1

ABOUT THE “ITALIAN METHOD OF THE VOLUME DI INVASO”

DOMENICO PIANESE(1), CARMINE COVELLI(1), LUCA COZZOLINO(2)

(1) Dipartimento di Ingegneria Idraulica, Geotecnica e Ambientale dell’Università degli Studi di Napoli Federico II - Via Claudio n.21 – 80125, Napoli (Italy).

(2) Università degli Studi di Napoli Parthenope, Facoltà di Ingegneria - Centro Direzionale di Napoli, 80142 Napoli, Italy.

1. INTRODUCTION

The “volume di invaso” (storage volume) method was firstly proposed by Fantoli (1904) for

sewer systems and then extended by Puppini (1923) for drainage networks. Supino (1947 and

1965) developed the approach in order to give the possibility to straightly and quickly carry out

the design of both sewer/rural drainage networks.

The approach proposed by Supino is one of the most used, in Italy, by the engineers working in

the field of design of runoff drainage systems, due to the circumstance that it is very easy to use

and, eventually, to implement in a software. On the other hand, it is able to furnish, very quickly,

also for very large systems, the results of computations, allowing to change very fast and easy, if

it is needed, one of more values of the decision variables involved in the design process

(longitudinal slopes, shapes and sizes of pipes, roughness parameters, upstream/downstream

elevations or crown/bed elevations, etc.). Because this method has, ‘in nuce’, the whole

characteristics of whatever semi-distributed, probabilistically based, hydrologic model, based on

a ‘extremal’, or ‘variational’, procedure (Della Morte et al. 2001; Denisov & Pack, 2009) for

choosing the ‘critical value’ for the rainstorm duration, it was also used, within several papers

involving the writers, just to show the procedures adopted (Palumbo et al., 2010).

2

2. FOUNDAMENTALS

Let’s consider the channel reach in Fig. 1.

h* h(t2)

h(t1)

(t2) (t1)

L t1

t2

Fig.1. Sketch of channel and channel cross section.

Generally speaking, if lateral inflow/outflows are left aside, the de Saint-Venant’s unsteady

flow equations could be written as:

0=∂∂

+∂Ω∂

xQ

t (1)

( ) 01=−+

∂∂

+∂∂

+∂∂ ji

xh

xu

gV

tu

g (2)

where i = senβ and β = angle between the bed and the horizontal (see Fig. 1).

If the kinematic wave approximation could be considered (Ponce et al., 1987; Yen, 1987), the

equation (2) becomes:

i = j (i.e.: locally and instantaneously uniform flow), and then a roughness formula such that

proposed by Chezy could be applied

αωµω ⋅≅⋅⋅⋅=⇒⋅⋅= iRkQiRkU cc (2’)

where the coefficient µ and the exponent α could be considered constant during the

filling/emptying process (with the value of the constants to define properly), and calibrated by

iterative procedure.

3

Applying the equation (1) to a discrete reach of length L (which could be assumed, for instance,

equal to the pipe’s length), the governing equations become:

[ ] 0=−

+⎟⎠⎞

⎜⎝⎛

Lqq

Lw

dtd inout

or ( )

dttdwqq outin =− (6)

and

αωµ ⋅=outq or νoutqkw ⋅= (7)

being w = Lω 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, α

ν 1= , ⎟⎟

⎠

⎞⎜⎜⎝

⎛= ν*

*

QWk , W

*= LΩ*

the maximum volume of water which could be stored in the reach when the discharge flowing out

from the reach itself is the maximum flow discharge capability Q*, and Ω* is the area of the reach

cross section when maximum flow discharge capability is attained.

By differentiating the first of equations (7), it gives:

outout dqqQ

d 11

1*

*−

⋅Ω

= α

ααω or outout dqq

QWdw

11

1*

* −= α

αα (8)

Substitution of Eq. (8) into the Eq. (6) gives:

( )dt

dqq

Q

Wqq outoutin out

11

1*

*−

⋅=− α

αα (9)

and then:

outoutin

dqqq

q

Q

Wdt out

−⋅=

−11

1*

* α

αα (10)

or

4

( )( )

outqq

outindq

qq

q

Q

Wtt out

out

out∫ −⋅=−

−

2

1 )(

11

1*12

α

αα (11)

If, at the beginning of the rainstorm (t1=0,) the discharge flowing out from the reach is (qout)1 =

0 and, at time Tf , the maximum permissible discharge flowing out from the reach is (qout)2 = Q,

the Eq. (11) becomes:

out

Q

outinf dq

qqq

Q

WT out∫ −⋅=

−

0

11

1*

*

)(

α

αα (12)

where Tf is the time needed to reach the maximum permissible discharge Q (Tf = Filling Time).

If qin = constant in time and the “attenuation ratios” z = qout/qin and Z* = Q*/qin are considered,

the time Tf could be evaluated as:

)(1

1 *

0

11

1*

*

ZqWdz

zz

ZqWT

in

Z

inf α

α

ααΦ⋅=

−⋅= ∫

−

(13)

where

dzzzzZ

dzz

z

ZZZ ZZ

∫∫ ++++=−

=Φ−

−

0211

1*0

11

1** ...)...1(1

11),( α

α

α

αα

αα (14)

The equation (13) describes the hydraulic response, to an input discharge qin constant in time, of

a reach where the volume ** Ω⋅= LW could be stored.

In order to evaluate qin, a hydrologic approach could be used. In the approach proposed by

Supino (1947 and 1965), the evaluation of discharge qin is made considering:

1) Storms characterized by a return period T, constant rainfall intensity id,T in the time

interval t[0,d] and constant rainfall intensity 0 for t>d;

5

2) Like input at most upstream cross section of channel 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 = Aimp/A, being Aimp and A,

respectively, the impervious and total drainage area of the basin;

4) 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

qin = c id A;

5) An Intensity-Duration-Frequency relationship having the simple structure

11,

1, )()( −−− ⋅=⋅⋅=⇔⋅⋅= n

Tn

TTdTn

Td dadkaikdai (15)

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 Tf (d>Tf), the Surface Runoff

SR=qind 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∗ (already established at

beginning of computations) would be exceeded (provoking, for instance, a pressurized flow

instead of a free surface flow).

On the other hand, if d<Tf, also SR<W*, and then the maximum cross sectional flow area and

the maximum flow depth attained during the filling process, respectively indicated by Ω and hmax,

should be Ω < Ω*, hmax<h*max and (Qout)max<(Q*out)max. In this case, the sizes and/or slope and/or

conveyance considered should be higher than those strictly needed, and the construction costs

should unnecessarily grow.

6

It is evident that, in order to avoid both under- and over- sizing of cross-

sections/slopes/conveyance capability, it would be right to consider the conditions d = Tf, Ω

= Ω*, hmax=h*max, *outout QQ = and *

*

Zq

Qq

QZ

in

out

in

out ===

As a consequence:

...1

...211

1)(),(2

* ++

+++

++

+=Φ=Φααααα j

ZZZZZZj

(16)

By substituting the equation (16) in the equation (13), together with the condition d = Tf , it

gives:

( ) ( )[ ] nn

n

nout ZZ

W

AacQ1

11

1 −

−Φ⋅⋅= α (17)

or

( ) ( )[ ]( )⎥

⎦

⎤⎢⎣

⎡−⋅−

⋅

−

−

⎟⎠⎞

⎜⎝⎛ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Φ⋅⋅=

αα

α

αµ11

1

11

1111 nn

nn

nn

nn

out ZZ

L

AacQ (17’)

If the ‘udometric coefficient’ (contribution of unit area of the basin to the formation of peak

discharge) AQu out= u = Q/A and the specific storage w = W/A (volume of water stored in the

reach of length L per unit basin area) are introduced, the equation (17) becomes:

( ) ( )[ ] nn

n

nZZ

w

acu1

11

1−

−Φ⋅⋅= α (18)

For a fixed value of Z, is it possible to evaluate:

- by the Eq. (17’), the values of outQ ;

- by the Eq. (18), corresponding value of u;

7

- by the following Equation

11−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

acZud (19)

corresponding rainstorm duration.

Then, by changing the Z value (i.e.: by varying the storm duration d and subsequent inflow

discharge qin), it becomes possible to evaluate the maximum ( )[ ]ZQQ outZmaxmax = . For this

reason, in technical literature, above procedure (and other similar) is very often indicated as

‘variational’ (or ‘extremal’) approach.

The duration dcrit for which outQ ≡ Qmax is given by

11

max −

⎥⎦

⎤⎢⎣

⎡=

n

critcrit AacZ

Qd (20)

and is usually defined as ‘critical storm duration’.

As a consequence, the maximum ( )[ ] ( )[ ] ( )critdZduduZuu === maxmaxmax could be evaluated by:

a) differentiating the equation (18) with respect to Z;

b) putting the derivative obtained equal to zero in order to evaluate Zcrit (and, then, the

critical rainstorm duration dcrit);

c) substituting this value of Z within the equation (18).

Following these steps, it is possible to have that, at “critical” conditions, it needs:

( ) ( ) ( )1111

−⋅+−

=Φ⋅−αα nnZZ critcrit (21)

8

Using the Eq. (21) for each couple of n and α values, it is possible to evaluate, by trial and error,

corresponding value of Zcrit and, then, corresponding value of the product ( )[ ] nn

critcrit ZZ1−

Φ⋅ α

present in the eq. (18)

A few values of Zcrit and ( )[ ] nn

critcrit ZZ1−

Φ⋅ α are summarized, for fixed α and n values, in the

Table 1

critZ ( )[ ] nn

critcrit ZZ1−

Φ⋅ α

n α = 1.00 α = 1.25 α = 1.50 α = 1.75 α = 2.00 α = 1.00 α = 1.25 α = 1.50 α = 1.75 α = 2.00 0.100 0.187 0.206 0.224 0.241 0.258 0.075 0.083 0.091 0.099 0.107 0.150 0.271 0.296 0.319 0.340 0.360 0.113 0.125 0.137 0.148 0.159 0.200 0.350 0.378 0.404 0.427 0.449 0.153 0.168 0.182 0.196 0.209 0.250 0.423 0.453 0.480 0.504 0.526 0.193 0.211 0.228 0.244 0.259 0.300 0.491 0.522 0.549 0.573 0.594 0.233 0.254 0.273 0.291 0.307 0.350 0.554 0.584 0.610 0.633 0.653 0.275 0.298 0.318 0.338 0.355 0.400 0.612 0.641 0.666 0.687 0.705 0.318 0.342 0.364 0.384 0.403 0.450 0.666 0.693 0.715 0.734 0.751 0.362 0.387 0.410 0.431 0.449 0.500 0.715 0.740 0.760 0.777 0.791 0.407 0.433 0.456 0.477 0.496 0.550 0.761 0.782 0.800 0.814 0.827 0.454 0.480 0.503 0.524 0.542 0.600 0.802 0.820 0.835 0.848 0.858 0.502 0.528 0.551 0.570 0.588 0.650 0.839 0.855 0.867 0.877 0.886 0.552 0.577 0.599 0.618 0.635 0.700 0.873 0.886 0.895 0.904 0.910 0.604 0.628 0.648 0.666 0.681 0.750 0.903 0.913 0.920 0.927 0.932 0.658 0.680 0.699 0.715 0.729 0.800 0.930 0.937 0.942 0.947 0.950 0.715 0.735 0.751 0.765 0.777

Table 1. A few values of critZ and ( )[ ] nn

critcrit ZZ1−

Φ⋅ α for fixed α and n values.

The values of the product ( )[ ] nn

critcrit ZZ1−

Φ⋅ α could be approximated by using different

interpolation formulas.

Puppini (1931) and Supino (1965) proposed the expression

( )[ ] ( ) nZZ nn

critcrit ⋅+≅Φ⋅−

21

1λαλα (22)

where λ1 = 0.259 and λ2 = 0.518 (with maximum errors about 5%) for Puppini and λ1=0.221 and

λ2 =0.574 for Supino (maximum errors lower than 3%).

9

Other and better approximation could also be obtained if the relation:

( )[ ] ( ) ( ) 21

5,682,477,403,43 nnZZ nn

critcrit ⋅+⋅−⋅+⋅≅Φ⋅−

ααα (23)

is considered (Supino, 1965), where the variables involved are expressed considering following

measure units: u [l/(s·ha)]; a [m/day] ; w [m].

Using the results obtained by the Eq. (21) in the ranges [ ]80.0,10.0∈n and [ ]2,1∈α , in order to

evaluate the maximum peak discharges, flow depths and volumes stored in the reach of length L,

the relationship which could be used at the first order of approximation (with errors ranging

within the interval [ ]%44.13%,80.7 +− , see Fig. 2):

( )[ ] ( )11

1

21 −⋅⋅+=

n

n

w

acnu λαλ (24)

with: λ1 = 0.140; λ2 = 0.709.

Fig. 2. Comparison between the experimental values of ( )[ ] nn

critcrit ZZ1−

Φ⋅ α obtained by numerical code and the

values calculated by the Eq. (24).

10

At second order of approximation (with errors ranging within the interval [ ]%26.1%,66.2 +− ,

see Fig. 3), this relationship could be used:

( ) ( )[ ] ( )11

1

6543 −⋅⋅⋅+++=

n

n

w

acnnu λαλλαλ (25)

with: λ3 =0.347; λ4 =0.364; λ5 = -0.340; λ6 = 0.566.

Fig. 3. Comparison between the experimental values of ( )[ ] nn

critcrit ZZ1−

Φ⋅ α obtained by numerical code and the

values calculated by the Eq. (25).

At third order of approximation (errors ranging within the interval [ ]%07.1%,44.0 +− , see Fig.

4), this relationship could be used:

( ) ( ) ( )[ ] ( )11

12

121110987 −⋅⋅⋅++⋅+++=

n

n

w

acnnnu λαλλαλλαλ (26)

with: λ7 = 0.358; λ8 = 0.389; λ9 = - 0.384; λ10 = 0.464; λ11 = 0.041; λ12 = 0.094.

11

Fig. 4. Comparison between the experimental values of ( )[ ] nn

critcrit ZZ1−

Φ⋅ α obtained by numerical code and the

values calculated by the Eq. (26).

Finally, at fourth order of approximation (errors ranging within the interval [ ]%82.0%,38.0 +− ,

see Fig. 5), this relation could be used:

( ) ( ) ( ) ( )[ ] ( )11

13

20192

181716151413 −⋅⋅⋅++⋅++⋅+++=

n

n

w

acnnnnu λαλλαλλαλλαλ (27)

with: λ13 =0.368; λ14 =0.364; λ15 = -0.456; λ16 = 0.646; λ17 = 0.193; λ18 = - 0.291; λ19 = 0.099;

λ20 = 0.250.

12

Fig. 5. Comparison between the experimental values of ( )[ ] nn

critcrit ZZ1−

Φ⋅ α obtained by numerical code and the

values calculated by the Eq. (27).

Summarizing:

a) the errors obtainable by each approximation are decreasing as the order of approximation

and formal complexity of used relationship increase (see Fig.6);

Fig. 6. Comparison among the errors of the approximation formulas (24), (25), (26), and (27).

13

b) the udometric coefficient u could be always evaluated, approximatively, by using a

relationship such as:

( ) ( )11

1

,−

⋅⋅=n

n

w

acnnfu α (28)

where ( )nf ,α is a simple polynomial, just function of α and n values.

Also the values of the Zcrit could be approximated by using different interpolation formulas. For

instance, the Zcrit values obtained by using the Eq. (21) could be considered depending on the α

and n values by means of following fitting formula:

( ) ( )762

5432

23

1 knknkknknknkZcrit +⋅+⋅+⋅+⋅+⋅+⋅≅ α (29)

with: k1 = 0.839; k2 = -1.434; k3 = 0.609; k4 = 0.026; k5 = -0.600; k6 = 1.700; k7 = -0.058 (and

errors ranging within the interval [ ]%02.0%,04.0 +− , see Fig.7).

Fig. 7. Comparison between the experimental values of critZ obtained by the. Eq. (29) – vertical axis - and the

values calculated by the Eq (21) - horizontal axis.

14

The equation (28) can be applied very easily, without other considerations, to evaluate main flow

characteristics in the whole source channels of the drainage network (i.e: maxima of discharge,

velocity and flow depth) for given characteristic of the reach (hydraulic, such as: longitudinal

slope, shape of the cross section, size/s characterizing the cross section, roughness parameter/s;

hydrological, such as: the drainage area of the catchment; the surface runoff coefficient; climatic,

such as: the coefficient a and the exponent n of the Intensity-Duration-Frequency relationship to

be considered for the area within the catchment falls; probabilistic, such as the return period T).

To do this, the calculations should be carried out following these indications:

1) the design return period T is assumed once, for which the value TaT Ka ⋅= µ is firstly

evaluated (being µa the mean of the annual maximum hourly rainfall and KT the growth

factor with the return period T, which is usually evaluated on regional basis);

2) the shape of channel, the size/s of cross section, the longitudinal slopes and the roughness

parameter/s have to be assigned, allowing the evaluation of the discharge outQ ;

3) a (first) tentative value α‘ of the exponent α present in the Eq. (28) has to be assumed (for

instance, if α‘ = α = 1, the routing is initially carried out by using the linear reservoir

conceptual model);

4) a tentative value u’ of u is considered;

5) thus, a first tentative maximum peak discharge value outQ' is evaluated (being

AuQ out ⋅= '' );

6) by using the steady and uniform roughness formula chosen, the first tentative 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;

15

7) the equation (28) is applied, and a new value of u (let say u”) is evaluated;

8) then, if uuuuu ε≤

⎟⎠⎞

⎜⎝⎛ +

−

2''''''

(30)

with 0→uε (for instance, εu = 0,0001), the iteration process is stopped; otherwise

9) a second tentative value of u (u = u”’) is considered, and the procedure is repeated

starting from the point 5 to the point 8 until the relation (30) is satisfied;

10) the value of hmax evaluated by this first iteration process is then subdivided in N steps (for

instance: N = 100), obtaining h1 = 1× hmax/N; h2 = 2 × hmax/N; …; hN = hmax, and Ω1 =

Ω1(h1); Ω2 = Ω2(h2); …; Ω = Ω(hmax);

11) by applying the roughness formula ( )hgq = chosen to carry out all the hydraulic

calculations, the discharges q1 = g(h1), q2 = g(h2), …, Qmax = g(hmax) are evaluated;

12) then, the values of couples (Qi, Ωi) are interpolated by using the expression αµ Ω⋅=q ,

thus obtaining a new tentative value for α (α = α'');

13) the procedure in the steps from the point 4) to the point 12) newly repeated;

14) then, if αεαααα

≤⎟⎠⎞

⎜⎝⎛ +

−

2''''''

(31)

with 0→αε (for instance εα = 0,0001), also the second stage iterative approach is

stopped; otherwise

15) a second tentative value of α (α = α'') is considered, and the procedure is repeated starting

from the point 3 to the point 14 until the relation (31) is also satisfied.

16

Usually, a few iterations have to be carried out to satisfy both the conditions (30) and (31), for

which, by using modern computers, the whole procedure described takes, at maximum, less than

(0.1 - 0.2) s (*)1.

3. EXTENSION OF THE ‘VOLUME DI INVASO METHOD’ TO THE WHOLE

CHANNEL NETWORK

In order to apply the approach illustrated above not only to source links of a channel network

but also for the channels downstream source links (and, then, to the whole channel network), the

approach itself is modified introducing in the equation, quite arbitrarily, a new parameter wo,

obtained by splitting the volume AWw = in two parts: 0w and rw , for which rwww += 0 :

- the first term, rrr AWw = , in which rW represents the maximum water volume actually

stored within the reach considered in the calculation, ad Ar is the area of the whole basin

drained from the most downstream cross-section of the reach r;

- the second term, rAWw 00 = , in which 0W represents the whole volume of water stored as

in the reaches hydraulically upstream the reach considered in the calculations, as in other

1 Please, note that, at the beginning of the computations, it could not be possible to assign a right tentative value of u. In fact, because the iteration processes obtain the solution oscillating around the final (true) values of u and α, it is possible that: i) by using a first value u’ too low, the second iteration could give a u” value too big, for which the second tentative Q”out value could be higher than the flow capability of the channel (and, then, the flow could overcome one or both the banks if the channel is open, or the flow could became ‘pressurized flow’ if the channel is closed); ii) on the contrary, that, by using a first value u’ too high, the second iteration could give a u” value too low, for which the second tentative Q”out value could be more and more lower than the flow capability of the channel (and, then, the channel itself becomes oversized). As a consequence, it could be suggested to perform initial computation of a ‘trial’ value of u’ . This value could be evaluated by taking in mind that the final (true) value of u is quite independent of: the sizes, the roughness parameter/s, the longitudinal slope and the shape of the channel. Thus, a ‘very good’ initial value u’ could be obtained by considering, at beginning of the computations carried out for the reach under consideration, a ‘virtual’ cross section, characterized by very large size/s, and by carrying out the whole iteration process for this ‘virtual channel’.

17

channels hydraulically linked to the reaches of main network (that formed by the reaches

actually considered in the calculations) but for which no specific calculations have been

carried out, as at the surface of basins constituting the catchment whose ending drain is just

the reach considered in the calculations. In particular:

[ ] [ ] [ ] [ ] [ ]( )rr

N

up

rup

N

up

rup

N

up

rup WWWWWW

rup

rup

rup

"0

'0

1

"

1

'

10 00

++⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∑∑∑

=== (32)

where:

• [ ]rupW is the volume of water stored within the link up of the network, located upstream

of the reach r considered (step by step) in the calculations and, then, which does not

flow to the reach r itself;

• [ ]rup

W '0 is the volume of water stored at the surface of the basin directly drained from the

reach up of the network and, then, which does not flow to the reach up itself;

• [ ]rup

W "0 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 to the reach r ;

• [ ]r

W '0 and [ ]

rW "

0 are the same of [ ]rup

W '0 and [ ]r

upW "

0 , but they apply to the reach r under

consideration;

• rupA is the area of the basin directly drained from the upth reach existing upstream the

reach r;

• rA is the area of the basin directly drained from the reach r;

18

• r

N

up

rup AAA

rup

+= ∑=1

is the whole area drained from the most downstream cross section of the

reach r considered in the calculation;

• rupN is the number of reaches located upstream the link under consideration.

As a consequence, the relationships (24), (25), (26), (27) and (28) became, respectively:

( )[ ] ( )( ) 11

1

21 −+⋅⋅+=

nro

n

wwacnu λαλ (24’)

( ) ( )[ ] ( )( ) 11

1

6543 −+⋅⋅⋅+++=

nro

n

wwacnnu λαλλαλ (25’)

( ) ( ) ( )[ ] ( )( ) 11

0

12

121110987 −+⋅⋅⋅++⋅+++=

nr

n

wwacnnnu λαλλαλλαλ (26’)

( ) ( ) ( ) ( )[ ] ( )( ) 11

0

13

20192

181716151413 −+⋅⋅⋅++⋅++⋅+++=

nr

n

wwacnnnnu λαλλαλλαλλαλ (27’)

( ) ( )( ) 11

0

1

,−+

⋅⋅=n

r

n

wwacnnfu α (28’)

In order to understand better the meaning of the introduction of 0w , it is possible to observe,

preliminarily, that, to arrive to the relationships (24’), (25’), (26’) and (27’), it needs to introduce

in the equation (6) the sum ( )[ ]twW r+0 instead of w(t), preserving 0W = constant in time during

the filling/empting processes afflicting the reach r under consideration.

19

Usually, the values of [ ] [ ]

[ ]rup

rup

rup

r AWW

w"'

,000

+= are considered such a parameters of the model

(though variable sub-basin by sub-basin) whereas the values of [ ]rupW are taken to be equal to the

values of rW already evaluated for all tha reaches hydraulically upstream 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

second (usually defined ‘independence hypothesis’), for which it is possible to evaluate, by using

the equation (28’), the performance of a reach upstream another (and also, the performance of a

reach upstream a confluence) without evaluate the eventual influence of the filling/empting

phenomena which contemporarily are 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.

REFERENCES

Della Morte, R., Iavarone, V., Pianese, D., 2001. Evaluation of maximum flow depths and discharges in drainage networks by a simplified model coupled to a variational approach, Proceedings of 29th IAHR Congress – Theme D2, Beijing, China,16-21 September, Qinghua University Press, ISBN 10 730204676X, 689-696.

Denisov, V.M., Pak, A.V., 2009. A Methodology of Determination of the Maximum Water Discharges and Volume of Rain Floods in Small Catchments, Russian Meteorology and Hydrology, 2009, Vol.34, No.12, Allerton Press, Inc., 818-827.

20

Fantoli, G., 1904. Final Report of Committee for the analysis of the sewer network of Milano City (in Italian. Original title: “Relazione della Commissione per lo studio della fognatura della città di Milano”). Milano, Italy.

Palumbo, A., Cimorelli, L., Mucherino, C., Covelli, C., Pianese, D., 2010. Using of semi-distributed hydrologic models for optimized drainage networks design (in Italian. Original title: “Utilizzazione di modelli idrologici semi-distribuiti per l’ottimizzazione delle reti di drenaggio”). Atti di IDRA2010, XXX Convegno di Idraulica e Costruzioni Idrauliche, 14-17 September 2010, Palermo. Palermo: Ed. Università di Palermo, Dipartimento di Ingegneria Idraulica ed Applicazioni Ambientali, Gruppo Italiano di Idraulica (CD-ROM).

Ponce, V.M., Li, R.M. e Simons, D.B. 1978, Applicability of kinematic and diffusion models, Journal of Hydraulic Division, ASCE, 104(3), pp. 353-360.

Puppini, F., 1923. Design of Rural Drainage Channels (in Italian. Original title: “Il calcolo dei canali di Bonifica”). Monitore Tecnico.

Supino, G., 1947. A few observations on the using of “volume di invaso” method to evaluate the specific flood contribution given by unit drained area (in Italian. Original title: “Alcune osservazioni sul metodo del volume di invaso per la determinazione del coefficiente udometrico”, but with french and english resumes in the Proceedings of XVIII Navigation Congress, Roma, Italy, 1953). L’Acqua.

Supino, G., 1965. The Hydraulic Networks (In italian. Original title: “Le Reti Idrauliche”). Casa Editrice Pàtron, Bologna (Italy), pp.227-231.