Luigi Cimorelli^{1}, Luca Cozzolino^{2}, Andrea D’Aniello^{2}, Francesco Morlando^{2*} and Domenico Pianese^{2}  
^{1}Department of Civil, Architectural and Environmental Engineering, University of Naples Federico II, Via Claudio n.21 – 80125, Napoli, Italy  
^{2}Department of Engineering, Parthenope University of Naples  Centro Direzionale di Napoli, 80142 Napoli, Italy  
Corresponding Author :  Francesco Morlando Department of Civil, Architectural and Environmental Engineering, University of Naples Federico II, Via Claudio n.21– 80125, Napoli, Italy Tel: +390817683463 Email: [email protected] 
Received May 30, 2015; Accepted June 29, 2015; Published July 14, 2015  
Citation: Cimorelli L, Cozzolino L, D’Aniello A, Morlando F, Pianese D (2015) A Simplified Approach for Stormwater Drainage Networks Sizing. J Climatol Weather Forecasting 3:136. doi:10.4172/2332 2594.1000136  
Copyright: © 2015 Cimorelli L, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.  
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In this work, a modification and a generalization of the “ItalianStorage method” (ISM), a renowned method for sizing rainstorm drainage systems, are proposed and applied. The approach adopts a steady nonlinear rainfallrunoff transformation probabilistic model within a “variational” or “maximizing” procedure. In this model, the transformation of rainfall in “excess (or effective) rainfall” is accomplished by means of the Rational Method, while the travel time is neglected. The flow within each link of the system, induced by a rectangular shaped discharge hydrograph for any rainfall duration, is assumed to be unsteady but locally uniform (Kinematic modeling). Combining the hydrological and hydraulic submodels, and considering different couples of rainfall heightsdurations, consistent with the local IDF curve, a “critical duration” is evaluated. Once the critical duration has been obtained, a maximum of maxima instantaneous flow discharges is evaluated for such duration, and subsequently, with such a discharge each link of the storm water drainage system can be designed.
Keywords 
Italianstorage method; Sewer/rural drainage networks 
Introduction 
In the last century, several methods have been proposed in the technical literature in order to allow the sizing and verification of storm water drainage networks, given a probabilistic framework. In particular, starting from several pioneering works [16], a few approaches, based on the adoption of optimization procedures, have been proposed for the optimal sizing of drainage networks (see, among others, [711]). Independently on the specific procedure adopted for the optimization process (e.g., linear or quadratic programming, heuristic algorithms, etc.), the considerable number of evaluations of network performances required in optimization problems imply the adoption of a tool for the fast evaluation of the network hydraulic features (e.g., discharges and flow depths values). As a consequence, very often simplified approaches are preferred to the most accurate, and meanwhile slow, ones [9]. 
Starting from these considerations, in this work a slight modification and generalization of a method for the sizing of rainstorm drainage networks, wellknown in Europe as the “ItalianStorage method” (ISM), is proposed and then applied to some case studies. This method was firstly proposed [12] for sewer systems and then extended [13] for drainage networks. Subsequently the approach was developed in order to give the possibility to straightly and quickly carry out the design of both sewer/rural drainage networks [14,15]. In particular, one of these approaches is nowadays widely adopted in Italy by technicians in the design of drainage systems, especially given the noticeable simplicity in software implementation [14]. Moreover, it allows the estimation of the hydraulic features of the whole design network in a very limited time, allowing its adoption in optimization procedures where it is usually needed the iterative modification of the network physical characteristics (e.g., longitudinal slopes, shapes and sizes of pipes, roughness parameters, upstream/downstream elevations or crown/bed elevations, etc.). 
Given that this method has, ‘in nuce’, the whole characteristics of whatever semidistributed, probabilistically based, hydrologic model, it has been already adopted, within several papers of these authors, together with an ‘extremal’, or ‘variational’, procedure [1618] for choosing the ‘critical value’ for the rainstorm duration, in order to allow the comprehension of the whole procedure. 
Fundamentals 
Generally speaking, if lateral inflow/outflows are left aside, the de SaintVenant’s unsteady flow equations could be written as: 
(1) 
(2) 
where i=sinβ and β=angle between the bed and the horizontal (Figure 1). 
If the kinematic wave approximation could be considered, equation (2) becomes i=j (i.e., locally and instantaneously uniform flow), and then a resistence formula, such as the one proposed by Chezy and shown hereinafter, could be applied: 
(3) 
where the coefficient μ and the exponent α could be considered constant during the filling/emptying process (with the value of the constants to be defined properly), and calibrated by an iterative procedure. 
Applying the equation (1) to a discrete reach of length L (which could be assumed, for instance, equal to the pipe length), the governing equations become: 
or (4) 
or (5) 
In the previous equations, w=Lω is the volume of water stored in the reach of length L when the discharge flowing out from the channel at time t is qout and the area of cross section is ω; W*= LΩ* is the maximum volume of water that can be stored in the reach when the discharge flowing out from the reach itself is the maximum flow discharge capacity Q*, and Ω* is the corresponding channel cross section. 
By differentiating the first of equations (7), one obtains: 
or (6) 
Substitution of Equation (8) into the Equation (6) gives: 
(7) 
and then: 
(8) 
(9) 
If, at the beginning of the rainstorm (t_{1}=0,) the discharge flowing out from the reach is (q_{out})_{1}=0 and, at time T_{f} , the maximum permissible discharge flowing out from the reach is (q_{out})_{2}=Q, the Equation (11) becomes: 
(10) 
where T_{f} , namely filling time, is the time needed to reach the maximum permissible discharge Q. 
If q_{in} = constant in time and the “attenuation ratios” are considered, the time T_{f} could be evaluated as: 
(11) 
where 
(12) 
Equation (13) describes the hydraulic response; to an input discharge q_{in} constant in time, of a reach where the volume can be stored. 
In order to evaluate q_{in}, a hydrologic approach could be used. In the approach proposed by Supino and Chow [14,15] the evaluation of discharge q_{in} is attained considering: 
1) Storms characterized by a return period T, constant rainfall intensity i_{d,T} in the time interval t[0,d] and constant rainfall intensity 0 for t>d; 
2) As the input values at most upstream cross section of channels considered in the calculations, the runoff discharges drained from the whole area A subtended by the most downstream cross section of the channel reach; 
3) The infiltration processes, by using the runoff coefficient c=A_{imp}/A, being A_{imp} and A, respectively, the impervious and total drainage area of the basin; 
Initially neglecting lag phenomenon due to the formation and subsequent arrival of the surface runoff to the upstream section of the reach considered in the analysis, for which is q_{in}=c i_{d} A; 
An IntensityDurationFrequency relationship having the simple structure 
(13) 
where a and n have to be evaluated by using regional and/or local rainfall data (partial series of the annual maximum rainfall depth in the duration d). 
If the storm duration d was higher than the channel filling time T_{f} (d>T_{f}), the Surface Runoff SR=q_{in}d would be higher than the storage availability W*, and, then, the maximum value of the flow depth h*_{ max}, corresponding to the maximum available flow area Ω*, would be exceeded (causing, for instance, a pressurized flow instead of a free surface flow). 
On the other hand, if d<T_{f} also SR<W*. As a direct consequence the maximum crosssectional flow area, Ω, and the maximum flow depth, hmax, both attained during the filling process, should follow the following relations: Ω< Ω*, h_{max}<h* _{max} and (Q_{out})_{max}<(Q*_{out})_{max}. In this case, the sizes and/or slope considered should be higher than those strictly needed, and, as a consequence, the construction costs would grow. 
Given the previous framework, it is evident that, in order to avoid both under and over sizing of crosssections/slopes/conveyance capability, it would be preferable to consider the following conditions: 
d=T_{f}, Ω< Ω*, h_{max}<h* _{max} and 
As a consequence, it would result: 
(14) 
By substituting equation (16) in equation (13), together with the condition d = T_{f} , one would obtain: 
(15) 
or 
(16) 
If the ‘udometric coefficient’ (i.e., the contribution of the basin unitary area to the formation of peak discharge), , and the specific storage (i.e., volume of water stored in the reach of length L per basin unitary area), w = W/A, are introduced, equation (17) becomes: 
(17) 
For a fixed value of Z, is it possible to evaluate: 
 by equation (17’), the values of ; 
 by equation (18), the corresponding value of u; 
 by the following equation 
(18) 
the corresponding rainstorm duration. 
Then, by changing the Z value (i.e.: by varying the storm duration d and, subsequently, inflow discharge q_{in}), the evaluation of the maximum discharge value, , becomes possible. This is the main reason why in the technical literature the procedure above described is known as ‘variational’ (or ‘extremal’) approach. In particular, the duration dcrit for which is given by: 
(19) 
and is usually defined as ‘critical storm duration’. 
As a consequence, the maximum could be evaluated by: 
a) differentiating equation (18) with respect to Z; 
b) putting the above obtained derivative equal to zero, in order to evaluate Z_{crit} (and, then, the critical rainstorm duration d_{crit}); 
c) Substituting this value of Z within equation (18). 
Following these steps, it is possible to show that, at “critical” conditions, it needs: 
(20) 
Using the Equation (21) for each couple of n and α values, it is possible to evaluate, by a trialanderror approach, corresponding value of Z_{crit} and, then, corresponding value of the product both present in equation (18). 
A few values of Z_{crit} and are summarized, for fixed α and n values, in Table 1. 
It is worth noticing that the values of the product can be approximated by using different interpolation formulas. Puppini and Supino [13,15] proposed the following expression: 
(21) 
where λ_{1}=0.259 and λ_{2}=0.518 (with maximum errors about 5%) for [13] formulation and λ_{1}=0.221 and λ_{2} =0.574 for [15] one, with a maximum error lower than 3%. 
Other and better approximation can also be obtained if the following relation is considered [15]: 
(22) 
where the variables involved are expressed considering the following units of measurement: u [l/(s·ha)]; a [m/day] and w [m]. 
In order to evaluate the maximum peak discharges, flow depths and volumes stored in the reach of length L, the following first order approximation relation can be adopted, starting from equation (21), (with errors ranging within the interval , (Figure 2): 
(23) 
with n∈[0.10, 0.80] , α ∈[1,2] , λ1= 0.140; λ2=0.709. 
A second order approximation is reported hereinafter, (with errors ranging within the interval , (Figure 3): 
(24) 
with: λ_{3}=0.347; λ_{4}=0.364; λ_{5}=0.340; λ_{6}=0.566. 
A third order of approximation has the following structure, (errors ranging within the interval [−0.44%, +1.07%] , (Figure 4): 
(25) 
with: λ_{7}=0.358; λ_{8}=0.389; λ_{9}=0.384; λ_{10}=0.464; λ_{11}=0.041; λ_{12}=0.094. 
Finally, a fourth order approximation leads to the following form (errors ranging within the interval [−0.38%, + 0.82%], see (Figure 5): 
(26) 
with: λ_{13}=0.368; λ_{14}=0.364; λ_{15}=0.456; λ_{16}=0.646; λ_{17}=0.193; λ_{18}= 0.291; λ_{19}=0.099; λ_{20}=0.250. 
Summarizing: 
a) the errors obtained by each approximation have a decreasing trend as the order of approximation and formal complexity increase (Figure 6); 
b) the udometric coefficient u can be expressed as it follows: 
(27) 
where is a simple polynomial, function of α and n. Furthermore, also Z_{crit} can be expressed by means of different interpolation formulas. Indeed, the Z_{crit} values obtained by equation (21) can be approximated by the following: 
(28) 
where k_{1}=0.839; k_{2}=1.434; k_{3}=0.609; k_{4}=0.026; k_{5}=0.600; k_{6}=1.700; k_{7}=0.058; (with errors ranging within the interval [−0.04%, + 0.02%] (Figure 7). 
Equation (28) can be easily applied in order to evaluate the main flow characteristics (i.e: maxima of discharge, velocity and flow depth) of the drainage network source channels. 
Once the hydraulic (i.e.: longitudinal slope; cross section; roughness parameter), the hydrological (i.e.: the drainage area of the catchment; the surface runoff coefficient), the climatic (i.e.: the coefficient a and the exponent n of the IDF curve) and the design (i.e.: the return period T) characteristics of the reach have been defined, the proposed design procedure follows the steps described hereinafter: 
1) a first guess value of α‘ of the exponent α in equation (28) has to be assumed (e.g.: if α‘=α=1, the routing is initially carried out by using the linear reservoir conceptual model); 
2) a guess value u’ of u is considered; 
3) a first guess maximum peak discharge value is evaluated (i.e.:) 
4) by means of the chosen steady and uniform roughness formula, the first guess values of the maximum peak flow depth h’_{max}, the maximum peak flow cross section area Ω’_{out}, the maximum water volume stored in the reach W’=Ω’_{out} L , and the maximum specific stored volume w’=W’/A are evaluated; 
5) equation (28) is applied, and a new value of u (let say u”) is evaluated; 
6) then, if (29) 
with (for instance, ε_{u}=0,0001), the iteration process is stopped; otherwise 
7) a second guess value of u (u=u”’) is considered, and the procedure is repeated from stage 1) to 6) until the relation (30) is satisfied; 
8) the value of h_{max} evaluated during the first iteration process is then subdivided in N steps (for instance: N=100), obtaining 
9) by means of the roughness formula q = g (h) , the discharges are evaluated; 
10) then, the values of couples (Q_{i}, Ω_{i}) are interpolated by using the expression , thus obtaining a new guess value for α (α = α’’); 
11) the iterative procedure is repeated again from step 1) to 11); 
12) then, if (30) 
with → (for instance =0,0001), the second iterative stage is stopped; otherwise 
13) a second guess value of α (α = α’’) is considered, and the procedure is repeated from stage 1) to 12) until convergence is reached, equation (31). 
Usually, after few iterations both the conditions (30) and (31) are satisfied. Thanks to modern computers, the whole procedure described usually takes less than (0.1  0.2) s ^{(*)1}. 
Extension of the italian storage method to the whole channel network 
In order to apply the approach illustrated above to the whole drainage network, the approach is modified introducing in the equation a new parameter wo, obtained by splitting the volume in two parts, namely and , for which 
 the first term, , where represents the maximum water volume actually stored within the reach considered in the calculation, and Ar is the area of the whole basin drained from the most downstream crosssection of the reach r; 
 the second term, , in which represents the whole volume of water stored either in the reaches upstream the reach considered in the calculations or in other channels hydraulically linked to the reaches of the main network or at the surface of basins constituting the catchment whose ending drain is just the reach considered in the calculations. In particular: 
(31) 
where: 
is the volume of water stored within the link up of the network, located upstream of the reach r considered in the calculations and, then, which does not flow into the reach r itself; 
is the volume of water stored on the surface of the basin directly drained from the reach up of the network and, then, which does not flow into the reach up itself; 
is the volume of water stored within the whole reaches/ tanks directly linked to the reach up of the network, located upstream of the reach r considered, but which were not explicitly considered in the calculations (“ghost reaches”); also this volume does not flow into the reach r ; 
and are the same of and , but they apply to the reach r under consideration; 
is the area of the basin directly drained from the upth reach existing upstream the reach r; 
is the area of the basin directly drained from the reach r; 
is the whole area drained from the most downstream cross section of the reach r considered in the calculation; 
is the number of reaches located upstream the link under consideration. 
As a consequence, the relationships (24), (25), (26), (27) and (28) become, respectively: 
is the number of reaches located upstream the link under consideration. 
As a consequence, the relationships (24), (25), (26), (27) and (28) become, respectively: 
(32) 
(33) 
(34) 
(35) 
(36) 
In order to better understand the meaning of , it is possible to observe, preliminarily, that to arrive to the relationships (33), (34), (35), (36) and (37), it needs to introduce in the equation (6) the sum instead of w(t), preserving = constant in time during the filling/empting processes afflicting the reach r under consideration. 
Usually, the values of are considered parameters of the model (though variable subbasin by subbasin) whereas the values of are taken to be equal to the values of r W already evaluated for all the reaches hydraulically upstream with respect to that considered in the calculations. 
The last hypothesis (usually defined ‘synchronism hypothesis’, because it implies that the peak time for all the reach of the networks has to be the same) has to be considered together with the second (usually defined ‘independence hypothesis’), which allows, by using equation (37), the performance of a reach upstream another (and also, the performance of a reach upstream a confluence) without evaluating the eventual influence of the filling/empting phenomena which are simultaneously developing in the downstream reaches. 
These hypotheses, though seeming (together with that related to not consider the lag times) the worst present in the model, could, in certain cases, balance one each other, because the backing up prorogued by filling phenomena occurring in the downstream reaches could enhance the flow depths in the upstream reaches at peak discharge flows. 
Application to a case study 
In this subsection the ISM is applied to a case study. The case study consists of a 17 branch rural drainage network with each branch having a trapezoidal cross section. The network geometrical characteristics are summarized in Table 2, while a schematic representation of the network is reported in Figure 8. 
The simulation with the ISM has been carried out by considering W_{0,i} = 40 m for each subcatchment while the following intensity duration frequency (IDF) curve has been considered: 
(37) 
For Comparisons purpose, the same network has been simulated with numerical solution of the Kinematic Wave Model (KWM) within a variational approach (see, for instance, [7]) using equation 38 as IDF curve and evaluating the rainstorm runoff with the runoff coefficient method (see point 4 at section 2) 
Results of the simulation are reported in Table 3 in terms of both peak discharge Q_{max} and maximum flow depth H_{max} obtained with ISM and KWM. 
By inspection of Table 3 it is evident that the ISM model can be successfully employed in the design of rainstorm drainage networks. 
Conclusion 
In this work a slight modification and a generalization of a wellknown method for sizing rainstorm drainage networks, very common in European contexts, the “ItalianStorage method” (ISM), are proposed and applied. The application to a case study has shown that the ISM is equivalent to the numerical solution of the Kinematic Wave Model, used within a variational approach in order to find the maximum discharges and flow depths flowing through the branch of a drainage network. For this reason, the ISM can be satisfactory employed in the design of both urban and rural rainstorm drainage network. 
References 

Table 1  Table 2  Table 3 
Figure 1  Figure 2  Figure 3  Figure 4 
Figure 5  Figure 6  Figure 7  Figure 8 