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Risk analysis of sustainable urban drainage and irrigation
Advances in Water Resources 83 (2015) 277–284
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Risk analysis of sustainable urban drainage and irrigation Nadia Ursino∗ Department ICEA, Civil Architectural and Environmental Engineering, University of Padova, via Loredan 20, Padova 35131, Italy
a r t i c l e
i n f o
Article history: Received 25 April 2014 Revised 23 June 2015 Accepted 26 June 2015 Available online 30 June 2015 Keywords: Landscape irrigation Detention Sustainability Reliability analysis Low impact development BMP
a b s t r a c t Urbanization, by creating extended impervious areas, to the detriment of vegetated ones, may have an undesirable influence on the water and energy balances of urban environments. The storage and infiltration capacity of the drainage system lessens the negative influence of urbanization, and vegetated areas help to re-establish pre-development environmental conditions. Resource limitation, climate, leading to increasing water scarcity, demographic and socio-institutional shifts promote more integrated water management. Storm-water harvesting for landscape irrigation mitigates possible water restrictions for the urban population in drought scenarios. A new probabilistic model for sustainable rainfall drainage, storage and re-use systems was implemented in this study. Risk analysis of multipurpose storage capacities was generalized by the use of only a few dimensionless parameters and applied to a case study in a Mediterranean-type climate, although the applicability of the model is not restricted to any particular climatic type.
1. Introduction Flood and drought have social, economic and environmental consequences which are exacerbated by urban development and climate change [28]. Urbanization, by creating extended impervious areas, to the detriment of vegetated ones, has an undesirable influence on the water and energy balances of urban environments [15], increasing peak discharge and total volume of surface runoff and reducing groundwater recharge. An important concern of urban hydrologists is thus to restore (at least partially) the pre-development water balance, thus, managing urban water in a more sustainable way. Nevertheless, since the increasing scarcity of clean water in many urban areas leads to intense competition for its use [4], the aim of sustainable urban water management cannot be restricted simply to restoring pre-development runoff conditions: rather, it should continue to evolve as urbanization increases and aim at better integrating urban, environmental, agricultural and industrial water use. There is a need to ascertain whether integrated strategies can achieve social and economic goals as well as good-quality ecosystem service and maintenance [22]. When rainfall falls over impervious land and does not infiltrate, it runs off becoming storm-water. Storm-water best management practices (BMP) preserve natural landscape features, minimize effective imperviousness, and treat storm-water as a resource rather than simply letting it go to waste [33]. The design criteria and performance of
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http://dx.doi.org/10.1016/j.advwatres.2015.06.011 0309-1708/© 2015 Elsevier Ltd. All rights reserved.
© 2015 Elsevier Ltd. All rights reserved.
BMP have gained much coverage in the technical literature [8,9,37]. Detention basins are effective controls which can attenuate peak discharge [31]. When they act as infiltration systems, or in series with such systems, they can help to restore the pre-development water balance, enhance groundwater recharge, delay and reduce surface runoff, and thus protect receiving water bodies, and sewer systems [11,38]. Plants growing on natural or artificial soils increase their infiltration capacity by altering the soil structure of the vadose zone [26]. Bioretention area must be sufficiently permeable to infiltrate storm-water and have adequate retention capacity to support healthy vegetation growth. Engineered soil mix may achieve both objectives. Vegetation is a crucial element in the mass and energy flux locally at the soil-vegetation-atmosphere interface, and in the earth’s climate in general [27,34]. Maintaining or restoring vegetated areas in urban environments may further help to re-establish the pre-development water and energy balances. Plants can control the urban microclimate through transpiration and soil sheltering, reducing rises in urban temperatures and increasing the “liveability” of cities [12]. Nevertheless, vegetated urban landscapes consume water and are drought-sensitive, and thus pose the problem of water saving in landscape irrigation [13]. Frequency of irrigation depends on soil properties, climate and vegetation type. Applied water should never exceed the water holding capacity of the soil. Typically, deep and infrequent irrigation favors root elongation, health and vigor of turfgrass. Irrigation of vegetated bioretention area must minimize water use and avoid underdrain flow during dry weather. Although storm-water harvesting and landscape irrigation systems may require pumping stations, the additional cost of energy
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N. Ursino / Advances in Water Resources 83 (2015) 277–284
distribution is expected to be offset by energy savings on cooling systems [30]. Storm-water harvesting and re-use schemes for non-potable purposes (such as landscape irrigation) may mitigate the risk of water restrictions for urban populations. Treated waste-water may also be used for landscape irrigation according to many local regulation, although the re-use of storm-water collected from impervious land, which is discussed here, have higher public acceptance than alternative water supplies such as waste-water recycling [7]. For example, swale adjacent to impervious areas can support grass growth on golf courses in arid areas [35], and properly designed, maintained, and treated storm-water harvested from roofs is low-cost, often less polluted than other sources of water in urban catchments, and may integrate existing water supplies [29]. Storm-water harvesting reduces growers’ dependence on the municipal water supply [6], however, storm-water re-use may meet water requirements for urban horticulture, if a sufficient volume is available during the growing season of the vegetation. It is not clear if, in climates of Mediterranean type, characterized by winter rains alternating with summer drought and high temperatures, which are often exacerbated by urbanization [25], landscape irrigation with reuse storm-water could be successful. In climates of Mediterranean type, due to the fact that rainfall and air temperature are out of phase and that the end of the rainy season corresponds to the beginning of the growing season, rain-fed vegetation often suffers water stress due to the long inter-storm interval between successive rainfall events in the dry-warm period. Collecting and reusing sufficient quantities of water of the required quality when the demand for irrigation is high is expected to be a problem, due to rainfall scarcity and variability [14]. The hydraulic design of a multipurpose storage tank aims at two contrasting goals: (i) to provide detention capacity and peak discharge reduction and (ii) to ensure a sufficient re-use water for irrigation. The first aim demands empty tanks at the beginning of each rain event. The second is to have the largest possible amount of water stored in the tanks during the inter-storm interval. If the system fails, this is due to overflow, when the volume of rainfall exceeds available storage capacity, or to water scarcity, when the stored rainfall volume is drawn before the next rainfall event. The probability of system failure is called risk. The main aim of this study is to examine if multipurpose tanks can store and provide sufficient storm-water volumes for landscape irrigation when required, whether this implies a significant increase in the risk of overflow, and to what extent the risk of drought or, conversely, overflow depends on climate, hydrology and/or management strategies. The probability approach [1] provides analytical equations to estimate the risk of failure of detention facilities and other elements of the urban drainage system in question [17,18]. Several studies have applied a probabilistic approach to BMP failure risk analysis and demonstrate that it matches many physically based methods [16,19,20,38,39]. The risk analysis of water-saving systems has so far received less attention than that of storage systems. Whether the storage capacity of a detention basin can also ensure a sufficient water supply for irrigation purposes, and whether multipurpose management strategies can affect the risk of overflow have not yet been studied on a probabilistic basis. A new integrated risk model was implemented in this study. Section 2 describes the conceptual model of a drainage, detention and re-use system. In Section 3, closed-form solutions are derived. In Section 4 the risk analysis is generalized by expressing risk as a function of a few dimensionless groups, and the influence of tank management strategy is evaluated analytically and validated numerically. In Section 5 the results are discussed and problems in sustainable water management are discussed.
RAINY SEASON
infiltration (1-imp)·f imp
1-imp Vs
controlled outflow imp·Q0
runoff overflow GROWING SEASON
irrigation (1-imp)·i
imp
runoff
1-imp Vi
overflow
Fig. 1. Conceptual model of drainage system.
2. Model 2.1. Conceptual model of drainage system A conceptual model is formulated for an urban catchment comprising one pervious (vegetated) and one completely impervious subcatchment area (Fig. 1). Impermeability imp is the ratio between the impervious area and the whole catchment area and identifies the stage of development of the catchment. The components of the drainage system (Fig. 1) are: (i) a network which transfers the rainfall volume falling on the impervious subcatchment instantaneously to (ii) an overflow divider which diverts flow to storage or overflow discharge; (iii) a multipurpose tank; (iv) a distribution system drawing water from the tank to the pervious sub-catchment at a constant rate in the period of time between two rainfall events; (v) an outflow pipe taking water, at controlled rate, to discharge. The network storage capacity should influence the rainfall-runoff transformation, which is kept as simple as possible, and indirectly the probability of overflow. However, the increasing storage capacity of the network with development, and thus imp, is neglected here for the sake of simplicity, possibly overestimating the risk of overflow. The tank is used for detention during the rainy season and for detention and landscape irrigation during the vegetation growing season. During the rainy season the pervious sub-catchment achieves in-place infiltration of storm water. The flow rate per unit surface in the infiltration system may reach soil infiltration capacity f during the rainy season, when the additional controlled outflow is Q and equals irrigation rate i during the growing season. The tank must be emptied as soon as possible after each rainfall event during the rainy season. Conversely, during the growing season, storm-water is stored for longer periods and re-used for irrigation at rate i < f . There is no controlled outflow during the growing season. This practice obviously increases the probability that storage capacity will not be available at the beginning of rare and intense showers during the growing season. Every time the rainfall volume exceeds the available storage capacity, the system fails. In this Section, the related probability of failure is called risk of overflow. The risk of water scarcity or drought is the probability of the tank being emptied during the irrigation season, before further rain falls (before the end of the inter-storm interval).
N. Ursino / Advances in Water Resources 83 (2015) 277–284
2.2. Rainfall distribution The meteorological input to the catchment is assumed to be represented by the probability density functions of three independent random variables: rainfall volume (h); duration (t) and inter-storm interval (b). Following the approach proposed by [17] h, t and b are modeled as random variables with exponential probability density functions. This assumption simplifies the mathematical tractability of risk analysis and allows its closed form analytical solution, even though there may be more suitable marginal distribution functions of the input rainfall [5]. The probability density functions of rainfall characteristics are:
279
general, Q = imp Q0 where Q0 is the maximum discharge ensuring channel protection, overbank flood protection or extreme flood protection. Flow discharge is expressed as volume per unit time, divided by catchment area. The aim of tank management during the rainy season is to empty the tank quickly to create space for the water of the next storm event. In order to accelerate the draw-down of Vs , part of the outflow discharge may be conveyed to the pervious sub-catchment which functions as an infiltration system. Soils that allow the realization of infiltration systems are sand, loamy sand sandy loam and loam [3], vegetated bioretention basins which must achieve high infiltration rate and water holding capacity, are often made out of engineered soil layers. Maximum outflow discharge of urban catchments with an infiltration system downward a storage tank is:
f h = ζ e−ζ h
(1)
ft = λ e−λt
(2)
Q = imp · Q0 + (1 − imp) · f
f b = ψ e−ψ b .
(3)
where f is the infiltration capacity of the pervious sub-catchment. The infiltration capacity f of the sub-catchment is selected as scale factor of the prescribed outflow discharge Q0 ,
The distribution parameters are: ζ = inverse of expected value of rainfall depth, λ = inverse of expected value of rainfall duration, ψ = inverse of expected value of rainfall inter-storm interval.
Q0 = α f
2.3. Storage capacity, outflow discharge and irrigation rate Tank storage capacity Vs is entirely used for peak flow control during the rainy season. The detention volume is reasonably expressed as a multiple of average rainfall volume ζ −1 at any development stage:
ζ −1
(6)
the upper limit of the outflow discharge from the reservoir is obtained by combining equations (5) and (6)
The characterization of the frequency distribution of rainfall is based on statistical analysis of records at a gauge stations. A definition of inter-event time (IETD) is selected to isolate storm events from continuum time series [1]. As IETD increases, the estimated mean of the isolated rainfall event volumes should equal the estimated variance. Once a suitable IETD is selected, statistics are performed on single rainfall volume duration and inter-storm interval [19]. Statistical analysis based on different IETDs results in different values of sample moments. It is impossible to observe an inter-storm interval b ≤ IETD on discretized rainfall records, thus Eq. (3) is defined for b ≥ IETD. Assessment of rainfall on small time-scales (from 1 to 10 min) is a prerequisite for urban runoff prediction in small urban catchments which typically have response times of less than 1 h [15], so that short IETDs would be more appropriate, although, too short IETDs would make the assumption that consecutive rain events are statistically independent questionable. The statistical analysis of site specific rainfall records and the influence of rainfall statistics on risk analysis are beyond the paper’s scope. Literature data are used here to characterize probability distributions (1)–(3) during the rainy and the growing season. The existence of two different eco-hydrological regimes supports the use of two different season-specific management strategies (Fig. 1). The rainy season priority objective is reducing the risk of overflow, the growing season priority objective is, at the same time, fulfillment of irrigation demand during the inter-storm interval, having a detention capacity for flood control. Quantitative risk analysis is performed independently for the two periods of the year, since hydrologic forcing and the tank management strategy are different, and may clarify to what extent water re-use can affect the risk of overflow.
Vs = imp β
(5)
(4)
where β is a design parameter depending on the catchment characteristics in the pre- and post-development scenarios. Volumes are expressed as equivalent depths across the whole catchment area. When vegetation is dormant, maximum outflow discharge Q is dictated by the site-specific storm-water drainage sizing criteria. In
Q = [1 + (α − 1) imp] f
(7)
α and β are design parameters which depend on design objectives, local regulation and site characteristics. Small impervious catchments with quick hydrological response times are expected to have higher α and β . For an urban catchment with a surface of a few hundred hectares, with reference to a return time of up to 100 years, and for f = 10 mm h−1 , α = 0.01 ÷ 2.5 and β = 1 ÷ 3 [16]. Typical stormwater recycling storage capacity has β = 0.2 ÷ 1 [22]. Irrigation avoids vegetation water stress and compensates evapotranspiration losses. Over-irrigation causes leakage and results in the waste of valuable water resources. Optimum irrigation rate i0 per unit surface depends on vegetation physiology, soil properties and climate [2]. If the storm-water volume collected during the growing season is not sufficient, the re-use irrigation rate is i ≤ i0 , and extra drinking water must be provided for irrigation. Assuming that the average rainfall volume during the growing season falling on the impervious area may be entirely collected and re-used, rainfall volume Vi available for irrigation is:
Vi = imp
1
ζ
TG
·
1
1 λ + ψ
= imp
λψ ζ λ+ψ
TG
(8)
where TG · λλ+ψψ is the expected number of storms during growing season TG . Under a very simplified assumption, average irrigation time Ti is estimated here as the average growing season dry time: 1
Ti = TG
ψ 1
1
λ + ψ
= TG
λ . λ+ψ
(9)
In the period between two consecutive rainfall events, trade-offs between immediate and future uses of water must be evaluated. The best management strategy should define an optimum irrigation rate which avoids large, long-lasting water shortages in the late interstorm period. In the following, two constant irrigation rates, i and i , are defined and compared with closed-form risk analysis solutions. If Vi is delivered in a period of time of Ti , the irrigation rate, expressed as the equivalent depth of water per unit time across the catchment surface, is i = i
i’ =
Vi = imp Ti
ψ . ζ
(10)
Alternatively, the irrigation rate is set at i = i , where:
i = Vs ψ = β i
(11)
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N. Ursino / Advances in Water Resources 83 (2015) 277–284 Table 1 Risk of overflow. Model parameters. Climate parameters (ζ , λ, ψ ) Rainy season
ζ R , λR , ψ R
Growing season
ζ G , λG , ψ G
imp Q0 + (1 − imp) · f (with infiltration) imp Q0 (without infiltration) i0 · (1 − imp) i’ = imp ψG ζG−1 (IS-1) i = β i (IS-2)
i corresponds to delivery of the maximum storage volume Vs = imp β ζ −1 in the average inter-storm period ψ −1 . 3. Risk analysis The risk and reliability of complex hydraulic systems and their components are here defined as the probability of failure and its complementary probability of success [23]. Assuming as certain that the network can transfer water from the impervious sub-catchment to the tank, and that the overflow divider performs correctly, only the tank is subject to failure.
R f = P[b ≥ Vs /Q] · P[h > Vs /imp] + P[b < Vs /Q] · P[h > Q b/imp]. (12) Equation (12) states that the risk of overflow Rf is the probability that rainfall volume imp h exceeds storage capacity Vs when interstorm interval b between two consecutive rainfall events is longer than draw-down time Vs Q −1 , or that it exceeds the empty fraction Q b of the storage volume if the second rainfall event occurs before draw-down is complete. Combining equations (12), (1), and (3), Rf may be expressed as a function of the average depth of rainfall and inter-storm interval. ∞ Vs Q
ψ e−ψ b
∞ Vs imp
ζ e−ζ h dh db +
Vs Q
IETD
ψ e−ψ b
∞ bQ imp
ζ e−ζ h dh db. (13)
Equation (13) is a stylized version of the risk analysis of [1], and may be further elaborated as follows:
Rf =
P[h ≤ Vs /imp] · P b > d h i−1
=
Ri
ψ ψ ζ ζ ψ imp ζQ e−IETD Q ( imp + Q ) + e−Vs ( imp + Q ) . ζ Q + ψ imp ζ Q + imp ψ
(14) Equation (14) with season-specific parameters (Table 1) is used to evaluate the risk of overflow during the growing and rainy seasons. 3.2. Risk of water shortage The aim of the growing season strategy is to retain storm-water in the reservoir, ready for re-use when irrigation demand is high. Retaining storm-water in the tank is convenient only if reducing the storage capacity as soon as possible is not so urgent as it is during the rainy season. During the growing season, an overflow is not only a failure of the drainage system but also the loss of water which could be re-used, and is thus an indirect cause of irrigation system failure.
+ P[h > Vs /imp] · P b < Vs i−1 /(1 − imp) Vs , IETD (1−imp)
For i <
According to [17] and [19], risk of overflow Rf is estimated, as the probability that either the rainfall volume is imp h > Vs or that two consecutive rainfall events occur in a short period of time. At the beginning of the second rainfall event, tank storage capacity is Vs < Vs and rainfall volume is imp h > Vs [18]. Under the most conservative assumption [24] that the reservoir is full at the end of the first of two consecutive rainfall events,
Risk of drought Ri is evaluated, under the most conservative assumption that the reservoir is empty at the beginning of the last rainfall event, the probability that rainfall volume imp h is less than storage capacity Vs and the time to the next rainfall event is greater than −1 −1 imp h (1 − imp) i−1 = imp d i−1 , with d = imp (1 − imp) , or that rainfall volume is greater than Vs but that the next rainfall event oc−1 curs after complete tank draw-down, i.e., Vs (1 − imp) i−1 .
Ri =
Vs imp IETD i d
ζ e−ζ h
∞ h d i
(15)
combining equations (15), (1) and (3), Ri may
be expressed as follows:
3.1. Risk of overflow
Rf =
Discharge (Q)
ψ e−ψ b db dh +
∞ Vs imp
ζ e−ζ h
∞ Vs i·(1−imp)
ψ e−ψ b db dh. (16)
By integration, Eq. (16) becomes the simple relationship:
Ri =
Vs iζ ψd · e−IETD (iζ +ψ d) + · e− imp i (iζ +ψ d) . iζ + ψ d iζ + ψ d
(17)
Depending on irrigation strategy (IS), in Eq. (17) the rate i is given by i , or i , according to Eqs. (10) and (11), respectively (in the analytical solutions), or by i0 · (1 − imp) (in numerical simulations). 4. Results In this Section, the results of dimensionless risk analysis are first presented for constant irrigation rates and different management strategies. Then tank performance is simulated numerically, under the assumption that irrigation volume is provided to compensate evapotranspiration losses, avoiding leakage. Land use change is viewed as a sequence of stationary scenarios with changing imp. Typical annual irrigation demand per surface unit of vegetated soil for open areas ranges between 200 and 800 mm, depending on climate, vegetation physiology and the type of irrigation system used. Daily evapotranspiration in Mediterranean-type climates is about 1 ÷ 5 mm d−1 . If TG = 5 months the estimated average rain volume falling during the growing season (Eq. (8)) is Vi = 180 mm, but intense rainfall events are often associated with substantial water losses due to runoff. This means that re-used storm-water may not fulfil irrigation demand, and more water for irrigation must be provided from the distribution system, if imp is not large enough. In Mediterranean-type climates, different values of ζ , λ and ψ may conveniently be attributed different values during the rainy season and during the vegetation growing season [32]. According to literature data ζR = 0.14 mm−1 ; ψR = 0.02 h−1 ; ζG = 0.16 mm−1 ; ψG = 0.008 h−1 ; λR = λG = 0.23 h−1 , where subscripts R and G refer to rainy and growing season parameters, respectively. 4.1. Dimensionless results of risk analysis – rainy season In order to derive generalized results of the risk analysis, Eqs. (17) and (14) may be rewritten as functions of few dimensionless groups
Rf =
e−E (F +1) + F · e−G (F +1) F +1
(18)
N. Ursino / Advances in Water Resources 83 (2015) 277–284
a
0.8
0.4
0.2
0.2
0.8
0 0.8 0 1
0.2
Rf
0.4
0.6
0.8
imp*=0.5
0.4
0.6
0.8
0 0.8 0 1
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
imp
1
0.2
0.4
0.6
imp
0.8
0 0 1
0.2
0.4
0.4
0.2
0.2
0.4
0.6
imp
0.8
=
F
=
G
=
(19)
IETD ψ
ζQ imp ψ Vs ψ Q iζ ψd Vs ψ i (1 − imp)
=
H
=
K
When vegetation is dormant, Ri = 0 and outflow rate Q is dictated by the channel protection design objective. Assuming that Eq. (4) holds, and that
Q = imp α f
(20)
according to a simplified version of Eq. (7), where no outflow is conveyed to the pervious sub-catchment, Rf may be rewritten as follows:
Rf =
ψ ·e
−IETD (ψ + f α ζ )
+ f αζ · ψ+ f αζ
β e− f α ζ (ψ + f α ζ )
.
(21)
Holding equation (20), Rf is p-independent, since:
ζαf ψ βψ . αζ
4
0 5 0
1
2
3
4
5
When vegetation is active, the average tank outflow in Eq. (14) is Q = i. In the following, the influence of irrigation strategy (IS) on Ri and Rf is analysed with reference to two theoretical constant values of i. Assuming that i = i (IS-1) according to Eq. (10)
F
=
1
G
=
β
H
=
1 − imp
K
=
Rf =
In view of possible scenarios of land use change, tank designers may take into account the expected future increase in the actual imp by setting Vs = imp∗ β ζ −1 in Eq. (18). In Fig. 2, Rf , according to Eq. (21), is plotted for different values of α (namely: α =0.01; 0.1; 1) and β = 1 and 5 (continuous line). The dashed lines show Rf as a function of imp for Vs = imp∗ β ζ −1 and , for imp∗ = 0.25, 0.5, 0.75, 1.
As expected, the risk of overflow decreases with increasing storage capacity (increasing β ). Increasing outflow discharge (increasing α ) reduces the risk of overflow due to short inter-storm interval between two consecutive events, although, when the draw down time becomes smaller than the IETD, no further reduction of Rf is appreciable. The sizing criteria for storm-water control (water treatment,
β 1 − imp
e−2 E + e−2 β 2
(24)
(1 − imp) · e−E imp(1+ 1−imp ) + e−β (1+ 1−imp ) 1
Ri =
1
2 − imp
.
(25)
Conversely, assuming that i = i (IS-2) according to Eq. (11)
β
F
=
G
=
1
H
=
β · (1 − imp)
K
=
1 1 − imp
Rf =
e−E (β +1) + β · e−(β +1) β +1
Ri =
β · (1 − p) · e−E imp(β + 1−imp ) + e−(β + 1−imp ) β · (1 − imp) + 1
(23)
f
imp∗ β ψ imp f α ζ
3
channel protection or groundwater recharge) dictates the upper limit of α . Rf does not change significantly for α > 1 (data not shown), whereas, when the tank is slowly emptied (α = 0.01) the risk of overflow is more affected by the IETD than by imp∗ and imp.
(22)
and
thus, G =
2
1
where:
E
1
4.2. Dimensionless results of risk analysis – growing season
H e−E d (H+1) + e−K (H+1) Ri = H+1
G=
0.6
Fig. 3. Risk of water scarcity Ri (dashed line) and risk of overflow Rf (circles) as a function of β for different developing stages (imp = 0, 0.25, 0.5, 0.75, 1) and IS.
Fig. 2. Risk of overflow Rf as a function of imp for different values of α and β . a: β = 1; b: β = 5. Continuous line: tank volume increasing with imp according to Eq. (4). ∗ ∗ Dashed line: fixed tank volume Vs = imp βζ −1 and imp = 0.25, 0.5, 0.75, 1.
F=
0.6
0 0
0 0
0.8
imp imp*=0.75 0.4
imp*=1.0.2
0.2
imp
0.8
imp*=0.25 0.4
IS-2
Ri , R f
0.6
0.6
0.6
b
IS-1
1
Rf
281
(26)
1
1
(27)
In Fig. 3, Rf and Ri are plotted vs β for different values of imp, namely imp = 0, 0.25, 0.5, 0.75, 1. Equations (26) and (27), which show IS-1, appear in the left panel, and (24) and (25) in the right panel (IS-2). Risk of water scarcity Ri (dashed line) decreases with imp, and approaches 0 when imp tends to 1. Risk of overflow Rf (circles) is not affected by the stage of development (as it does not depend on imp) under the assumption that both Vs and Q are linear functions of imp. In the left panel of Fig. 3 (IS-1): (1) the risk of overflow (circles) and the risk of water scarcity (dashed lines) evaluated for imp = 0 coincide; (2) when β increases, Rf and Ri approach irreducible limits, i.e., R f lim = 0.5 e−2E and Ri lim = e−E (imp+d) (1 − imp)/(2 − imp),
N. Ursino / Advances in Water Resources 83 (2015) 277–284
1 1
0.8
0.8
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.8 0.6
R f , Ri
20
5
1
15
IS-1
0 0 1 0.8 0.6
imp 0.2
0.4
0.60,67 0.8
0 1 0 1
imp 0.2
0.4
0.60,67 0.8
0.8
R f , Ri
0 0 1 1
10
5
imp 0.2
0.4
0.60,67 0.8
1
0 0
0.8
0.6
0.6
0.4
0.4
0.2
0.2 0 0
imp 0.2
0.4
0.6
0.8
0 0,95 0 1
1
0.2
imp 0.2
0.4
0.60,67 0.8
0 10
50
100
150
100
150
ET0
IS-2 0.4
h
Vs
imp 0.20,28 0.4
0.6
(1-imp)· nH
0.1 1
imp· h
282
0.8
1
Fig. 4. Risk of overflow during rainy season (continuous line); risk of drought (dashed lines) and risk of overflow (circles) during dry growing season, as a function of imp for different β and IS. Upper panels: irrigation rate i = min[i’ ; i0 · (1 − imp)]. Lower panels: irrigation rate i = min[i’’ ; i0 · (1 − imp)].
respectively; (3) the risk of water scarcity is already close to the asymptotic limit for β ≈ 1.5, meaning that larger storage capacity would not improve the reliability of the re-use system. This result encourages the use of medium-sized tanks (Vs ≈ 1.5 · ζ −1 ) for detention and re-use. In the right panel of Fig. 3, i ∝ β ∝ Vs and Rf may be reduced indefinitely at the cost of increasing the risk of water scarcity by increasing β , and thus Vs . When β ≈ 1.5, IS-1 is more reliable than IS-2 with respect to the irrigation objective and approximatively equally reliable with respect to flood control. Increasing β further does not lead to any additional benefit with IS-1, but to a lack of reliability of the re-use system with IS-2. 4.3. Sustainable water drainage and re-use in Mediterranean-type climates As land development increases, rainfall volume becomes Vi ≥ Ti · i0 and the irrigation rate may reach i = i0 . At the same time, Rf and Ri change. Fig. 4 shows the risks of overflow during the rainy season (continuous line), drought (dashed line) and overflow during the dry growing season (circles) as a function of imp, for α = 2 and β = 0.1, 1 and 5, corresponding to small, medium and large storage capacities, respectively. Re-use water is applied at a rate i = min[i0 · (1 − imp); i], where i0 = 0.02 f, and f = 10 mm h−1 . The irrigation rate is i = i (IS-1) in the upper panels of Fig. 4 and i = i (IS-2) in the lower panels; i and i are evaluated according to Eq. (10) and Eq. (11) respectively. Rainy season outflow discharge Q is calculated according to Eq. (7). Rf and Ri both decrease with β (increasing storage capacity Vs ) but obviously display opposite behavior as a function of imp because, as the land is developed, larger detention capacity is used as storage capacity for irrigation purposes, but the drainage system loses reliability, due to increased runoff. As imp increases, the system switches from: (a) almost constant risk of overflow (circles) and slowly decreasing risk of water scarcity (dashed line), to (b) rapidly increasing Rf and decreasing Ri . The switching point from (a) to (b) is dictated by i = i0 and occurs at imp = 0.67 when i = i (upper panels) and at imp values which are inversely proportional to β when i = i (lower panels). IS-1 leads to higher Rf and lower Ri than IS-2, when storage capacity is large (β = 5 upper and lower right panels), whereas the influence of the IS on Rf and Ri is reversed for small storage capacity (β = 0.1 left upper and lower panels).
0
50
TG
Fig. 5. Basic elements of tank and soil water balance.
4.4. Comparison between analytical and numerical simulation results – irrigation rate depends on soil saturation The closed-form solutions proposed in the previous ssections hold for constant irrigation rates. These rates are also linked to water availability and thus to tank design parameters, rather than to soil moisture and vegetation characteristics. The timing and amount of irrigation is typically dictated by climate and plant physiology, because irrigation water must be provided when plants need it. In order to validate the proposed closed-form solutions, the daily irrigation rate was evaluated by solving numerically vegetated soil moisture and tank water balances, on a daily time-scale. Fig. 5 shows the functions and concepts of the numerical model. Rainfall series for the growing season are generated stochastically, according to Eqs. (1)–(3), to simulate daily rainfall series of a statistically significant number of years. Rainfall imp h falling on the impervious subcatchment is stored in the re-use tank. Water volume in the re-use tank is V. If the available tank volume is Vs − V < imp h, excess rainfall is lost and the tank overflows. Rain falling on the vegetated subcatchment (1 − imp) h when soil saturation is S∗ ≤ S ≤ 1, in)h filtrates and provides additional soil moisture S = (1−imp , where nH nH is the amount of soil moisture that can be stored in the root zone. After rainfall, soil saturation is S + S ≤ 1 or 1. Excess rain falling on the vegetated subcatchment is lost. Evapotranspiration ET0 S increases linearly with S (to account for vegetation water stress) and varies during the growing season due to the vegetation stage of development, according to Fig. 5. When S = S∗ , plants cannot extract water from the soil and either irrigation is provided by the re-use tank or the plants undergo a water crisis. The irrigation volume is the minimum between the required nH and the available V/(1 − imp). After V irrigation soil moisture is S = 1 if V ≥ nH(1 − imp), or S = nH (1−imp )
if irrigation volume is V/(1 − imp) < nH. The frequency of tank overflow is estimated as the average ratio between the numbers of overflows and rainy days in a growing season. The frequency of water scarcity is estimated as the average ratio between the number of days in a growing season when S = S∗ and the number of days in TG . Risk of overflow (dashed line), risk of drought (continuous line) and the average simulated frequencies of overflow (squares) and water scarcity (circles) are plotted in Fig. 6 vs imp for β = 0.1; 1; 5, ET0 = 5 mm d−1 and nH = 360; 60 mm, corresponding to deep and shallow root zones, respectively. Risks of overflow and drought correspond to those shown in Fig. 4. In this case study, i0 ≈ ET0 . The risk of drought is lower and closer to the analytical prediction when nH = 60 mm. When nH = 360 mm the analytical solution greatly underestimates the risk of drought. The
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to drought-tolerant or shallow-rooted plants, may reduce the risk of drought with less effect on the risk of overflow. Optimal reservoir release policies generally depend on initial growing season storage and growing season inflow [10,21,36]. However, in this case study, initial storage during the growing season has less effect on the risk of overflow and thus justifies the use of two independent management strategies during the rainy and growing seasons.
References 0.2
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Fig. 6. Risk of overflow Rf (dashed line) and water scarcity Ri (continuous line) evaluated as in Fig. 4, and average simulated frequency of overflow (squares) and water scarcity (circles) vs imp (soil-saturation-dependent irrigation rate). ET0 = 5 mm d−1 ; a: nH = 360 mm; b: nH = 60 mm. Filled symbols: tank full at beginning of the growing season. Empty symbols: tank empty at beginning of the growing season.
risk of overflow is in general overestimated by the analytical model, and major differences arise after switching point i = i0 when i < i0 . The analytical model is in fact based on the over-conservative assumption that the tank is full at the end of the first of two consecutive rainfall events, whereas water is used for irrigation after each rain event. The simulation was run twice for all sets of model parameters. Under assumption (i), the tank was full at the beginning of the rainy season (filled symbols) and, under (ii), it was completely empty (empty symbols). The results show that water storage at the beginning of the rainy season has less effect on the results: that is, of the contrasting aims of having water stored in the tank at the end of the rainy season and emptying it as soon as possible for safety reasons, the latter may be privileged for imp up to 0.9.
5. Discussion and conclusions Compared with sewage water re-use, collecting and re-using storm-water and by so doing achieving ecological, social and economic goals, requires the development of minor treatment technologies [14,22]. However, the availability of water resources may become a major limiting factor to building multipurpose tanks, and certainly deserves attention. Climate, the demand for irrigation dictated by the ecosystem composition [13] and irrigation management strategies [21,36] all affect social and economic goals as well as the ecosystem service that may be achieved. Risk analysis [1], expressed in the closed form for the risk of overflow during the rainy and growing seasons and for the risk of water shortage during the growing season, was derived for a multipurpose storm-water tank. The interrelation between risk of overflow and of water scarcity is demonstrated here for the first time with an analytical probabilistic model. Results encourage the use of medium-sized storage capacities, because larger multipurpose tanks would not improve the reliability of the re-use system. Hedging rules reduce deliveries in order to retain water in storage, and reduce the risk of large water shortages at the cost of more frequent small-scale ones [10]. Although the model presented here is conceptually simple, it shows whether irrigation demand can be satisfied during the vegetation growing season and, at the same time, whether the drainage system is reliable. To achieve these contrasting goals is a challenging task, involving the right choice of appropriate urban water conservation methods [13]. In the case study presented here, even though the closed form solution of probabilistic analysis overestimates the risk of overflow and underestimates that of drought, it suggests that slow, parsimonious storm-water release
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Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres
Risk analysis of sustainable urban drainage and irrigation Nadia Ursino∗ Department ICEA, Civil Architectural and Environmental Engineering, University of Padova, via Loredan 20, Padova 35131, Italy
a r t i c l e
i n f o
Article history: Received 25 April 2014 Revised 23 June 2015 Accepted 26 June 2015 Available online 30 June 2015 Keywords: Landscape irrigation Detention Sustainability Reliability analysis Low impact development BMP
a b s t r a c t Urbanization, by creating extended impervious areas, to the detriment of vegetated ones, may have an undesirable influence on the water and energy balances of urban environments. The storage and infiltration capacity of the drainage system lessens the negative influence of urbanization, and vegetated areas help to re-establish pre-development environmental conditions. Resource limitation, climate, leading to increasing water scarcity, demographic and socio-institutional shifts promote more integrated water management. Storm-water harvesting for landscape irrigation mitigates possible water restrictions for the urban population in drought scenarios. A new probabilistic model for sustainable rainfall drainage, storage and re-use systems was implemented in this study. Risk analysis of multipurpose storage capacities was generalized by the use of only a few dimensionless parameters and applied to a case study in a Mediterranean-type climate, although the applicability of the model is not restricted to any particular climatic type.
1. Introduction Flood and drought have social, economic and environmental consequences which are exacerbated by urban development and climate change [28]. Urbanization, by creating extended impervious areas, to the detriment of vegetated ones, has an undesirable influence on the water and energy balances of urban environments [15], increasing peak discharge and total volume of surface runoff and reducing groundwater recharge. An important concern of urban hydrologists is thus to restore (at least partially) the pre-development water balance, thus, managing urban water in a more sustainable way. Nevertheless, since the increasing scarcity of clean water in many urban areas leads to intense competition for its use [4], the aim of sustainable urban water management cannot be restricted simply to restoring pre-development runoff conditions: rather, it should continue to evolve as urbanization increases and aim at better integrating urban, environmental, agricultural and industrial water use. There is a need to ascertain whether integrated strategies can achieve social and economic goals as well as good-quality ecosystem service and maintenance [22]. When rainfall falls over impervious land and does not infiltrate, it runs off becoming storm-water. Storm-water best management practices (BMP) preserve natural landscape features, minimize effective imperviousness, and treat storm-water as a resource rather than simply letting it go to waste [33]. The design criteria and performance of
∗
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© 2015 Elsevier Ltd. All rights reserved.
BMP have gained much coverage in the technical literature [8,9,37]. Detention basins are effective controls which can attenuate peak discharge [31]. When they act as infiltration systems, or in series with such systems, they can help to restore the pre-development water balance, enhance groundwater recharge, delay and reduce surface runoff, and thus protect receiving water bodies, and sewer systems [11,38]. Plants growing on natural or artificial soils increase their infiltration capacity by altering the soil structure of the vadose zone [26]. Bioretention area must be sufficiently permeable to infiltrate storm-water and have adequate retention capacity to support healthy vegetation growth. Engineered soil mix may achieve both objectives. Vegetation is a crucial element in the mass and energy flux locally at the soil-vegetation-atmosphere interface, and in the earth’s climate in general [27,34]. Maintaining or restoring vegetated areas in urban environments may further help to re-establish the pre-development water and energy balances. Plants can control the urban microclimate through transpiration and soil sheltering, reducing rises in urban temperatures and increasing the “liveability” of cities [12]. Nevertheless, vegetated urban landscapes consume water and are drought-sensitive, and thus pose the problem of water saving in landscape irrigation [13]. Frequency of irrigation depends on soil properties, climate and vegetation type. Applied water should never exceed the water holding capacity of the soil. Typically, deep and infrequent irrigation favors root elongation, health and vigor of turfgrass. Irrigation of vegetated bioretention area must minimize water use and avoid underdrain flow during dry weather. Although storm-water harvesting and landscape irrigation systems may require pumping stations, the additional cost of energy
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distribution is expected to be offset by energy savings on cooling systems [30]. Storm-water harvesting and re-use schemes for non-potable purposes (such as landscape irrigation) may mitigate the risk of water restrictions for urban populations. Treated waste-water may also be used for landscape irrigation according to many local regulation, although the re-use of storm-water collected from impervious land, which is discussed here, have higher public acceptance than alternative water supplies such as waste-water recycling [7]. For example, swale adjacent to impervious areas can support grass growth on golf courses in arid areas [35], and properly designed, maintained, and treated storm-water harvested from roofs is low-cost, often less polluted than other sources of water in urban catchments, and may integrate existing water supplies [29]. Storm-water harvesting reduces growers’ dependence on the municipal water supply [6], however, storm-water re-use may meet water requirements for urban horticulture, if a sufficient volume is available during the growing season of the vegetation. It is not clear if, in climates of Mediterranean type, characterized by winter rains alternating with summer drought and high temperatures, which are often exacerbated by urbanization [25], landscape irrigation with reuse storm-water could be successful. In climates of Mediterranean type, due to the fact that rainfall and air temperature are out of phase and that the end of the rainy season corresponds to the beginning of the growing season, rain-fed vegetation often suffers water stress due to the long inter-storm interval between successive rainfall events in the dry-warm period. Collecting and reusing sufficient quantities of water of the required quality when the demand for irrigation is high is expected to be a problem, due to rainfall scarcity and variability [14]. The hydraulic design of a multipurpose storage tank aims at two contrasting goals: (i) to provide detention capacity and peak discharge reduction and (ii) to ensure a sufficient re-use water for irrigation. The first aim demands empty tanks at the beginning of each rain event. The second is to have the largest possible amount of water stored in the tanks during the inter-storm interval. If the system fails, this is due to overflow, when the volume of rainfall exceeds available storage capacity, or to water scarcity, when the stored rainfall volume is drawn before the next rainfall event. The probability of system failure is called risk. The main aim of this study is to examine if multipurpose tanks can store and provide sufficient storm-water volumes for landscape irrigation when required, whether this implies a significant increase in the risk of overflow, and to what extent the risk of drought or, conversely, overflow depends on climate, hydrology and/or management strategies. The probability approach [1] provides analytical equations to estimate the risk of failure of detention facilities and other elements of the urban drainage system in question [17,18]. Several studies have applied a probabilistic approach to BMP failure risk analysis and demonstrate that it matches many physically based methods [16,19,20,38,39]. The risk analysis of water-saving systems has so far received less attention than that of storage systems. Whether the storage capacity of a detention basin can also ensure a sufficient water supply for irrigation purposes, and whether multipurpose management strategies can affect the risk of overflow have not yet been studied on a probabilistic basis. A new integrated risk model was implemented in this study. Section 2 describes the conceptual model of a drainage, detention and re-use system. In Section 3, closed-form solutions are derived. In Section 4 the risk analysis is generalized by expressing risk as a function of a few dimensionless groups, and the influence of tank management strategy is evaluated analytically and validated numerically. In Section 5 the results are discussed and problems in sustainable water management are discussed.
RAINY SEASON
infiltration (1-imp)·f imp
1-imp Vs
controlled outflow imp·Q0
runoff overflow GROWING SEASON
irrigation (1-imp)·i
imp
runoff
1-imp Vi
overflow
Fig. 1. Conceptual model of drainage system.
2. Model 2.1. Conceptual model of drainage system A conceptual model is formulated for an urban catchment comprising one pervious (vegetated) and one completely impervious subcatchment area (Fig. 1). Impermeability imp is the ratio between the impervious area and the whole catchment area and identifies the stage of development of the catchment. The components of the drainage system (Fig. 1) are: (i) a network which transfers the rainfall volume falling on the impervious subcatchment instantaneously to (ii) an overflow divider which diverts flow to storage or overflow discharge; (iii) a multipurpose tank; (iv) a distribution system drawing water from the tank to the pervious sub-catchment at a constant rate in the period of time between two rainfall events; (v) an outflow pipe taking water, at controlled rate, to discharge. The network storage capacity should influence the rainfall-runoff transformation, which is kept as simple as possible, and indirectly the probability of overflow. However, the increasing storage capacity of the network with development, and thus imp, is neglected here for the sake of simplicity, possibly overestimating the risk of overflow. The tank is used for detention during the rainy season and for detention and landscape irrigation during the vegetation growing season. During the rainy season the pervious sub-catchment achieves in-place infiltration of storm water. The flow rate per unit surface in the infiltration system may reach soil infiltration capacity f during the rainy season, when the additional controlled outflow is Q and equals irrigation rate i during the growing season. The tank must be emptied as soon as possible after each rainfall event during the rainy season. Conversely, during the growing season, storm-water is stored for longer periods and re-used for irrigation at rate i < f . There is no controlled outflow during the growing season. This practice obviously increases the probability that storage capacity will not be available at the beginning of rare and intense showers during the growing season. Every time the rainfall volume exceeds the available storage capacity, the system fails. In this Section, the related probability of failure is called risk of overflow. The risk of water scarcity or drought is the probability of the tank being emptied during the irrigation season, before further rain falls (before the end of the inter-storm interval).
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2.2. Rainfall distribution The meteorological input to the catchment is assumed to be represented by the probability density functions of three independent random variables: rainfall volume (h); duration (t) and inter-storm interval (b). Following the approach proposed by [17] h, t and b are modeled as random variables with exponential probability density functions. This assumption simplifies the mathematical tractability of risk analysis and allows its closed form analytical solution, even though there may be more suitable marginal distribution functions of the input rainfall [5]. The probability density functions of rainfall characteristics are:
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general, Q = imp Q0 where Q0 is the maximum discharge ensuring channel protection, overbank flood protection or extreme flood protection. Flow discharge is expressed as volume per unit time, divided by catchment area. The aim of tank management during the rainy season is to empty the tank quickly to create space for the water of the next storm event. In order to accelerate the draw-down of Vs , part of the outflow discharge may be conveyed to the pervious sub-catchment which functions as an infiltration system. Soils that allow the realization of infiltration systems are sand, loamy sand sandy loam and loam [3], vegetated bioretention basins which must achieve high infiltration rate and water holding capacity, are often made out of engineered soil layers. Maximum outflow discharge of urban catchments with an infiltration system downward a storage tank is:
f h = ζ e−ζ h
(1)
ft = λ e−λt
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Q = imp · Q0 + (1 − imp) · f
f b = ψ e−ψ b .
(3)
where f is the infiltration capacity of the pervious sub-catchment. The infiltration capacity f of the sub-catchment is selected as scale factor of the prescribed outflow discharge Q0 ,
The distribution parameters are: ζ = inverse of expected value of rainfall depth, λ = inverse of expected value of rainfall duration, ψ = inverse of expected value of rainfall inter-storm interval.
Q0 = α f
2.3. Storage capacity, outflow discharge and irrigation rate Tank storage capacity Vs is entirely used for peak flow control during the rainy season. The detention volume is reasonably expressed as a multiple of average rainfall volume ζ −1 at any development stage:
ζ −1
(6)
the upper limit of the outflow discharge from the reservoir is obtained by combining equations (5) and (6)
The characterization of the frequency distribution of rainfall is based on statistical analysis of records at a gauge stations. A definition of inter-event time (IETD) is selected to isolate storm events from continuum time series [1]. As IETD increases, the estimated mean of the isolated rainfall event volumes should equal the estimated variance. Once a suitable IETD is selected, statistics are performed on single rainfall volume duration and inter-storm interval [19]. Statistical analysis based on different IETDs results in different values of sample moments. It is impossible to observe an inter-storm interval b ≤ IETD on discretized rainfall records, thus Eq. (3) is defined for b ≥ IETD. Assessment of rainfall on small time-scales (from 1 to 10 min) is a prerequisite for urban runoff prediction in small urban catchments which typically have response times of less than 1 h [15], so that short IETDs would be more appropriate, although, too short IETDs would make the assumption that consecutive rain events are statistically independent questionable. The statistical analysis of site specific rainfall records and the influence of rainfall statistics on risk analysis are beyond the paper’s scope. Literature data are used here to characterize probability distributions (1)–(3) during the rainy and the growing season. The existence of two different eco-hydrological regimes supports the use of two different season-specific management strategies (Fig. 1). The rainy season priority objective is reducing the risk of overflow, the growing season priority objective is, at the same time, fulfillment of irrigation demand during the inter-storm interval, having a detention capacity for flood control. Quantitative risk analysis is performed independently for the two periods of the year, since hydrologic forcing and the tank management strategy are different, and may clarify to what extent water re-use can affect the risk of overflow.
Vs = imp β
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(4)
where β is a design parameter depending on the catchment characteristics in the pre- and post-development scenarios. Volumes are expressed as equivalent depths across the whole catchment area. When vegetation is dormant, maximum outflow discharge Q is dictated by the site-specific storm-water drainage sizing criteria. In
Q = [1 + (α − 1) imp] f
(7)
α and β are design parameters which depend on design objectives, local regulation and site characteristics. Small impervious catchments with quick hydrological response times are expected to have higher α and β . For an urban catchment with a surface of a few hundred hectares, with reference to a return time of up to 100 years, and for f = 10 mm h−1 , α = 0.01 ÷ 2.5 and β = 1 ÷ 3 [16]. Typical stormwater recycling storage capacity has β = 0.2 ÷ 1 [22]. Irrigation avoids vegetation water stress and compensates evapotranspiration losses. Over-irrigation causes leakage and results in the waste of valuable water resources. Optimum irrigation rate i0 per unit surface depends on vegetation physiology, soil properties and climate [2]. If the storm-water volume collected during the growing season is not sufficient, the re-use irrigation rate is i ≤ i0 , and extra drinking water must be provided for irrigation. Assuming that the average rainfall volume during the growing season falling on the impervious area may be entirely collected and re-used, rainfall volume Vi available for irrigation is:
Vi = imp
1
ζ
TG
·
1
1 λ + ψ
= imp
λψ ζ λ+ψ
TG
(8)
where TG · λλ+ψψ is the expected number of storms during growing season TG . Under a very simplified assumption, average irrigation time Ti is estimated here as the average growing season dry time: 1
Ti = TG
ψ 1
1
λ + ψ
= TG
λ . λ+ψ
(9)
In the period between two consecutive rainfall events, trade-offs between immediate and future uses of water must be evaluated. The best management strategy should define an optimum irrigation rate which avoids large, long-lasting water shortages in the late interstorm period. In the following, two constant irrigation rates, i and i , are defined and compared with closed-form risk analysis solutions. If Vi is delivered in a period of time of Ti , the irrigation rate, expressed as the equivalent depth of water per unit time across the catchment surface, is i = i
i’ =
Vi = imp Ti
ψ . ζ
(10)
Alternatively, the irrigation rate is set at i = i , where:
i = Vs ψ = β i
(11)
280
N. Ursino / Advances in Water Resources 83 (2015) 277–284 Table 1 Risk of overflow. Model parameters. Climate parameters (ζ , λ, ψ ) Rainy season
ζ R , λR , ψ R
Growing season
ζ G , λG , ψ G
imp Q0 + (1 − imp) · f (with infiltration) imp Q0 (without infiltration) i0 · (1 − imp) i’ = imp ψG ζG−1 (IS-1) i = β i (IS-2)
i corresponds to delivery of the maximum storage volume Vs = imp β ζ −1 in the average inter-storm period ψ −1 . 3. Risk analysis The risk and reliability of complex hydraulic systems and their components are here defined as the probability of failure and its complementary probability of success [23]. Assuming as certain that the network can transfer water from the impervious sub-catchment to the tank, and that the overflow divider performs correctly, only the tank is subject to failure.
R f = P[b ≥ Vs /Q] · P[h > Vs /imp] + P[b < Vs /Q] · P[h > Q b/imp]. (12) Equation (12) states that the risk of overflow Rf is the probability that rainfall volume imp h exceeds storage capacity Vs when interstorm interval b between two consecutive rainfall events is longer than draw-down time Vs Q −1 , or that it exceeds the empty fraction Q b of the storage volume if the second rainfall event occurs before draw-down is complete. Combining equations (12), (1), and (3), Rf may be expressed as a function of the average depth of rainfall and inter-storm interval. ∞ Vs Q
ψ e−ψ b
∞ Vs imp
ζ e−ζ h dh db +
Vs Q
IETD
ψ e−ψ b
∞ bQ imp
ζ e−ζ h dh db. (13)
Equation (13) is a stylized version of the risk analysis of [1], and may be further elaborated as follows:
Rf =
P[h ≤ Vs /imp] · P b > d h i−1
=
Ri
ψ ψ ζ ζ ψ imp ζQ e−IETD Q ( imp + Q ) + e−Vs ( imp + Q ) . ζ Q + ψ imp ζ Q + imp ψ
(14) Equation (14) with season-specific parameters (Table 1) is used to evaluate the risk of overflow during the growing and rainy seasons. 3.2. Risk of water shortage The aim of the growing season strategy is to retain storm-water in the reservoir, ready for re-use when irrigation demand is high. Retaining storm-water in the tank is convenient only if reducing the storage capacity as soon as possible is not so urgent as it is during the rainy season. During the growing season, an overflow is not only a failure of the drainage system but also the loss of water which could be re-used, and is thus an indirect cause of irrigation system failure.
+ P[h > Vs /imp] · P b < Vs i−1 /(1 − imp) Vs , IETD (1−imp)
For i <
According to [17] and [19], risk of overflow Rf is estimated, as the probability that either the rainfall volume is imp h > Vs or that two consecutive rainfall events occur in a short period of time. At the beginning of the second rainfall event, tank storage capacity is Vs < Vs and rainfall volume is imp h > Vs [18]. Under the most conservative assumption [24] that the reservoir is full at the end of the first of two consecutive rainfall events,
Risk of drought Ri is evaluated, under the most conservative assumption that the reservoir is empty at the beginning of the last rainfall event, the probability that rainfall volume imp h is less than storage capacity Vs and the time to the next rainfall event is greater than −1 −1 imp h (1 − imp) i−1 = imp d i−1 , with d = imp (1 − imp) , or that rainfall volume is greater than Vs but that the next rainfall event oc−1 curs after complete tank draw-down, i.e., Vs (1 − imp) i−1 .
Ri =
Vs imp IETD i d
ζ e−ζ h
∞ h d i
(15)
combining equations (15), (1) and (3), Ri may
be expressed as follows:
3.1. Risk of overflow
Rf =
Discharge (Q)
ψ e−ψ b db dh +
∞ Vs imp
ζ e−ζ h
∞ Vs i·(1−imp)
ψ e−ψ b db dh. (16)
By integration, Eq. (16) becomes the simple relationship:
Ri =
Vs iζ ψd · e−IETD (iζ +ψ d) + · e− imp i (iζ +ψ d) . iζ + ψ d iζ + ψ d
(17)
Depending on irrigation strategy (IS), in Eq. (17) the rate i is given by i , or i , according to Eqs. (10) and (11), respectively (in the analytical solutions), or by i0 · (1 − imp) (in numerical simulations). 4. Results In this Section, the results of dimensionless risk analysis are first presented for constant irrigation rates and different management strategies. Then tank performance is simulated numerically, under the assumption that irrigation volume is provided to compensate evapotranspiration losses, avoiding leakage. Land use change is viewed as a sequence of stationary scenarios with changing imp. Typical annual irrigation demand per surface unit of vegetated soil for open areas ranges between 200 and 800 mm, depending on climate, vegetation physiology and the type of irrigation system used. Daily evapotranspiration in Mediterranean-type climates is about 1 ÷ 5 mm d−1 . If TG = 5 months the estimated average rain volume falling during the growing season (Eq. (8)) is Vi = 180 mm, but intense rainfall events are often associated with substantial water losses due to runoff. This means that re-used storm-water may not fulfil irrigation demand, and more water for irrigation must be provided from the distribution system, if imp is not large enough. In Mediterranean-type climates, different values of ζ , λ and ψ may conveniently be attributed different values during the rainy season and during the vegetation growing season [32]. According to literature data ζR = 0.14 mm−1 ; ψR = 0.02 h−1 ; ζG = 0.16 mm−1 ; ψG = 0.008 h−1 ; λR = λG = 0.23 h−1 , where subscripts R and G refer to rainy and growing season parameters, respectively. 4.1. Dimensionless results of risk analysis – rainy season In order to derive generalized results of the risk analysis, Eqs. (17) and (14) may be rewritten as functions of few dimensionless groups
Rf =
e−E (F +1) + F · e−G (F +1) F +1
(18)
N. Ursino / Advances in Water Resources 83 (2015) 277–284
a
0.8
0.4
0.2
0.2
0.8
0 0.8 0 1
0.2
Rf
0.4
0.6
0.8
imp*=0.5
0.4
0.6
0.8
0 0.8 0 1
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
imp
1
0.2
0.4
0.6
imp
0.8
0 0 1
0.2
0.4
0.4
0.2
0.2
0.4
0.6
imp
0.8
=
F
=
G
=
(19)
IETD ψ
ζQ imp ψ Vs ψ Q iζ ψd Vs ψ i (1 − imp)
=
H
=
K
When vegetation is dormant, Ri = 0 and outflow rate Q is dictated by the channel protection design objective. Assuming that Eq. (4) holds, and that
Q = imp α f
(20)
according to a simplified version of Eq. (7), where no outflow is conveyed to the pervious sub-catchment, Rf may be rewritten as follows:
Rf =
ψ ·e
−IETD (ψ + f α ζ )
+ f αζ · ψ+ f αζ
β e− f α ζ (ψ + f α ζ )
.
(21)
Holding equation (20), Rf is p-independent, since:
ζαf ψ βψ . αζ
4
0 5 0
1
2
3
4
5
When vegetation is active, the average tank outflow in Eq. (14) is Q = i. In the following, the influence of irrigation strategy (IS) on Ri and Rf is analysed with reference to two theoretical constant values of i. Assuming that i = i (IS-1) according to Eq. (10)
F
=
1
G
=
β
H
=
1 − imp
K
=
Rf =
In view of possible scenarios of land use change, tank designers may take into account the expected future increase in the actual imp by setting Vs = imp∗ β ζ −1 in Eq. (18). In Fig. 2, Rf , according to Eq. (21), is plotted for different values of α (namely: α =0.01; 0.1; 1) and β = 1 and 5 (continuous line). The dashed lines show Rf as a function of imp for Vs = imp∗ β ζ −1 and , for imp∗ = 0.25, 0.5, 0.75, 1.
As expected, the risk of overflow decreases with increasing storage capacity (increasing β ). Increasing outflow discharge (increasing α ) reduces the risk of overflow due to short inter-storm interval between two consecutive events, although, when the draw down time becomes smaller than the IETD, no further reduction of Rf is appreciable. The sizing criteria for storm-water control (water treatment,
β 1 − imp
e−2 E + e−2 β 2
(24)
(1 − imp) · e−E imp(1+ 1−imp ) + e−β (1+ 1−imp ) 1
Ri =
1
2 − imp
.
(25)
Conversely, assuming that i = i (IS-2) according to Eq. (11)
β
F
=
G
=
1
H
=
β · (1 − imp)
K
=
1 1 − imp
Rf =
e−E (β +1) + β · e−(β +1) β +1
Ri =
β · (1 − p) · e−E imp(β + 1−imp ) + e−(β + 1−imp ) β · (1 − imp) + 1
(23)
f
imp∗ β ψ imp f α ζ
3
channel protection or groundwater recharge) dictates the upper limit of α . Rf does not change significantly for α > 1 (data not shown), whereas, when the tank is slowly emptied (α = 0.01) the risk of overflow is more affected by the IETD than by imp∗ and imp.
(22)
and
thus, G =
2
1
where:
E
1
4.2. Dimensionless results of risk analysis – growing season
H e−E d (H+1) + e−K (H+1) Ri = H+1
G=
0.6
Fig. 3. Risk of water scarcity Ri (dashed line) and risk of overflow Rf (circles) as a function of β for different developing stages (imp = 0, 0.25, 0.5, 0.75, 1) and IS.
Fig. 2. Risk of overflow Rf as a function of imp for different values of α and β . a: β = 1; b: β = 5. Continuous line: tank volume increasing with imp according to Eq. (4). ∗ ∗ Dashed line: fixed tank volume Vs = imp βζ −1 and imp = 0.25, 0.5, 0.75, 1.
F=
0.6
0 0
0 0
0.8
imp imp*=0.75 0.4
imp*=1.0.2
0.2
imp
0.8
imp*=0.25 0.4
IS-2
Ri , R f
0.6
0.6
0.6
b
IS-1
1
Rf
281
(26)
1
1
(27)
In Fig. 3, Rf and Ri are plotted vs β for different values of imp, namely imp = 0, 0.25, 0.5, 0.75, 1. Equations (26) and (27), which show IS-1, appear in the left panel, and (24) and (25) in the right panel (IS-2). Risk of water scarcity Ri (dashed line) decreases with imp, and approaches 0 when imp tends to 1. Risk of overflow Rf (circles) is not affected by the stage of development (as it does not depend on imp) under the assumption that both Vs and Q are linear functions of imp. In the left panel of Fig. 3 (IS-1): (1) the risk of overflow (circles) and the risk of water scarcity (dashed lines) evaluated for imp = 0 coincide; (2) when β increases, Rf and Ri approach irreducible limits, i.e., R f lim = 0.5 e−2E and Ri lim = e−E (imp+d) (1 − imp)/(2 − imp),
N. Ursino / Advances in Water Resources 83 (2015) 277–284
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0.8
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0.4
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0.2
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0.8 0.6
R f , Ri
20
5
1
15
IS-1
0 0 1 0.8 0.6
imp 0.2
0.4
0.60,67 0.8
0 1 0 1
imp 0.2
0.4
0.60,67 0.8
0.8
R f , Ri
0 0 1 1
10
5
imp 0.2
0.4
0.60,67 0.8
1
0 0
0.8
0.6
0.6
0.4
0.4
0.2
0.2 0 0
imp 0.2
0.4
0.6
0.8
0 0,95 0 1
1
0.2
imp 0.2
0.4
0.60,67 0.8
0 10
50
100
150
100
150
ET0
IS-2 0.4
h
Vs
imp 0.20,28 0.4
0.6
(1-imp)· nH
0.1 1
imp· h
282
0.8
1
Fig. 4. Risk of overflow during rainy season (continuous line); risk of drought (dashed lines) and risk of overflow (circles) during dry growing season, as a function of imp for different β and IS. Upper panels: irrigation rate i = min[i’ ; i0 · (1 − imp)]. Lower panels: irrigation rate i = min[i’’ ; i0 · (1 − imp)].
respectively; (3) the risk of water scarcity is already close to the asymptotic limit for β ≈ 1.5, meaning that larger storage capacity would not improve the reliability of the re-use system. This result encourages the use of medium-sized tanks (Vs ≈ 1.5 · ζ −1 ) for detention and re-use. In the right panel of Fig. 3, i ∝ β ∝ Vs and Rf may be reduced indefinitely at the cost of increasing the risk of water scarcity by increasing β , and thus Vs . When β ≈ 1.5, IS-1 is more reliable than IS-2 with respect to the irrigation objective and approximatively equally reliable with respect to flood control. Increasing β further does not lead to any additional benefit with IS-1, but to a lack of reliability of the re-use system with IS-2. 4.3. Sustainable water drainage and re-use in Mediterranean-type climates As land development increases, rainfall volume becomes Vi ≥ Ti · i0 and the irrigation rate may reach i = i0 . At the same time, Rf and Ri change. Fig. 4 shows the risks of overflow during the rainy season (continuous line), drought (dashed line) and overflow during the dry growing season (circles) as a function of imp, for α = 2 and β = 0.1, 1 and 5, corresponding to small, medium and large storage capacities, respectively. Re-use water is applied at a rate i = min[i0 · (1 − imp); i], where i0 = 0.02 f, and f = 10 mm h−1 . The irrigation rate is i = i (IS-1) in the upper panels of Fig. 4 and i = i (IS-2) in the lower panels; i and i are evaluated according to Eq. (10) and Eq. (11) respectively. Rainy season outflow discharge Q is calculated according to Eq. (7). Rf and Ri both decrease with β (increasing storage capacity Vs ) but obviously display opposite behavior as a function of imp because, as the land is developed, larger detention capacity is used as storage capacity for irrigation purposes, but the drainage system loses reliability, due to increased runoff. As imp increases, the system switches from: (a) almost constant risk of overflow (circles) and slowly decreasing risk of water scarcity (dashed line), to (b) rapidly increasing Rf and decreasing Ri . The switching point from (a) to (b) is dictated by i = i0 and occurs at imp = 0.67 when i = i (upper panels) and at imp values which are inversely proportional to β when i = i (lower panels). IS-1 leads to higher Rf and lower Ri than IS-2, when storage capacity is large (β = 5 upper and lower right panels), whereas the influence of the IS on Rf and Ri is reversed for small storage capacity (β = 0.1 left upper and lower panels).
0
50
TG
Fig. 5. Basic elements of tank and soil water balance.
4.4. Comparison between analytical and numerical simulation results – irrigation rate depends on soil saturation The closed-form solutions proposed in the previous ssections hold for constant irrigation rates. These rates are also linked to water availability and thus to tank design parameters, rather than to soil moisture and vegetation characteristics. The timing and amount of irrigation is typically dictated by climate and plant physiology, because irrigation water must be provided when plants need it. In order to validate the proposed closed-form solutions, the daily irrigation rate was evaluated by solving numerically vegetated soil moisture and tank water balances, on a daily time-scale. Fig. 5 shows the functions and concepts of the numerical model. Rainfall series for the growing season are generated stochastically, according to Eqs. (1)–(3), to simulate daily rainfall series of a statistically significant number of years. Rainfall imp h falling on the impervious subcatchment is stored in the re-use tank. Water volume in the re-use tank is V. If the available tank volume is Vs − V < imp h, excess rainfall is lost and the tank overflows. Rain falling on the vegetated subcatchment (1 − imp) h when soil saturation is S∗ ≤ S ≤ 1, in)h filtrates and provides additional soil moisture S = (1−imp , where nH nH is the amount of soil moisture that can be stored in the root zone. After rainfall, soil saturation is S + S ≤ 1 or 1. Excess rain falling on the vegetated subcatchment is lost. Evapotranspiration ET0 S increases linearly with S (to account for vegetation water stress) and varies during the growing season due to the vegetation stage of development, according to Fig. 5. When S = S∗ , plants cannot extract water from the soil and either irrigation is provided by the re-use tank or the plants undergo a water crisis. The irrigation volume is the minimum between the required nH and the available V/(1 − imp). After V irrigation soil moisture is S = 1 if V ≥ nH(1 − imp), or S = nH (1−imp )
if irrigation volume is V/(1 − imp) < nH. The frequency of tank overflow is estimated as the average ratio between the numbers of overflows and rainy days in a growing season. The frequency of water scarcity is estimated as the average ratio between the number of days in a growing season when S = S∗ and the number of days in TG . Risk of overflow (dashed line), risk of drought (continuous line) and the average simulated frequencies of overflow (squares) and water scarcity (circles) are plotted in Fig. 6 vs imp for β = 0.1; 1; 5, ET0 = 5 mm d−1 and nH = 360; 60 mm, corresponding to deep and shallow root zones, respectively. Risks of overflow and drought correspond to those shown in Fig. 4. In this case study, i0 ≈ ET0 . The risk of drought is lower and closer to the analytical prediction when nH = 60 mm. When nH = 360 mm the analytical solution greatly underestimates the risk of drought. The
N. Ursino / Advances in Water Resources 83 (2015) 277–284
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(a) 0.8 0.6
Rf Ri
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Rf Ri
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to drought-tolerant or shallow-rooted plants, may reduce the risk of drought with less effect on the risk of overflow. Optimal reservoir release policies generally depend on initial growing season storage and growing season inflow [10,21,36]. However, in this case study, initial storage during the growing season has less effect on the risk of overflow and thus justifies the use of two independent management strategies during the rainy and growing seasons.
References 0.2
0.4
0.6
imp
0.8
1
Fig. 6. Risk of overflow Rf (dashed line) and water scarcity Ri (continuous line) evaluated as in Fig. 4, and average simulated frequency of overflow (squares) and water scarcity (circles) vs imp (soil-saturation-dependent irrigation rate). ET0 = 5 mm d−1 ; a: nH = 360 mm; b: nH = 60 mm. Filled symbols: tank full at beginning of the growing season. Empty symbols: tank empty at beginning of the growing season.
risk of overflow is in general overestimated by the analytical model, and major differences arise after switching point i = i0 when i < i0 . The analytical model is in fact based on the over-conservative assumption that the tank is full at the end of the first of two consecutive rainfall events, whereas water is used for irrigation after each rain event. The simulation was run twice for all sets of model parameters. Under assumption (i), the tank was full at the beginning of the rainy season (filled symbols) and, under (ii), it was completely empty (empty symbols). The results show that water storage at the beginning of the rainy season has less effect on the results: that is, of the contrasting aims of having water stored in the tank at the end of the rainy season and emptying it as soon as possible for safety reasons, the latter may be privileged for imp up to 0.9.
5. Discussion and conclusions Compared with sewage water re-use, collecting and re-using storm-water and by so doing achieving ecological, social and economic goals, requires the development of minor treatment technologies [14,22]. However, the availability of water resources may become a major limiting factor to building multipurpose tanks, and certainly deserves attention. Climate, the demand for irrigation dictated by the ecosystem composition [13] and irrigation management strategies [21,36] all affect social and economic goals as well as the ecosystem service that may be achieved. Risk analysis [1], expressed in the closed form for the risk of overflow during the rainy and growing seasons and for the risk of water shortage during the growing season, was derived for a multipurpose storm-water tank. The interrelation between risk of overflow and of water scarcity is demonstrated here for the first time with an analytical probabilistic model. Results encourage the use of medium-sized storage capacities, because larger multipurpose tanks would not improve the reliability of the re-use system. Hedging rules reduce deliveries in order to retain water in storage, and reduce the risk of large water shortages at the cost of more frequent small-scale ones [10]. Although the model presented here is conceptually simple, it shows whether irrigation demand can be satisfied during the vegetation growing season and, at the same time, whether the drainage system is reliable. To achieve these contrasting goals is a challenging task, involving the right choice of appropriate urban water conservation methods [13]. In the case study presented here, even though the closed form solution of probabilistic analysis overestimates the risk of overflow and underestimates that of drought, it suggests that slow, parsimonious storm-water release
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