–17.6 m width at the top and 6.0–6.4 m at the bottom of the canal. The side slopes are 3:1 and 1.5:1 or 1.75:1 for composite trapezium sections (first 37 km) and 2:1 for the simple ones. The cross section of the canal later becomes narrower with a rectangular shape: open (width range: 6.8–5.6 m, elevation range: 3–2.7 m) or closed (width range: 6.4–5.6 m, elevation range: 2.1–1.9 m) and made of reinforced concrete. The CER receives no inflow from surface runoff, drainage, or different types of discharges, but it has several offtakes. From the canal, the water is offtake using pumps or gates, and it is conveyed to the irrigated fields through secondary channels that are managed by Associated Consortia. The irrigation offtakes have a seasonal variability, and therefore the maximum permitted discharge at the main pumping station varies from 68 m³/s (from May to September) to 25 m³/s (the rest of the year). Moreover, discharges are also affected by the meteorological issues (e.g., long dry seasons), the type of cultivated crops, and the irrigation practices. The Consortium of the CER is in charge of: (1) maintenance operations (geometric and functioning repairs, periodic cleanings); (2) collection of quantitative and qualitative measurements; and, (3) supply of irrigation services to farmers (by means of several irrigation Associated Consortia that distributes water to final users).

2.2. Investigation Period and Available Data

This study focuses on the period of full operation of the CER i.e., the irrigation season (June–August) that is characterized by the highest water demand and irrigation frequency. The irrigation period selected comprises 73 days (20 June–31 August) of the years from 2012 to 2015. These years were characterized by different average daily rainfall. For example, 2013 (1.30 mm/day) and 2015 (0.94 mm/day) had daily rainfall that was close to the decennial (2005–2015) average value (1.1 mm/day), while 2014 (2.22 mm/day) and 2012 (0.13 mm/day) were especially rainy and dry, respectively.

The main available data for this study are: (1) water volumes at offtakes (calculated indirectly); (2) crop water requirements (estimated); and, (3) water levels at the main canal (measured); (4) functioning data of pumping stations along the CER (measured).

In particular, for each irrigation offtake, calculated and estimated water amounts were provided. The former refers to monthly cumulated volumes indirectly calculated by the Associated Consortia on the basis of flow rates and working times of offtakes pumps or the opening gate area, the opening time, and the water level at offtakes manual gates.

On the other hand, estimated water volumes were based on the crop water requirements provided by the IRRINET management DSS, which was developed by the Consortium of the CER [45]. IRRINET is identified as the reference tool for the estimation of irrigation volumes in the Emilia Romagna Region [46], and it provides to farmers a day-by-day information on how much and when to irrigate crops [47]. It is based on a daily water balance of soil-plant-atmosphere system. IRRINET processes a huge quantity of information related to: areas (meteorological, water table depth and soil data) and farms (types of irrigated crops, start and stop crops dates). Since 2012, at the end of every irrigation period, the Consortium of the CER has collected daily optimum crop water requirement (CWR) values for all the crops that are served by IRRINET. For every type of crop (i) and for every day, these values are averaged; afterwards, they are cumulated on a decadal time scale giving CWR_i (Section 2.4.1).

Along the CER, the only hydraulic measurements available are water levels. In total, forty cross-sections are equipped with ultrasonic level transmitters (The Probe PL-517, Terry Ferraris &C. S.p.A., Milan, Italy). These instruments are generally located near two types of infrastructures: (a) culverts (passing under different rivers; in total, 29 instruments); and (b) pumping stations (in suctions and/or delivery tanks; in total 11 instruments). After direct field surveys, the measurement accuracy of both types of transmitters was estimated to be lower than the original instrument accuracy (±0.02 m), in particular, ±0.05 m and ±0.10 m, respectively. The transmitters located near culverts serve for management purposes, and their accuracy was probably affected by flow disturbances (sediment build up and depressions next to the edges of culverts entrances due to velocity changes) [48]. On the other hand, the transmitters near pumping stations are used for operational purposes and they are strongly influenced by the pumps functioning.

At each of the 40 cross-sections, the water level value is transmitted and is registered with a time step of 30 min. Because of the offtake data time scale (monthly or decadal) and because of the general water level series incompleteness, the 30 min available measures were averaged on a daily time scale.

Finally, the daily measured functioning data at one pumping station (Pieve di Cento) were investigated (Section 2.6). Every time that the installed pumps would turn on or turn off the following parameters were measured: voltage (V), electric current (A), functioning time (h), discharge (m³/s), volume (m³), suction, and delivery tanks water level (m).

Table 1. The available data and their characteristics.

Available Data	Type	Unit	Time Step	Source
Offtake Volumes	Indirectly calculated	m3	Monthly (cumulated values)	Associated Consortia
CWRi	Estimated	mm	Decadal (cumulated values)	IRRINET
Water Levels	Measured	m	Daily (average values)	CER
Water Levels at Suction/Delivery Tanks	Measured	m	Pumps on/off (single values)	CER

provides a summary of all the available data used in the present application.

2.3. Description of the Pilot Segment (PS)

The multi-disciplinary modelling approach was developed on a 7 km long Pilot Segment (PS) of the CER.

The PS extremities coincide with two concrete culverts called Culv_1 (upstream) and Culv_2 (downstream) (Figure 2). They are characterized by rectangular flow sections of 36 m² and of 31.5 m², respectively, and by submerged entrances and surface or/and piped-flow conditions. PS has three different trapezium cross sections with width ranges of 22.8–25.8 m (at the top) and 3.3–7 m (at the bottom). The side slopes are 3:1 and 1.5:1 for the first composite cross section and 2:1 for the other two simple sections. For the first 700 m along the segment, the bed altimetry goes from 12.81 m to 13.74 m above the sea level. After that part, the canal has a constant slope with a final value of 13.32 m above sea level.

Six offtakes of PS serve a large irrigated area (8385 ha) through a network of not-pressurized irrigation channels. Over the four years of analysis, the biggest amount of water diverted from the segment (70% of the total offtake), was always diverted by the same three offtakes out of the six mentioned. The water gauges present at the segment are four: two at Culv_1 and two at Culv_2. They are located a few meters away from the entrance and the exit of both culverts (Figure 2).

2.4. Elaboration of the Multi-Disciplinary Modelling Approach on PS

The methodology was developed on a 7 km long Pilot Segment (PS), which was characterized by a simple geometry and a quite significant availability of water level measurements. The offtake discharges were estimated and verified also while considering the daily optimum CWR at field scale [46], that was estimated by the IRRINET, a regional irrigation DSS [45]. Combining hydraulic modelling of the CER with the optimization process of the hydraulic variables (Manning’s coefficient and gate discharge coefficient) allowed for determining the flowing discharges. The simulations were run under steady flow conditions using the hydraulic software SIC² (5.38c, UMR G-eau IRSTEA, Montpellier, France) [49].

The methodology developed on PS was later applied on a 22 km Extended Segment (ES), that apart from a more complex geometry and hydraulic functioning (especially because of the presence of four culverts), also has a lower hydraulic data availability and lower accuracy when compared to PS. The methodology was tested on this particular segment, since it was characterized by different issues that are common in irrigation networks [19].

2.4.1. Reconstruction of the Unmeasured Offtake Discharges

The offtakes that were not measured were reconstructed using the indirectly calculated and the estimated data provided by the Associated Consortia and by the IRRINET service, respectively. In the following description, in order to distinguish these two data sources, different indexes are used: D for the former (Associated Consortia) and T for the latter (IRRINET). The index C indicates the results that were obtained by calculations done by the authors with the available data. The T-data aim to refine the time scale of the D-data and to verify them by a comparison with agronomic values, such as crop water requirements. Therefore, the obtained C-results (Equations (1)–(3)) have a decadal time scale instead of a monthly one; moreover, their values include agronomic aspects (e.g., optimum crop water requirement), the intensity, and the efficiency of the irrigation practices (Equation (4)).

During the decade n, the discharge exiting from a generic offtake k, q_kCn (m³/s) can be written as:

$$q_{kCn}=q_{rDm} w_{kCn}$$

(1)

where q_rDm (m³/s) is the average discharge diverted from the reference offtake during the month m (m = 1, 2, 3), and w_kCn is the weight of the offtake k during the decade n (n = 1, ..., 7).

The reference offtake was identified every year as the one diverting the greatest irrigation water volume. q_rDm was calculated as:

$$q_{rDm}=\frac{V_{rDm}}{D_m}$$

(2)

where V_rDm (m³) is the indirectly calculated cumulated volume of the reference offtake for the month m, while D_m (s) is the duration of the month m.

The weight was obtained comparing the offtake k and the reference offtake in volumetric terms. The approach considered w_kCn, as follow:

$$w_{kCn}=\frac{\left(w_{kDm}+w_{kTn}\right)}{2}; w_{kDm}=\frac{V_{kDm}}{V_{rDm}}; w_{kTn}=\frac{V_{kTn}}{V_{rTn}}$$

(3)

where w_kDm (-) and w_kTn (-) are the weights of the offtake k obtained using the D-data and the T-data, respectively, V_kDm (m³) is the indirectly calculated volume of the offtake k during the month m, V_kTn (m³), and V_rTn (m³) are the volumes of the offtake k and of the reference offtake, respectively, calculated during the decade n using IRRINET.

In particular, for the decade n, the calculated volume of the generic offtake k (V_kTn) was determined by the expression [14]:

$$V_{kTn}=\left[{\displaystyle \sum }_{i=1}^n\left(\frac{CW{R}_i A_i I{I}_i}{E{I}_i}\right)\right] \frac{1}{ED}$$

(4)

where A_i (m²) is the area covered by the crop i per each year, CWR_i (mm) is the decadal cumulated optimum water requirement for the crop i, II_i (-) is the irrigation intensity of the crop i, EI_i (-) is the efficiency of the irrigation method for the crop i, and ED (-) is the efficiency of the delivery system.

If the generic offtake k is the reference offtake, the Equation (4) gives the quantity V_rTn.

The CWR values were provided by the Consortium of the CER, as already said in Section 2.2 for extensive cultivations (maize, soy, and alfa-alfa), for vegetables (beet, onion, melon, potato, and tomato), and for orchards (pear-tree, peach-tree, and vine).

The coefficient II indicates the intensity of irrigation, in other words, the ratio between the irrigated area and the area that potentially could be irrigated [50,51,52]. Its values were determined through field studies at the regional scale [53,54,55,56]. In particular, for the involved case-study crops, II ranges from 0.25 to 1, as shown in

Table 2. The values of the coefficients intensity of irrigation (II) and efficiency of the irrigation method (EI) for the irrigated crops served by PS and extended segment (ES).

Irrigated Crops	IIi (-)	EIi (-)
Extensive crops
Maize	0.75	0.75
Soy	0.50	0.75
Alfa-Alfa	0.25	0.75
Vegetables
Beet	0.60	0.75
Onion	1.00	0.75
Melon	1.00	0.85
Potato	1.00	0.75
Tomato	1.00	0.85
Orchards
Pear	1.00	0.85
Peach	1.00	0.85
Vine	0.50	0.85

.

The coefficient EI indicates the efficiency of the irrigation method [57]. In Emilia Romagna, the considered value ranges are: 0.85–0.90 for drip irrigation and 0.70–0.80 for sprinkling irrigation [58]. In Table 2, the values of 0.85 and 0.75 were adopted for crops that were under the former and the latter irrigation efficiency, respectively.

The coefficient ED indicates the efficiency of the system that conveys water from the offtakes on the banks of the CER to the fields. For the present case-study, it was considered to be 0.50 [59,60]. In the area, in fact, 1122 km of channels (for both irrigation and drainage) and only 235 km of pipes provide water for crops. In particular, non-lined channels realize the 88% of the irrigation distribution [61].

2.4.2. Reconstruction of the Unmeasured Flowing Discharges

The hydraulic modelling combined with hydraulic variables optimization processes allowed for reconstructing the unmeasured flowing discharges along the segment.

SIC² (Simulation and Integration of Control for Canals) was selected as the most appropriate irrigation canal modelling software. It has been developed at IRSTEA (previously CEMAGREF, Montpellier, France) [62] and it enables describing the dynamics of rivers, drainage networks, and irrigation canals [63]. For the latter, devices (i.e., sills and gates) and irrigation offtakes can be specified in geometric and functioning terms [49]. SIC² can run steady flow computations under boundary conditions for discharge and/or water level [64]. In fact, it can consider several combinations of settings for devices and offtakes. The software provides the water level and the discharge profiles along the analyzed hydraulic system [29]. SIC² models also unsteady flow for initial conditions that were obtained from steady state computations [64] in discharge and water level terms. It can be used for water demand and control operations [19,65]. SIC² describes the dynamic behavior of water (discharge and water level) with the complete one-dimensional (1-D) Saint Venant equations in a bounded system [49]. This is the case of the CER in which the flow can be considered as mono-dimensional with a direction sufficiently rectilinear.

The 1-D Saint Venant equations are mathematically expressed as [66]:

$$\frac{\partial Q}{\partial x}+\frac{\partial S}{\partial t}=0$$

(5)

$$\frac{\partial Q}{\partial t}+\frac{\partial (\raisebox{1ex}{$Q^2$}\!\left/ \!\raisebox{-1ex}{$S)$}\right.}{\partial x}+g S \frac{\partial Z}{\partial x}+g S J=0$$

(6)

where Q (m³/s) is the discharge, S (m²) is the wetted area, g (m/s²) is the acceleration due to gravity, Z (m) is the water level, J (m/m) is the friction slope, x (m) is the longitudinal abscissa, and t (s) is the time.

The friction slope is obtained by the Manning-Strickler formula:

$$J=\frac{n^2 Q^2}{S^2 R^{4/3}}$$

(7)

where n (m^1/3/s) is the Manning’s coefficient and R (m) is the hydraulic radius.

The continuity (Equation (5)) and the momentum (Equation (6)) equations are completed by boundary conditions for which SIC² provides a large range of options. They can be imposed in discharge, elevation, or rating curve terms. Lateral inflows and weir and gate equations can also be inserted. For example, the flow through a gate structure can be expressed by several classical or advanced equations, such as the submerged flow equation:

$$Q=C_d \sqrt{2 g} L u \sqrt{Z_{up}-Z_{dn}}$$

(8)

where C_d (-) is the gate discharge coefficient, L (m) is the gate width, u (m) is the gate opening, Z_up (m), and Z_dn (m) are the water levels at the upstream and at the downstream of the gate, respectively.

The Saint Venant equations are non-linear partial differential equations and an analytical solution is restricted to problems of simple geometry. For all other cases, implicit finite difference approximations and a Preissmann scheme are used, as in the case of SIC² [66,67,68].

After the PS geometry entry, several hydraulic aspects were evaluated in SIC². The hydraulic variables values were set according to the literature: The Manning’s coefficient presented a constant value of 0.013 (m^1/3/s) along the segment and within the two culverts [68] and the gate discharge coefficient that characterizes the entrances of each culvert was 0.6 [16,49,69]. The offtakes were modelled as “nodes” and they were characterized in discharge terms. In particular, the q_kCn values were inserted and were linearly interpolated in time.

For the year y, the vectors Z1_obs,y, Z2_obs,y, Z3_obs,y, and Z4_obs,y can be defined. They contain the daily measured water levels at the four gauges: WL IN_1, WL OUT_1, WL IN_2, and WL OUT_2, respectively (Figure 2).

$$Z{1}_{obs,y}=\left(\begin{array}{c}Z{1}_{ob{s}_1}\\ Z{1}_{ob{s}_2}\\ Z{1}_{ob{s}_j}\\ ..\\ Z{1}_{ob{s}_e}\end{array}\right);Z{2}_{obs,y}=\left(\begin{array}{c}Z{2}_{ob{s}_1}\\ Z{2}_{ob{s}_2}\\ Z{2}_{ob{s}_j}\\ ..\\ Z{2}_{ob{s}_e}\end{array}\right);Z{3}_{obs,y}=\left(\begin{array}{c}Z{3}_{ob{s}_1}\\ Z{3}_{ob{s}_2}\\ Z{3}_{ob{s}_j}\\ ..\\ Z{3}_{ob{s}_e}\end{array}\right);Z{4}_{obs,y}=\left(\begin{array}{c}Z{4}_{ob{s}_1}\\ Z{4}_{ob{s}_2}\\ Z{4}_{ob{s}_j}\\ ..\\ Z{4}_{ob{s}_e}\end{array}\right)$$

(9)

where j is the index for the examined day of the year y (j = 1, ..., e).

The software SIC² can compute the values of discharge and water level along PS under two boundary conditions only in water level terms; for PS they were represented by Z1_obs,y, and Z4_obs,y. The daily simulated water level values at WL OUT_1, and WL IN_2 (Z2_sim,y and Z3_sim,y) were compared to those that were measured (Z2_obs,y and Z3_obs,y) in order to demonstrate the reliability and accuracy of the hydraulic model, and therefore, of the computed discharge values. The vectors Z2_sim,y and Z3_sim,y can be defined as:

$$Z{2}_{sim,y}=\left(\begin{array}{c}Z{2}_{si{m}_1}\\ Z{2}_{si{m}_2}\\ Z{2}_{si{m}_j}\\ ..\\ Z{2}_{si{m}_e}\end{array}\right); Z{3}_{sim,y}=\left(\begin{array}{c}Z{3}_{si{m}_1}\\ Z{3}_{si{m}_2}\\ Z{3}_{si{m}_j}\\ ..\\ Z{3}_{si{m}_e}\end{array}\right)$$

(10)

where j is the index for the examined day of the year y (j = 1, ..., e).

The simulations can be run under steady or unsteady state. The use of the former can be justified by the slow dynamics in the CER and the time and CPU (Central Processing Unit) memory saving. In particular, SIC² allows implementing a series of steady state simulations. The year 2015 was examined as a first test. The hydraulic model was run under a series of one-day steady state simulations and under one-day and 10-min unsteady state simulations.

A refined hydraulic model can be obtained after an optimization process. It allows for minimizing the differences in water level terms at WL OUT_1 and WL IN_2 playing on the values of the hydraulic variables and of a scaling factor for the offtakes; they were set as parameterized variables.

The optimization process consisted in a set of parameters to be evaluated, a criterion to be minimized, and a minimization function; it was based on the dialogue between SIC² and Matlab^® (version 9.1, The MathWorks, Inc., Natick, MA, USA).

In SIC², the parameterized hydraulic variables were explicit Cd1 and Cd2, gate discharge coefficients of Culv_1 and Culv_2; n, n1 and n2, Manning’s coefficients along PS, within Culv_1 and Culv_2.

In Matlab^®, this hydraulic set was recalled and the scaling factor Cq allowed multiplying the offtake discharge values from Section 2.4.1. In the math code, the criterion and the minimization function were implemented.

The vectors diff2_y and diff3_y can be defined as:

$$diff{2}_y=Z{2}_{sim,y}-Z{2}_{obs,y} and diff{3}_y=Z{3}_{sim,y}-Z{3}_{obs,y}$$

(11)

Therefore, the criterion to be minimized J was expressed as:

$$J=\sqrt{{\displaystyle \sum }_{j=1}^e\left[\frac{{\left(diff{2}_y\right)}^2}{\sigma 2_y^2}+\frac{{(diff{3}_y)}^2}{\sigma 3_y^2}\right]}$$

(12)

where j is the index for the examined day of the year y (j = 1, ..., e), σ2_y, and σ3_y are the vectors containing the weights (values of 10 or 1), indicating whether a measure is affected by errors or not.

The iterative play on the parameterized hydraulic variables influenced the elements of the vectors diff2_y and diff3_y, and consequently, the criterion J.

The minimization function considered was based on the Nelder-Mead simplex direct search algorithm, already implemented in Matlab^® [70]. In Figure 3, the iterations on J are shown for the year 2015.

At the end of the process, the minimization function identified parameterized hydraulic variables values that represent real minimum for the criterion (Figure 4).

For every year, these values were used for running the hydraulic simulations in SIC². The obtained model was called “optimized” and it returned the simulated discharges and water levels along PS. Finally, the optimization process was characterized by the cost of J that indicated the criterion value at the end of the iterations.

Within the overall methodology, the measurement reliability represented a significant issue. The measures that are probably affected by errors (called “suspicious measures”) can be contained in WL IN_1 and WL OUT_2 (boundary conditions), as in WL OUT_1 and WL IN_2 (optimization conditions) data series. The former affected the hydraulic model, while the latter the optimization process.

The days that are affected by suspicious measures were weighted in the optimization process through the elements of σ2_y and σ3_y. In particular, if a day j is affected by a suspicious measure, the weight (σ2_j; σ3_j) was set as 10; otherwise, it was equal to 1.

A detection method was elaborated considering the vectors Z1_obs,y, Z2_obs,y, Z3_obs,y, Z4_obs,y, Q2_sim,y, and Q3_sim,y. The latter two contained simulated values of discharge (output of the optimized hydraulic model) at the Culv_1 and Culv_2, respectively.

They can be expressed as:

$$Q{2}_{sim,y}=\left(\begin{array}{c}Q{2}_{si{m}_1}\\ Q{2}_{si{m}_2}\\ Q{2}_{si{m}_j}\\ ..\\ Q{2}_{si{m}_e}\end{array}\right); Q{3}_{sim,y}=\left(\begin{array}{c}Q{3}_{si{m}_1}\\ Q{3}_{si{m}_2}\\ Q{3}_{si{m}_j}\\ ..\\ Q{3}_{si{m}_e}\end{array}\right)$$

(13)

where j is the index for the examined day of the year y (j = 1, ..., e).

The method was based on the vectors:

$$delt{a}_y=Z{2}_{obs,y}-Z{3}_{obs,y;}$$

$$delta{1}_y=Z{1}_{obs,y}-Z{2}_{obs,y};$$

(14)

$$delta{2}_y=Z{3}_{obs,y}-Z{4}_{obs,y};$$

For the day j, their elements represented the differences in water level terms along the segment and at the Culv_1 and Culv_2, respectively. The plots of delta_y-delta1_y, and delta_y-delta2_y were used to evaluate in which vector the suspicious measures were located. The outliers of the data linear fitting were investigated. If the element j of delta_y results as an outlier in both plots, a suspicious measure was in Z2_obsj or in Z3_obsj. If the element j of delta_y results as an outlier in the first plot but not in the second, the suspicious measure was in Z1_obsj. If the element j of delta_y is an outlier in the second plot but not in the first, the suspicious measure was in Z4_obsj. To evaluate if a suspicious measure is in Z2_obsj or Z3_obsj, Q2_sim,y-delta1_y, and Q3_sim,y-delta2_y were plotted. For both, a data quadratic fitting of data was considered. If the j-th element of delta1_y results as an outlier, the suspicious measure was in Z2_obsj while if the element j results as an outlier of delta2_y, the suspicious measure was in Z3_obsj.

The most significant results obtained are given in Section 3.1.

2.5. Description of the Extended Segment (ES)

The multi-disciplinary modelling approach was then applied over a 22 km Extended Segment (ES) of the CER (Figure 5). Its downstream corresponds to WL IN_1 and its upstream is located in a delivery tank, few meters away from the pumping station Pieve di Cento exit. The latter counts seven pumps with a maximum capacity of 50 m³/s and a maximum head of 4.5 m. For the first 33 m along the segment, the trapezium cross section top width is higher (85 m) and the bed altimetry varies from 10.79 m to 13.50 m above the sea level. Later, ES presents three different composite trapezium cross sections (top width from 26.4 m to 22.8 m; bottom width from 5.0 m to 3.3 m; side slope 3:1 and 1.5:1) and a constant slope (bed altimetry from 13.50 m to 12.81 m above the sea level). Four culverts under passing two roads (Road crossing_1 and Road crossing_2), the Navile Canal (Culv_3), and the Savena River (Culv_4) are characterized by a rectangular flow section of 36 m² (Figure 5). The road crossings present a modest length (about 20 m), while Culv_3 and Culv_4 are about 63 m and 86 m, respectively. The 12 occurring offtakes serve a total irrigated area of about 12,580 ha. The water gauges involved are only two at the ES extremities: WL OUT_0 (at the upstream) and WL IN_1 (at the downstream).

2.6. Application of the Multi-disciplinary Modelling Approach on ES

ES was characterized by a high complexity in geometric and functioning terms (Section 2.5). Moreover, the hydraulic data availability was poor; in fact, only two locations were equipped with water gauges. The multi-disciplinary modelling approach was applied over this segment in order to test its validity in more difficult conditions, representing a typical configuration in irrigation networks and with a significant lack of hydraulic measurements [19].

The offtake discharges decadal values were estimated as in Section 2.4.1. Due to the lack of available data, the PS flowing discharge resulted at WL OUT_1, was used to calculate the flowing discharges over ES, to run the optimized hydraulic model, and to compare the simulated and measured water level values at WL OUT_0. It was considered to be reliable due to the values of the parameterized hydraulic variables, of the linear interpolation parameters, and of the RMSE (Section 3.1.3). In particular, the PS flowing discharge values were used to define the ES upstream boundary conditions.

For the year y, the vector containing the calculated discharge values of a generic offtake k (Section 2.4.1) can be expressed as:

$$q_{kC,y}=\left(\begin{array}{c}{q}_{kC{n}_1}\\ q_{kC{n}_2}\\ q_{kC{n}_j}\\ ..\\ q_{kC{n}_e}\end{array}\right)$$

(15)

where j is the index for the examined day of the year y (j = 1, ..., e).

Defining as q_totC,y, the total offtake discharges vector:

$$q_{totC,y}=\left(\begin{array}{c}{q}_{tot{C}_1}\\ q_{tot{C}_2}\\ q_{tot{C}_j}\\ ..\\ q_{tot{C}_e}\end{array}\right)$$

(16)

Its element q_totCj was calculated as:

$$q_{tot{C}_j}={\displaystyle \sum }_{k=1}^{12}{q}_{kC{n}_j}$$

(17)

where k is the index of the generic offtake (k = 1, …, 12).

For the year y, the vector Q0_y represented the ES upstream boundary conditions. It was obtained as:

$$Q{0}_y=Q{2}_{sim,y}+q_{totC,y}$$

(18)

whereas, the Z1_obs,y values reported were used as the downstream boundary conditions. The hydraulic model was implemented under a series of one-day steady state simulations. For every year, the vector Z0_obs,y contains the daily measured water levels values at WL OUT_0. They were used for testing the model performances and for evaluating the optimization process. Z0_obs,y can be defined as:

$$Z{0}_{obs,y}=\left(\begin{array}{c}Z{0}_{ob{s}_1}\\ Z{0}_{ob{s}_2}\\ Z{0}_{ob{s}_j}\\ ..\\ Z{0}_{ob{s}_e}\end{array}\right)$$

(19)

where j is the index for the examined day of the year y (j = 1, ..., e).

The optimized parameterized hydraulic variables set was larger than that of PS. It consisted in Cd3, Cd4, Cd5, and Cd6, gate discharge coefficients of Culv_3 and Culv_4 and of the two road crossings; n, n3, n4, n5, and n6, Manning’s coefficients along ES, within the two culverts and the two road crossings. The significant uncertainty that affects the measured water levels at WL OUT_0 (Section 2.2) was reflected in the larger parameterized hydraulic variables set size. The high degree of freedom allowed for obtaining physically possible values of the parameters and the lower cost of J at the optimization process end. The gate discharge coefficients values could not be imposed as those of PS because the geometric and functioning characterization difference. Moreover, if the Manning’s coefficients are imposed, the optimization process gives higher gate discharge coefficients values (>1) that are not physically correct. The offtake discharges scaling factor was not considered, as explained in Section 3.1.3.

For the year y, the vector Z0_sim,y contained the daily simulated water levels at WL OUT_0:

$$Z{0}_{sim,y}=\left(\begin{array}{c}Z{0}_{si{m}_1}\\ Z{0}_{si{m}_2}\\ Z{0}_{si{m}_j}\\ ..\\ Z{0}_{si{m}_e}\end{array}\right)$$

(20)

where j is the index for the examined day of the year y (j = 1, ..., e).

The optimization criterion was based on the definition of the vectors diff0_y and σ0_y. The former contained the values of the daily differences between simulated and measured water levels at WL OUT_0, as:

$$diff{0}_y=Z{0}_{obs,y}-Z{0}_{sim,y}$$

(21)

The vector σ0_y weighted the measures probably affected by errors (“suspicious”) located in Z0_obs,y. The detection involved the Pieve di Cento pumps functioning data. In particular, for the year y, the vectors Z0_pmax,y and Z0_pmin,y contained the daily maximum and minimum values of the delivery tank water level that is registered by the pumps functioning, as:

$$Z{0}_{pmax,y}=\left(\begin{array}{c}Z{0}_{pma{x}_1}\\ Z{0}_{pma{x}_2}\\ Z{0}_{pma{x}_j}\\ ..\\ Z{0}_{pma{x}_e}\end{array}\right); Z{0}_{pmin,y}=\left(\begin{array}{c}Z{0}_{pmi{n}_1}\\ Z{0}_{pmi{n}_2}\\ Z{0}_{pmi{n}_j}\\ ..\\ Z{0}_{pmi{n}_e}\end{array}\right)$$

(22)

where j is the index for the examined day of the year y (j = 1, ..., e).

For every day j, the functioning range Z0_pmaxj-Z0_pminj was identified. If Z0_obsj do not belong to it, it is defined as a suspicious measure.

The expression of the criterion J was:

$$J=\sqrt{{\displaystyle \sum }_{j=1}^e\frac{{\left(diff{0}_y\right)}^2}{\sigma 0_y^2}}$$

(23)

The minimization function is that of PS (Nelder-Mead simplex direct search algorithm).

The most significant results obtained are given in Section 3.2.

3. Results and Discussion

3.1. Pilot Segment (PS)

3.1.1. Unmeasured Offtake Discharges

For every year, the values of w_kDm, w_kTn, and w_kCn were calculated, as in Section 2.4.1. Out of these weights, the first one resulted generally higher than the second one; the V_rDm-V_kDm, in fact, differed considerably from V_rTn-V_kTn. When considering the year 2015 as an example, the maximum values were 23.04 × 10⁴ m³ and 13.68 × 10⁴ m³, respectively. For the same year, Figure 6a underlines the monthly variability of w_kDm as compared to the decadal one of w_kTn for two offtakes: Offtake₁ (reference offtake) and Offtake₅ (Figure 2). The weights w_kCn were obtained averaging D-data and T-data according to Equation (3), and they were reported in Figure 6b. The averaging of those values was needed to minimize the possible measurement errors in D-data, and also, to take into account that CWR from IRRINET are “optimal requirements”, when considering that water was always fully available.

Over the four years of analysis, the trend of the offtake discharge values (q_kCn) was mainly coherent with the yearly meteo-climatic conditions (i.e., average daily rainfall). The reference offtake discharge values ranged from 0 m³/s to 0.24 m³/s. q_kCn of all other offtakes varied from 0 m³/s to 0.17 m³/s.

Figure 7a and Figure 7b show that the two offtakes (Offtake₁ and Offtake₅) had the lowest values in 2014 (mean values of 0.021 m³/s and 0.032 m³/s, respectively) and the highest mainly in 2012 (mean values of 0.137 m³/s and 0.082 m³/s, respectively). If the month of July is considered, the discharge values of the reference offtake were lower in 2012 than in 2013 and 2015. This can be explained not only by meteo-climatic conditions (that resulted in crop stress), but also by insufficient machine, manpower, or energy availability at the field. Moreover, among the years that were analysed, the cultivated crops differ.

3.1.2. Steady State Flow Condition

To evaluate if the hydraulic models should be run under steady or unsteady state conditions, the results of the year 2015 were analysed. They consisted in discharge and water level values at WL OUT_1 and WL IN_2. The hydraulic model of PS was run under a series of one-day steady state (Steady-1d) simulations, and under one-day (Unsteady-1d), and 10-min (Unsteady-10mn) unsteady state simulations.

The vectors Z2sim-2015 and Z3sim-2015 for Steady-1d and Unsteady-1d were completely overlaid. The differences obtained by comparing these vectors for Steady-1d and Unsteady-10mn reported mean values of 0.285 m and 5.347 × 10⁻⁴ m, respectively.

For Steady-1d and Unsteady-1d the vectors Q2sim-2015 and Q3sim-2015, so as the simulated water levels, were completely overlaid. If the simulations of steady state and those of unsteady state with time step 10 min are compared, the resulted maximum and mean differences were 3.843 m³/s and 0.279 m³/s, respectively.

For the optimization of the hydraulic model, the results in water level and discharge terms can be considered to be approximatively identical for the three flow conditions that are considered.

The series of one-day steady simulations was adopted for running the hydraulic models of both PS and ES. This assumption was justified by the slow dynamics occurring in the CER, and it is also coherent with the time scale of calculated offtake discharges (decadal) and of measured water level (daily) data. The use of steady state saves time and CPU memory that is an important point, since this hydraulic calculation is embodied into an optimization loop. Using only one run, SIC² computes 73 steady state simulations; one for every day of the irrigation period. The hydraulic variables on a daily basis are not function of time.

3.1.3. PS optimized Model

The optimized hydraulic model returned the flowing discharges along the PS. For example, in Figure 8, the Q2_sim,y values are reported (values at the upstream of PS). For every year, they are grouped into two vectors: Q2_simc (output from measured water levels not affected by errors) and Q2_sims (output from measured water levels probably affected by errors, Section 2.4.2).

The lowest values of flowing discharge were calculated for the rainier year (2014) and they were 17.560 m³/s (Q2_sim,2014) and 16.660 m³/s (Q3_sim,2014), with standard deviations of 2.694 m³/s and 3.204 m³/s, respectively. When considering Q2_sim,y as an example, the years 2013 and 2015 were characterized by higher flowing discharge mean values (23.930 m³/s and 21.710 m³/s) and standard deviation values (4.230 m³/s and 4.841 m³/s) as compared to those of the year 2012 (20.700 m³/s and 2.538 m³/s, respectively). This can be justified by the limiting factors that are mentioned in Section 3.1.1. Therefore, the years with extreme climatic conditions (2014 and 2012) presented less variability in relation to flowing discharge mean values when compared to the years 2013 and 2015 characterized by the alternation of dry and rainy intervals.

The values of the flowing discharge were the result of many factors: offtake discharges (that followed characterization, as explained in Section 3.1.1), the modelled functioning of culverts and the measured water levels.

For the year y, the optimized hydraulic model performances were evaluated through the values of the parameterized variables and of the differences between water levels simulated and measured at WL OUT_1 and WL IN_2. At the end of the optimization process, the values of the hydraulic variables should be physically correct and coherent with literature [68].

For Cq, the yearly values that were obtained resulted close to 1. If the optimization process were cut around these values, they would not represent real minimum. The offtake discharges impact on the water levels at WL OUT_1 and WL IN_2 around their nominal values was less than the measurement accuracy considered (±0.05 m); the offtake discharges represented small rates if compared to flowing discharges. When considering the year 2015 as an example, the flowing discharge maximum and minimum values were 29.66 m³/s and 12.88 m³/s, respectively, while the reference offtake discharge ranged from 0.24 m³/s (0.81% of the flowing discharge maximum) to 0.07 m³/s (0.24% of the flowing discharge minimum). Cq cannot be considered as one of the parameters for the optimization process since it did not have any influence on it.

The parameterized hydraulic variables and the cost of the criterion are reported in

Table 3. The values of the five parameterized variables and the cost of the criterion obtained from the optimization process: Without (above) and with the weights of suspicious measures (below).

Year	Parameterized Hydraulic Variables					Cost of the Criterion
	Cd1 (-)	Cd2 (-)	n (m1/3/s)	n1 (m1/3/s)	n2 (m1/3/s)	J Cost (m)
Without Suspicious Measures Weights
2012	0.37	0.64	0.014	0.015	0.015	0.1742
2013	0.68	0.76	0.015	0.009	0.011	0.2799
2014	0.39	0.82	0.016	0.015	0.008	0.2331
2015	0.49	0.69	0.015	0.012	0.013	0.1036
With Suspicious Measures Weights
2012	0.37	0.65	0.014	0.014	0.015	0.1460
2013	0.71	0.74	0.016	0.009	0.011	0.1480
2014	0.44	0.80	0.015	0.013	0.010	0.1101
2015	0.50	0.71	0.014	0.012	0.012	0.0808

for every year of analysis. The results that were obtained with the suspicious measures weights are discussed in the following paragraphs.

The gate discharge coefficients (Cd1 and Cd2) refer to submerged flow for both culverts. The Cd2 values were coherent with the range 0.60–0.85 that was reported in literature [69,71,72,73]. For all years, surface flow occurred within Culv_2. For some years, the Cd1 values significantly differed from the literature range, and it can be explained by applying Equation (8) to the two gates at Culv_1 and Culv_2. For example, when Equation (8) was applied on the year 2012 (Cd1 = 0.37, Cd2 = 0.64), for Culv_1, the term (Z_up-Z_dn) reported maximum and minimum values of 0.13 m and 0.03 m, respectively. They were higher than those at Culv_2 that were 0.07 m and 0.01 m, respectively. Due to the modest impact of the offtakes, the values of the discharges at the two culverts were similar, and therefore the gate discharge coefficient at Culv_1 has to be lower than the one at Culv_2. Within Culv_1, both flow types (surface and piped) occurred. The years 2012 and 2015 were characterized by 39 days of surface and 34 days of piped flows. The years 2013 and 2014, on the other hand, presented mainly surface flow (57 and 59 days, respectively).

The n values that were obtained were coherent with the reported literature range for concrete canals (0.010–0.020 m^1/3/s) [68]. Over the last four years, the mean value was 0.0147 m^1/3/s and the maximum difference attested was 0.002 m^1/3/s (2012–2013). The n1 and n2 values were coherent with the literature range for concrete culverts (0.010–0.014 m^1/3/s) [74] and both had a mean of 0.012 m^1/3/s. When considering the analysis period, the maximum difference among the years was 0.005 m^1/3/s (between 2012 and 2013 for n1, and between 2012 and 2014 for n2). The Manning’s coefficient is the result of many factors: Basic value (roughness of the material that was used to line the canal), irregularities of the canal bed, cross sections variations, obstacles, vegetation growth, and meandering [75,76]. Along the PS, the Manning’s coefficient was stable; its variations can be related to the presence of obstacles (debris, downed plants, and dropped obstacles) and algae growth. For the culverts, it showed more variability and it was the result of many possible factors, such as the grids at the culverts entrances, which involve head losses, the gates modelling approximations, and the additional head losses due to the change of geometry between open and closed flow cross sections.

For every year, the performances of the optimized hydraulic model were evaluated through the elements contained in Z2_sim,y and Z3_sim,y (Figure 9).

The differences between simulated and measured water levels at WL OUT_1 and at WL IN_2 affected the cost of the criterion and the linear interpolation parameters. The former was also influenced by the σ2_y and σ3_y vectors, as shown in Table 3. The maximum difference for the cost of the criterion was 0.1319 m (0.2799–0.1480 m) for 2013.

For every year, in order to compare simulated and observed water level values, the former were plotted in the X-axis, while the latter in the Y-axis [77,78,79,80,81]. In this plot format, the points on the Y = X line represent the perfect correspondence between model-predicted and measured values; therefore, the intercept and the slope are 0 and 1, respectively [82]. Points below or above that line indicate over or under-estimations of the model [77]. In Figure 10, the elements of the vector Z2_sim,y were plotted versus those of the vector Z2_obs,y. The former were reported for optimized (Opt) and non-optimized (Non-Opt) models.

The validity of the optimized model was verified because of the line interpolation parameters values were closer to the optimum ones, especially in line intercept terms (i.e., 0.029 instead of 0.429 for Z2sim-2012). Over the four years, the mean values of intercept and slope line were 0.031 and 0.998, respectively. The same evaluation method was applied to Z3_sim,y and Z3_obs,y (Figure 11). Also, in this case, the optimized model shows an excellent fit, reporting mean values of intercept and slope line of 0.105 and 0.994, respectively.

The performances of the optimization process have been also evaluated in terms of the root mean square error (RMSE) (

Table 4. The root mean square error (RMSE) values for both optimized and non-optimized models at WL OUT_1 and WL IN_2.

Year	RMSE (m)
	Non-Optimized Hydraulic Model	Optimized Hydraulic Model
WL OUT_1
2012	0.0586	2.9 × 10−4
2013	0.0220	6.1 × 10−4
2014	0.0216	5.8 × 10−4
2015	0.0185	3.9 × 10−4
WL IN_2
2012	0.0318	6.1 × 10−3
2013	0.0340	11.2 × 10−3
2014	0.0316	8.3 × 10−3
2015	0.0265	7.0 × 10−3

). For the optimized model, the RMSE was calculated at WL OUT_1 and at WL IN_2 reporting mean values of 4.661 × 10⁻⁴ m and 8.150 × 10⁻³ m, respectively. They significantly differ from those of the non-optimized one (mean value of 0.0302 m at WL OUT_1 and 0.0285 m at WL IN_2).

Overall, the comparison among simulations highlighted the fact that the optimized model achieved excellent results, which are very close to the measured values. In the RMSE terms, the differences between the two models (non-optimized vs optimized) had maximum value of 0.0583 m, that was recorded at WL OUT_1 for the dry year (2012). Moreover, the mean differences were 0.0297 m and 0.0228 m at WL OUT_1 and WL IN_2, respectively. When considering the measurements accuracy order of magnitude (±0.05 m), the optimization process significantly improved the obtained results.

3.2. Extended Segment (ES)

The ES offtake discharges were calculated as in Section 2.4.1, and the q_kCn obtained were mainly coherent with the yearly meteo-climatic condition. In particular, the ES reference offtake reported minimum (0.02 m³/s) and maximum (1.17 m³/s) values during the rainy and the dry years, respectively. Moreover, q_kCn for all other offtakes varied from 0 m³/s (2014) to 0.87 m³/s (2012). This range was larger than those of PS (0–0.24 m³/s for the reference offtake and 0–0.17 m³/s for all other offtakes). In fact, the ES irrigated land supplied is 1.5 times larger (12,580 ha) than that of PS.

At WL OUT_0, the flowing discharges calculated with the Equation (18) are reported in Figure 12. For every year, the Q0 elements were grouped in Q0_s and Q0_c vectors in order to distinguish the flowing discharge values that are based on Q2_sims and Q2_simc, respectively.

The lowest values of flowing discharges resulted for the rainy year with a mean value of 18.46 m³/s (standard deviation of 2.70 m³/s); the highest values were related to 2012 (24.13 m³/s on average with a standard deviation of 3.22 m³/s) and 2013 (25.81 m³/s on average, standard deviation of 4.569 m³/s).

For the year y, the performances of the ES optimized hydraulic model were evaluated through the values of the parameterized hydraulic variables and the differences between simulated and measured water levels at WL OUT_0. When considering the hydraulic variables, the optimization process returned physically possible values while only using a larger set of parameters. In particular, four gate discharge coefficients and five Manning’s coefficients were investigated to characterize ES (in roughness terms) and every culvert (in roughness and head loss terms). For three years, the values of the parameterized hydraulic variables obtained are reported in

Table 5. The values of the nine parameterized variables and of the cost of the criterion obtained from the optimization process.

Year	Parameterized Hydraulic Variables									Cost of the Criterion
	Cd3 (-)	Cd4 (-)	Cd5 (-)	Cd6 (-)	n (m1/3/s)	n3 (m1/3/s)	n4 (m1/3/s)	n5 (m1/3/s)	n6 (m1/3/s)	J Cost (m)
2012	0.60	0.45	0.62	0.52	0.020	0.020	0.013	0.011	0.010	0.5057
2013	0.58	0.60	0.58	0.58	0.014	0.013	0.013	0.013	0.013	0.4667
2015	0.42	0.59	0.43	0.45	0.011	0.019	0.019	0.019	0.010	0.3465

.

For every year of analysis, the optimization process was run in order to obtain the parameterized hydraulic variables values. For the year 2014, it could not end and it tended to minimized the criteria assigning negative values to the Manning’s coefficients and high values (>1) to the gate discharge coefficients. So, for this year, the optimization loop was not finalized.

All gate discharge coefficients referred to submerged flow. The values obtained presented less variability than those of PS (Table 3). They were around 0.60, except for the year 2015 (mean value of 0.47). In 2012, the four culverts were mainly characterized by piped flow (as for Culv_2 of PS), while the year 2013, except for Culv_4, presented mainly free flow conditions.

As for PS, the n values that were obtained were coherent with the literature range reported for concrete canals 0.010–0.020 m^1/3/s [68]. Over the three years, the mean value was 0.015 m^1/3/s (very similar to PS n) and the maximum difference of 0.009 m^1/3/s was between the years 2012 and 2015 (0.002 m^1/3/s in PS). The n3, n4, n5 and n6 values were coherent with the range 0.010–0.014 m^1/3/s for concrete culverts [74], except for the year 2015, for which the values were higher (0.019 m^1/3/s maximum). As said before, the Manning’s coefficient differences can be attributed to several factors, such as geometric irregularities or variations of the canal bed and of cross sections, obstacles, vegetation growth, and meandering. Moreover, the field survey estimated the accuracy of Z0_obs,y values (±0.10 m) to be lower than those in PS.

When considering the same year y, the ES J cost quite significantly differed from PS, and the values of the elements contained in diff2_y, diff3_y and diff0_y can justify these results. In fact, considering the year 2015 as an example, the maximum absolute values (only for days not affected by suspicious measures) are 0.015 m at WL OUT_1 (diff2_y) and 0.011 m at WL IN_2 (diff3_y), while at WL OUT_0 (diff0_y), it was much higher and very close to the accuracy threshold (0.102 m). In any case, Figure 13 shows that all the single elements of Z0_sim,y are within the Z0_obs,y accuracy range (±0.10 m), except for few days (eight for 2012, nine for 2013, and two for 2015) that are related to Z0_obsj or Q2_simj and that were probably affected by errors.

As for PS, the vectors Z0_sim,y and Z0_obs,y were plotted (Figure 14) in order to detect the modelling impacts of the diff0_y elements. The intercept and the slope of the linear correlation were evaluated, and they were compared with the optimum values (i.e., perfect fitting) and with those reported for PS.

The results can be considered excellent for the years 2013 and 2015, with RMSE values of 0.09 m and 0.05 m, respectively. For 2012, especially the intercept of the linear interpolation (6.047) was significantly different from the optimal value (0). As already reported by Mesplé [83], the modelling overestimation/underestimation was probably combined with the proportionality of the gap between the measured and simulated values. Therefore, the RMSE for 2012 (0.09 m) was similar to the other years.

Overall, the RMSE values in ES simulations were higher than those reported in PS, indicating that the model worked better in a segment that was characterized by simpler geometry and with higher availability and reliability of measured hydraulic data. When the model was tested on a more complex reality i.e., ES, it had to face two critical aspects: A scarce number and a lower accuracy of the hydraulic measured data. The latter affected the optimization process, especially for the years with extreme climatic conditions. In fact, the dry 2012 was characterized by an intense functioning of pumps with a maximum daily difference in water level terms of 1.23 m. On the contrary, the rainy 2014 presented lower irrigation demands and therefore the functioning of pumps was more intermittent. This implies that, due to the slope of ES (3.8 × 10⁻⁵), a backwater flow occurred affecting the optimization process and leading to a poor representation of the reality. The multi-disciplinary modelling approach developed in this study presented satisfying results for the two remaining years (2013 and 2015).

4. Conclusions

A low availability of hydraulic data can seriously affect efficient management of irrigation canals. Therefore, this paper presents a novel approach that can be applied to reconstruct the missing hydraulic data by combining hydraulic modelling and an irrigation DSS (that was developed at a regional scale).

The approach was developed on a Northern Italian canal, more specifically, on its 7 km long segment (PS), which is characterized by a quite simple geometry and full availability of water levels, and it gave very good results. Its application on a more complex segment (ES) with a poor data availability and accuracy, confirmed that the approach can be successfully used to reconstruct data for years with standard meteo-climatic conditions, while years with extreme climatic conditions are more difficult to be simulated. It was found that the measuring point and consequently instrument accuracy are key factors for obtaining a model that can well represent the reality.

Moreover, the results showed that the offtake discharges can be estimated on the base of crop water delivery schedules and combining them with measured water levels could enable calculating the discharges that are flowing through the use of an optimized hydraulic model.

However, this approach was developed on a lined concrete canal. Therefore, its application on secondary channels, often on earth with considerable infiltration losses, have to be further studied in order to optimize the hydraulic model and to increase its relevance.

Since the approach proposed allows quantifying discharges and water levels along an irrigation canal, it can be integrated with water qualitative analysis (e.g., microbiological aspects), thus widening its multi-disciplinarity.