The SD water balance model of the Lake Bracciano system was developed with the free software INSIGHT MAKER, which allows the creation and use of SD-based models from a WEB browser [29]. The study takes up, develops, and refines a previous experience [17], in which a lake water balance was carried out based on a 30-year time interval (1961–1990), with a time scale of one year and a single reference station (Vigna di Valle).
The SD model improves understanding of the inner workings of a complex socio-hydrological system with its various direct and indirect interconnections [30]. The challenge of this approach is to use a limited dataset, estimating flow values as closely as possible where measured data are not available.
Water balance equation
According to the principles of conservation of mass in a closed system, the inflows into a water system are equal to its outflows plus the change in storage during a time interval. The general water balance equation is:
P = S + ET + ΔS [equation 1]
where P is precipitation, S is streamflow, ET is evapotranspiration, and ΔS is the positive or negative change in storage. In the case of a lake, especially if its resource is used as a water supply, the equation is more complex: the runoff factor is divided into inflow by tributaries and outflow by emissaries, and in the right-hand side of the equation are the different types of human withdrawals. The communication between the water stored in the lake and the groundwater stored in the aquifer below the lake further complicates the balance equation. However, the lake level is the direct indicator of how much water is stored in the lake system and well represents the variation factor ΔS. Consequently, the level of Lake Bracciano, as an expression of the Sabatino lake-aquifer system (Fig. 4), can be represented as follows:
LL = Pl + R + Bf – Eb – Ri – A ± Gf [equation 2]
where:
-
LL – Lake Level: level of Lake Bracciano above or below a fixed hydrometric reference;
-
Pl – Precipitation: direct rainfall in the lake area only;
-
R – Runoff: flow along the slopes and the small tributaries (streamflow) towards the lake;
-
Bf – Base flow: persistent flow in streams even in dry phases or seasons;
-
Eb – Evaporation: direct evaporation from the surface of Lake Bracciano;
-
Ri – River: surface outflow from the system by the Arrone River, which is the only emissary of the lake;
-
A – Anthropic: all withdrawals that take water directly from the lake;
-
Gf – Groundwater flow: indicates the groundwater interaction between the aquifer and the lake.
The streamflow (S) in [equation 1] is divided in two distinct factors:
-
the runoff (R), part of the rainfall falling within the catchment area (Pc), carried to the lake along hillsides or through small streams in the hydrographic network;
-
the base flow (Bf), flow within streams fed by linear springs due to the intersection of the volcanic water table with the hydrographic network.
Groundwater interaction between lake and aquifer may change in intensity and direction, depending on the level of the lake and other flows within the hydrogeological basin. Rainfall infiltration (I), estimated as part of total rainfall in the entire hydrogeological basin sector (Pb, including the catchment area but excluding the lake surface), feeds the aquifer. Withdrawal from the several wells for local use and direct evaporation from the surface of Lake Martignano (Em) remove water from the aquifer and stimulate groundwater flow from Lake Bracciano.
By adapting a generic but versatile approach to a specific problem, the application of SD to hydrological analysis necessarily requires the simplification of some physical dynamics.
The absence of the evapotranspiration factor (ET in [equation 1]) from the mass balance [equation 2] is a consequence of the approximate rainfall-runoff transformation used in the estimation of runoff and infiltration. Assuming the latters as constant rates, invariable in time and space, of the rainfall, the evapotranspiration constitutes the part of rainfall that does not “touch the ground” and does not constitute flow in the system.
A further unavoidable approximation of the model, based on currently available data and information, is to consider the lake-aquifer system as closed and therefore not to consider, either in quantity or direction, any flow between the Sabatino aquifer and adjacent ones. Another way of interpreting this approximation is to consider the groundwater inflows into the Sabatino aquifer equal to the outflows and, therefore, their sum equal to zero.
GIS analysis
The study area of the lake-aquifer system can be schematically subdivided into three distinct and concentric sectors: the Lake Bracciano area, the territory included in the Lake Bracciano catchment area and the territory included in the Lake Bracciano hydrogeological basin (Fig. 4). Through an elementary geospatial analysis (calculation of Thiessen/Voronoi polygons), the influence factor of each of the four weather-climate stations (Castello Vici, Vigna di Valle, Bracciano and Trevignano Romano) on the three sectors was defined. The rainfall, temperature and relative humidity values used in the model are the sum of the measurements of each station, weighted by its specific influence factor (Table 1).
The GIS analysis of the third level classes of the Corine Land Cover project permitted estimating the extent of the different land cover types in the four available years: 1990, 2000, 2006, 2012 and 2018.
Table 1 - Percentage weight of available stations for the period 2012–2019
Weather-climate stations
|
Lake
|
Catchment area
|
Hydrogeological basin
|
Municipality
|
Locality
|
Area (km2)
|
Weight (%)
|
Area (km2)
|
Weight (%)
|
Area (km2)
|
Weight (%)
|
Bracciano
|
Vigna di Valle
|
10.18
|
17.77
|
5.41
|
6.17
|
7.71
|
5.60
|
Trevignano Romano
|
|
21.19
|
36.98
|
42.45
|
48.39
|
59.42
|
43.18
|
Bracciano
|
Bracciano
|
12.19
|
21.27
|
31.94
|
36.41
|
43.22
|
31.41
|
Anguillara Sabazia
|
Castello Vici
|
13.74
|
23.98
|
7.92
|
9.03
|
27.25
|
19.80
|
Total
|
57.30
|
100
|
87.72
|
100
|
137.60
|
100
|
Available flow data
The monthly dataset used to implement the model is fairly continuous and complete from January 1970 to December 2019 for lake level and for rainfall, temperature and relative humidity.
In addition, public data are available for the two main direct water abstraction plants (uptake) from the lake, operated by ACEA private enterprise to supply the metropolitan area of Rome: the Paolo Aqueduct and the New Aqueduct of Bracciano (NAB), operational since 1990 (Fig. 5).
Estimated flows
Temperature and relative humidity were used to estimate direct evaporation from the two lakes using a site-specific equation [31]. By assigning homogeneous characteristics and values of soil permeability and evapotranspiration in the area, runoff (R) and rainfall infiltration (I) were estimated as part of the total rainfall.
The base flow contribution to the lake was estimated through stream flow measurements taken after more than 5 dry days since the last rainfall event. A measurement campaign on an approximately quarterly basis, from December 2017 to June 2019, supplemented a series of available measurement made between 1992 and 2009. The entire dataset was averaged, and a monthly value was consistently assigned and repeated for each year of the study period.
The flow rate of the Arrone river was estimated according to the outflow formula:
Q = 0.4 * h * L * √2g h [equation 3]
where:
-
h – height of the water head (lake level) above the top of the dam;
-
L – length of the dam (15 m);
-
g – gravity acceleration.
Although it is assumed that the lake contributes to the Arrone river through the riverbed, the model assumes that the outflow occurs exclusively through the dam overflow in the months when the lake level exceeds the hydrometric zero, represented by the top of the dam (163.04 m a.s.l.).
In addition, the ACEA supply system and other outflows for human consumption (A) are considered in the model:
-
minor withdrawals directly from the lake, mainly used for the irrigation of the surrounding areas;
-
the Trajan aqueduct that captures the springs located within the lake’s catchment area;
-
the sewerage collector (COBIS) that collects part of the runoff water around the lake;
-
the various public and private wells that take groundwater from the entire hydrogeological basin for domestic and productive use.
The minors abstractions were roughly estimated based on the results of the law enforcement investigation conducted during the summer crisis of 2017.The two indirect withdrawals, the COBIS collector and the Trajano aqueduct, were estimated as the percentage of runoff and rainfall infiltration, respectively.
The water consumption related to production and domestic activities in the study area is based on the National Institute of Statistic (ISTAT) censuses on agriculture (1990, 2000 and 2010) and population trends (decennial between 1971 and 2001 and then annual). Unfortunately, the data on different types of land cover do not cover the first two decades of the analysis period. Among the agricultural land in the area, the only ones of interest for the assessment of water consumption for irrigation purposes are the “permanent crops” of orchards (222) and olive groves (223) and the “heterogeneous agricultural areas” (242 and 243).
Model structure
The Bracciano SD model organises the information on water resources in a schematic structure, with the aim of replicating the flow and storage dynamics of the Sabatino hydrogeological system with the least possible approximation (Fig. 6).
The different inflows and outflows were expressed in terms of metres of monthly contribution (m/month, positive or negative) to the lake level. The unit of measurement of the flows expresses the thickness of the monthly volume distributed above (inflows) or subtracted below the lake surface. The volumetric quantities of runoff and infiltration flows imply the normalisation over catchment and hydrogeological basin, respectively. Considering that the area of Lake Bracciano corresponds to 57,300,000 m2, a flow rate of 1 m3/s results in a change in the lake level of about 51 cm per month (0.051 m/month).
The Bracciano SD model is based on thirteen main elements consisting of two intercommunicating reservoirs, represented by Lake Bracciano and the main volcanic aquifer, four inflows and six outflows and a bi-directional flow between the two reservoirs, called “Groundwater flow”, whose direction depends on the relative reciprocal level (Fig. 7).
The variation in time of the main elements is a function of eighty secondary elements, consisting of converters, variables and constants, interconnected by links that reproduce the physical dynamics of the system and complete the structure. Twelve data sets feed the converter and the model flows, conditioned by eighteen independent variables (Table 2) and thirteen constants (Table 3). Thirty-three variables of three types are added: dependent (directly or indirectly, on the eighteen independent variables), conditional (needed to handle discontinuous flows) and “temporal” (necessary to manage population and livestock data series).
In addition to the twelve “real” data sets, there are four “instrumental” ones. While one is necessary for the correct calculation of daily data (number of days per month), the other three link the monthly average value of the consumption variables to a specific dimensionless seasonal variability factor (Table 4): “Human seasonal factor”, “Agricultural seasonal factor”, and “Livestock seasonal factor”.
Table 2
Unit and significance of each of the eighteen independent variables used in the model
Variables
|
Significance
|
Initial groundwater level
|
Assumed height of the water table at lake level zero in January 1970
|
Runoff coefficient
|
Percentage value of the rainfall
|
Infiltration coefficient
|
Average Sabatini conductivity
|
Representative value for all lithotypes in the area
|
Infiltration delay
|
Average time taken by the rainfall to cross the unsaturated zone between the surface and the aquifer
|
Visentini correction factor
|
Adjustment factor of the original formula developed for Lake Bracciano to the different area, depth and coastal morphology of Lake Martignano
|
COBIS factor
|
Percentage value of “Runoff coefficient”
|
Trajan factor
|
Percentage value of “Infiltration coefficient”
|
Instant average minor withdrawals
|
Total flow value of all the supposed minor supply systems feeding directly from the lake
|
Average daily domestic consumption
|
Daily water demand
|
Average daily poultry consumption
|
Average daily bovine consumption
|
Average daily swine/ovine consumption
|
Monthly average heterogeneous agricultural consumption
|
Monthly water demand
|
Monthly average fruit/olive consumption
|
Domestic use factor
|
Compensation factors for the lack of accuracy of the data concerning the inhabitants, agricultural activities and livestock withdrawals actually fed by the Sabatino aquifer groundwater
|
Livestock use factor
|
Agricultural use factor
|
Table 3
Datasets and constants used in the System Dynamics model for this work
Dataset
|
Unit
|
|
Elements
|
Unit
|
Value
|
Rainfall
|
mm
|
|
Initial lake Bracciano level
|
metre
|
-0.134
|
Temperature
|
°C
|
|
Lake Bracciano area
|
square metre
|
57,300,000
|
Relative humidity
|
%
|
|
Catchment/lake area ratio
|
dimensionless
|
1.531
|
Base flow
|
l/s
|
|
Hydrogeological basin/lake area ratio
|
dimensionless
|
2.403
|
Paolo aqueduct
|
m3/s
|
|
Martignano/Bracciano lake area ratio
|
dimensionless
|
0.039
|
NAB aqueduct
|
m3/s
|
|
Lake Bracciano perimeter
|
metre
|
31,703
|
Fruit trees and olive grooves
|
km2
|
|
Arrone dam constant 1
|
1/second
|
0.4
|
Heterogeneous agricultural areas
|
km2
|
|
Arrone dam constant 2
|
metre
|
19.62
|
Swine and ovine
|
n°
|
|
Arrone dam width
|
metre
|
15
|
Bovine
|
n°
|
|
Visentini constant
|
metres/month
|
1/1,000
|
Poultry
|
n°
|
|
Seconds per day
|
seconds/day
|
86,400
|
Population
|
n°
|
|
Zero level
|
metre
|
0
|
|
|
|
Zero flow
|
cubic metres/month
|
0
|
Table 4
Seasonal factors for human, agricultural and livestock uses
Month
|
Human seasonal factor
|
Agricultural seasonal factor
|
|
Temperature (°C)
|
Livestock seasonal factor
|
January
|
0.5
|
0
|
|
19
|
1
|
February
|
0.6
|
0
|
|
20
|
1.3
|
March
|
0.8
|
0
|
|
25
|
1.5
|
April
|
1
|
0.1
|
|
30
|
2
|
May
|
1.2
|
0.5
|
|
|
|
June
|
1.3
|
1
|
|
|
|
July
|
1.5
|
1.5
|
|
|
|
August
|
1.5
|
1.5
|
|
|
|
September
|
1.3
|
0.8
|
|
|
|
October
|
1
|
0.1
|
|
|
|
November
|
0.8
|
0
|
|
|
|
December
|
0.7
|
0
|
|
|
|
Apart from rainfall, base flow and Paolo and NAB aqueducts, fed directly from the data series, all others depend on several variables, datasets and constants, constituting complex sub-components, linked to the main structure. While some sub-components are simple, having to represent only a function (evaporation) or a reduction factor of an original flow (runoff, infiltration, Trajan, COBIS; Fig. 8), others are substantially more complex.
Seven dataset and nine variables form the basis for estimating local demand, the most complex sub-component (Fig. 9). For each of the six datasets (orchards/olive groves, heterogeneous agricultural areas, pigs/sheep, cattle, poultry, and population), a monthly value (dependent variable) is provided based on a unit consumption value (independent variable), a “Seasonal effect” factor and a “Utilisation factor” that contributes to the calibration of the flow weight on the system.
There are four “feedback loop” dynamics in the model that control the reciprocal relationships between the different elements of the system: two concern the “Groundwater” flow between the lake and the aquifer (Fig. 10) and two concern the “Arrone” river, all dependent on the level of “Lake Bracciano”.
The infiltration flow towards the water table is subjected to a delay, which acts through the variable “Infiltration delay”, representative of the time required for rain falling on the surface to cross the “unsaturated” zone and reach the aquifer.
Model calibration
The eighteen independent variables were adjusted through a process of model calibration to better match the model output with historical data. The model calibration was performed by analysing the degree of alignment of the curve produced by the model for the “Lake Bracciano” tank level with the curve of historical lake level measurements from the Castello Vici hydrometric station, available for the entire 50-year observation time window. The difference between the calculated and observed levels indicates the reliability of the model and is expressed in terms of the square of the difference between the estimated and measured value.
The first step in calibrating the model consisted of running several simulations and producing graphs showing the variation over time of each of the model’s variables, as well as the data series with which the model was trained. The next step was to use the optimisation function (Optimization and Goal Seek) of the INSIGHT MAKER software, applying the least squares method to the values of the independent variables to achieve the desired goal.
In order to reduce the discretion left to the software, data and information were found that define solid references for the variability intervals between minimum and maximum, for runoff and infiltration coefficients, human [32], livestock [33, 34] and irrigation water consumption [35, 36]. Some other parameters have limits of variability based mainly on expert judgement, approximate information, and rough estimates (Table 5).
Table 5
– Unit, minimum and maximum values and level of accuracy used in the optimisation process and final value of each variable used in the model
Variables
|
Unit
|
Minimum value
|
Maximum value
|
Accuracy
|
Final value
|
Initial groundwater level
|
metre
|
0
|
20
|
0.5
|
10
|
Runoff coefficient
|
dimensionless
|
0.08
|
0.18
|
0.001
|
0.173
|
Infiltration coefficient
|
0.29
|
0.45
|
0.001
|
0.301
|
Average Sabatini conductivity
|
metres/second
|
1x10− 7
|
1x10− 6
|
1x10− 5
|
4.0*10− 6
|
Infiltration delay
|
month
|
0.4
|
6
|
0.1
|
0.4
|
Visentini correction factor
|
dimensionless
|
0.1
|
2
|
0.01
|
1.05
|
COBIS factor
|
0.1
|
0.5
|
0.005
|
0.375
|
Trajan factor
|
0.1
|
0.5
|
0.05
|
0.3
|
Instant average minor withdrawals
|
cubic metres/second
|
0.02
|
0.25
|
0.01
|
0.16
|
Average daily domestic consumption
|
cubic metres/day
|
0.2
|
0.5
|
0.02
|
0.5
|
Average daily poultry consumption
|
0.0001
|
0.005
|
0.0001
|
0.0046
|
Average daily bovine consumption
|
0.02
|
0.15
|
0.001
|
0.145
|
Average daily swine/ovine consumption
|
0.005
|
0.03
|
0.005
|
0.03
|
Monthly average heterogeneous agricultural consumption
|
cubic metres/month/square kilometre
|
20,000
|
150,000
|
1,000
|
85,000
|
Monthly average fruit/olive consumption
|
15,000
|
120,000
|
1,000
|
120,000
|
Domestic use factor
|
dimensionless
|
0.1
|
1
|
0.01
|
1
|
Livestock use factor
|
0.1
|
1
|
0.01
|
0.46
|
Agricultural use factor
|
0.1
|
1
|
0.01
|
0.55
|