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,
qkCn (m
3/s) can be written as:
where
qrDm (m
3/s) is the average discharge diverted from the reference offtake during the month
m (
m = 1, 2, 3), and
wkCn 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.
qrDm was calculated as:
where
VrDm (m
3) is the indirectly calculated cumulated volume of the reference offtake for the month
m, while
Dm (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
wkCn, as follow:
where
wkDm (-) and
wkTn (-) are the weights of the offtake
k obtained using the
D-data and the
T-data, respectively,
VkDm (m
3) is the indirectly calculated volume of the offtake
k during the month
m,
VkTn (m
3), and
VrTn (m
3) 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 (
VkTn) was determined by the expression [
14]:
where
Ai (m
2) is the area covered by the crop
i per each year,
CWRi (mm) is the decadal cumulated optimum water requirement for the crop
i,
IIi (-) is the irrigation intensity of the crop
i,
EIi (-) 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 VrTn.
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 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.
SIC
2 (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]. SIC
2 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]. SIC
2 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]. SIC
2 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]:
where
Q (m
3/s) is the discharge,
S (m
2) is the wetted area,
g (m/s
2) 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:
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 SIC
2 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:
where
Cd (-) is the gate discharge coefficient,
L (m) is the gate width,
u (m) is the gate opening,
Zup (m), and
Zdn (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 SIC
2 [
66,
67,
68].
After the PS geometry entry, several hydraulic aspects were evaluated in SIC
2. 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
qkCn 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).
where
j is the index for the examined day of the year
y (
j = 1, ..., e).
The software SIC
2 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:
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:
Therefore, the criterion to be minimized
J was expressed as:
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 diff2y and diff3y, 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 σ2y and σ3y. In particular, if a day j is affected by a suspicious measure, the weight (σ2j; σ3j) was set as 10; otherwise, it was equal to 1.
A detection method was elaborated considering the vectors Z1obs,y, Z2obs,y, Z3obs,y, Z4obs,y, Q2sim,y, and Q3sim,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:
where
j is the index for the examined day of the year
y (
j = 1, ..., e).
The method was based on the vectors:
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 deltay-delta1y, and deltay-delta2y 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 deltay results as an outlier in both plots, a suspicious measure was in Z2obsj or in Z3obsj. If the element j of deltay results as an outlier in the first plot but not in the second, the suspicious measure was in Z1obsj. If the element j of deltay is an outlier in the second plot but not in the first, the suspicious measure was in Z4obsj. To evaluate if a suspicious measure is in Z2obsj or Z3obsj, Q2sim,y-delta1y, and Q3sim,y-delta2y were plotted. For both, a data quadratic fitting of data was considered. If the j-th element of delta1y results as an outlier, the suspicious measure was in Z2obsj while if the element j results as an outlier of delta2y, the suspicious measure was in Z3obsj.
The most significant results obtained are given in
Section 3.1.