(b)
Corrado Gisonni–Roberto Greco
Dipartimento di Ingegneria Civile – Seconda Università di Napoli (DIC-SUN)
via Roma, 29 – 81031 Aversa (CE) - Italy
Tel. +39-081-5010207 , Fax. +39-081-5037370
E-mail: gisonni@unina.it, robgreco@unina.it
Rudy Gargano
Dipartimento di Meccanica, Strutture, Ambiente e Territorio (DIMSAT)
Università di Cassino
via G. Di Biasio, 43 – 03043 Cassino (FR) - Italy
Tel. +39-0776-299751 – E-mail: gargano@ing.unicas.it
Abstract: In 1997 the city of Naples was heavily hit by tragic event due to sewer failure and consequent formation of large cavities below the road level. The failure mechanism is strongly depending on particular soils involved. In this paper the phenomenon is highlighted, with respect to leakage from sewer networks subject to pressurised flow. A simple procedure is proposed, in order to estimate the risk correlated to pressurised flow in sewer channels buried in unsaturated pyroclastic soils. The procedure defines a critical event with respect to the risk of excavations over channel top, by studying the time dependent flow in unsaturated soil caused by leaching from pressurised sewer channel. A dimensionless abacus is derived, allowing to identify critical events for general situations. Some criteria to evaluate the reliability of an existing sewer network are presented, with special respect to the probability of a structural collapse of the channel together with the surrounding soils. Finally, an application of the procedure to a channel of Naples sewer network is presented.
Keywords: urban hydrology, sewer failure, unsaturated pyroclastic soil, reliability index
Urban areas are continuously expanding due to population growth. Consequently it is very important to estimate the reliability of existing sewer networks subject to new operating conditions which are usually very different from the design conditions. In fact, when designing a sewer channel some important parameters must be fixed such as maximum filling ratio h/D, where h is the design flow depth, and maximum velocity Vmax. Design discharge is usually estimated by means of hydrologic approach and refers to a fixed return period T. Variations of operating conditions are normally due to connection of new urban areas to an existing sewer system as well as increase of impervious areas (increase of runoff coefficients). Of course these factors should be taken into account when designing a new sewer system, but this prediction is not satisfactory when dealing with some decades old sewer networks. In this case sewer channels are subject to pressurised flow with a frequency higher than assumed during design; this condition can still be tolerable if consequences are only consisting in poor hydraulic performance. Unfortunately, for old sewers pressurised flow often leads to heavy damages in terms of structural failures or roads instability. This is the case for the city of Naples (Italy) where, in 1997, after heavy storms, large failures occurred and lives losses were recorded.
The failure mechanism is due to the dependency on water content of mechanical characteristics of pyroclastic soils in which hydraulic networks are buried. Strong reduction of shear strength have been observed for these soils in wet conditions, even up to almost complete disappearance of cohesion already at water potential height of about -1 m. This is due to mechanical actions exerted by air-water menisci on soil particles. When water fills up the smallest voids, these actions weaken and, even under small loads, soil particles collapse reaching a more compact structure. Increase of pyroclastic soil bulk density up to 40% has been indeed measured at saturation (Scotto di Santolo et al., 2000).
Pressurised flow in sewer channels may cause leakage of water into unsaturated pyroclastic soil layers. Consequently collapsed incoherent soil can be scoured and flow away through channel cracks when pipe flow conditions end up, causing excavations around the top of the conduit. Several examples of excavations have been observed in sewer network channels of Naples (Fig. 1), where a number of branches of the original last century network are today inadequate and undergo pipe flow conditions.
In the following, a new simple procedure is introduced in order to assess the reliability of a sewer network with respect to the excavation. The procedure consists of the following steps:
l Hydrologic and hydraulic study of i-th sewer channel, in order to define the duration D and the pressure head hp of pipe flow conditions (Q>QF,i, whith QF,i maximum open channel discharge) during a flood;
l Study of time dependent unsaturated flow in the soil surrounding a generic channel, in order to introduce, for each value of pressure head hp* inside the channel, a critical duration D* of leaching giving rise to dangerous excavation;
l Identification of the critical event as the critical flood (hp,i*,Di*) associated with the smallest return period, T*;
l Reliability assessment with local and overall indices.
In a quite original way the last step of the methodology allows, through a reliability study, to identify channels more frequently working in a critical condition, making possible planning of rehabilitation works and improvement of ancient drainage networks.
(a)
(b)
Fig. 1 Excavation over the top of the sewer channel: (a) photo, (b) scheme of the phenomenon
So far, several and valuable papers about reliability of hydraulic distribution systems have been proposed (e.g.: Mays, 1989; Cabrera and Vela, 1995). Conversely, technical literature is still poor of contributions dealing with reliability of sewer systems, although operating conditions of sewer networks strongly depend on random variables.
In order to state a synthetic and objective evaluation of reliability, a local reliability for singular sewer channels Ri is defined. Besides, an overall index is defined to estimate reliability for entire sewer system Rnet.
In this paragraph a methodology to perform hydraulic analysis of a sewer network is exposed. For the sake of clarity, the procedure is explained with special reference to the city of Naples, but it can be easily extended to general contexts.
The first part of the study consists in a detailed investigation, in order to define the main characteristics of sewer network and corresponding basins. The depth-duration-frequency (DDF) curve for the rainfalls has been estimated by means of a specific hydrological analysis. As an example, for the city of Naples, the return period of the DDF curve has been computed by referring to a frequency factor KT derived from a regional analysis in Campania, based on a Two Component Extreme Value (TCEV) distribution. More precisely, KT is equal to 0.87, 1.16, 1.38, 1.64, 1.80, 2.03, 2.36 for T equal to 2, 5, 10, 20, 30, 50 and 100 years respectively.
The DDF curve is given by the following relationship (Del Giudice, 2000):
[mm]
(1)
For the case of Naples, the linear reservoir model was used to estimate effective runoff hydrographs from the outlet section of any urban catchment served by the sewer network. The linear reservoir constant Ki was estimated by means of Desbordes formula, which was calibrated on urban catchments similar to the Neapolitan ones; this gives:
[min]
(2)
with Ai catchment extension (ha), si main channel average slope (m/m) and Imi ratio between impervious and total catchment area, referred to the i-th channel.
According to the phenomenon analysed in this paper, it is more important to consider the duration of pressurised flow in the sewer, rather than the peak discharge. Thus, once the maximum open channel discharge QF,i is given for a certain section, it is possible to evaluate, for different return periods T, duration D during which discharge exceeds the limit QF,i. It can be easily demonstrated that D is expressed by means of the following relationship:
(3)
where φi is the runoff coefficient, and the rainfall
intensity is depending on the return period T and the duration tp through the relation (1).
Consequently, it is possible to compute the average discharge Qav during the duration D. Fig.2 shows the plots (tp, D) and (tp, Qav) for different return periods T, referred to the
channel considered in Fig. 1a.
When Q>QF,i,
pipe flow establishes in the channel and, assuming uniform motion, it is
possible to associate unambiguously a gradeline slope J to a given T.
Conversely, the position of gradeline, and thus the pressure head over the top
of the channel, is undetermined, since it mainly depends on flow conditions in
the downstream branches of the network.
Fig. 2 Plots (tp, D) (continuous line) and (tp, Qav) (dotted line) for different return periods T (years)
A rough estimation of gradeline position can be obtained by assuming open channel flow in the subsequent downstream channel. In this hypothesis, Fig. 3a shows the position of gradeline in a channel with slope s0,i.
The first upstream section, where maximum pressure head hp=Li(J-s0,i)+lv2/(2g) (with 0£l£1) is attained over the top of conduit, is the most dangerous with respect to leaching into surrounding unsaturated soil.
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(b)
Fig. 3 Scheme of the pressurised flow (longitudinal section) and unsaturated soil flow (transverse section)
In unsaturated pyroclastic soils, leaching of water causes excavations over channel top. In this paper soil potentially subjected to excavation is conventionally defined as the zone where water capillary potential h(x, z)>-1.0 m. Every hydrograph characterised by duration D* and pressure head hp* giving rise to height of this zone over the top of the conduit equal to D/2 is defined as critical. The critical event is represented by the critical hydrograph having the smallest return period T*.
Water capillary potential field h(x, z) in a homogeneous unsaturated soil layer around a long leaching pipe (see Fig. 3b) is described by 2D Richards equation:
(4)
with θ volumetric water content and k unsaturated hydraulic conductivity; the relevant initial and boundary conditions are:
(5)
In (5) Ω represents the half plane in which h(x, z) is defined and G=G(x, z) is conduit boundary equation.
Soil water capillary potential tends asymptotically to a steady distribution, which can be analytically expressed if exponential hydraulic conductivity curve, k(h)=ksat·eah, and water retention curve, θ(h), without hysteresis are assumed (Philip and Knight, 1997). Steady water potential distribution depends on channel through D and z0, and on soil hydraulic characteristics only through the sorptive number a, a parameter close to 2 m-1 for a wide range of soils (Philip, 1968). Saturated hydraulic conductivity ksat, varying over several orders of magnitude for different soils, only affects the time at which steady distribution is reached. Fixing the value of a, it is thus possible to study transient flow in dimensionless variables by introducing D as characteristic length and D/ksat as characteristic time.
Numerical simulations performed by means of a finite elements algorithm show that, for usual duration of critical rainfalls in a sewer network, rarely exceeding few hours, steady distribution is far from being reached, even with the highest values of ksat. Thus, numerical simulations of transient flow have been carried out, for a typical pyroclastic soil from Neapolitan area, with k(h)=ksat·eah (a=2 m-1) and θ(h) according to Brooks and Corey model (Scotto di Santolo et al., 2000), by neglecting hysteresis.
Results are given in Fig. 4a in dimensionless plane (ksatD/D, hp/D), in which a point represents a flood event with known duration D and pressure head hp inside the conduit. Curves plotted in the diagram represent the relationship between dimensionless critical duration ksatD*/D and relevant pressure head hp*/D inside the conduit, for various values of initial soil water capillary potential h0/D. Points falling on and above plotted curves represent critical hydrographs.
(a)
(b)
Fig. 4 Dimensionless critical curves (a) and their application to a channel of Naples sewer network (b)
For sewer networks buried in unsaturated pyroclastic soils, according to the above described phenomenon, it is possible to define the unreliability of the i-th channel as the probability that a critical event happens in a defined time interval. Since the probability of a critical event is equal to 1/Ti*, by supposing that proper working condition and failure are two independent and exhaustive events for each channel, unreliability Fi and reliability Ri are respectively given by:
;
(6)
The second expression (6) evaluates reliability as the probability that working conditions don’t give rise to a critical event. Thus, for the i-th channel (D, hp) curve expressing hydraulic working conditions falls entirely below the threshold (D*, hp*) curve.
If Ti* is expressed in years, expressions (6) evaluate unreliability and reliability of i-th channel with reference to year time interval.
Expression (6) alone cannot estimate the influence of each channel on the overall sewer network. In particular, failure of a channel may cause several inefficiencies in the upstream channels. The more important is the hierarchical position in the network of the failing channel, the larger may be the number of upstream channels subject to outworking condition because of it. Consequently, weight has to be defined, in order to consider the incidence of each channel on the overall sewer network operating conditions.
It is possible to suppose that the extension of the drained area represents the most important parameter to be considered as a weight. So, the weight for the i-th section is:
(7)
where N is the total number of channels forming the sewer network analysed; the numerator represents the overall area drained by the i-th section; the denominator evaluates the sum of all areas drained by the N sections.
The weighted average of the local reliability indices gives the overall reliability Rnet of a sewer network:
(8)
Expression (8) allows to easily evaluate the sewer system reliability whatever the topologic scheme of the network is.
The proposed methodology has been applied to the case of the same channel chosen for the hydrologic study. With reference to Qav curves plotted in Fig. 2, corresponding hp values have been calculated by means of the above introduced expression hp=Li(J-s0,i)+lv2/(2g), by assuming a safe value of l=1. The obtained (D, hp) curves (continuous lines), for different return periods T, are plotted in Fig. 4b together with the (D*, hp*) curve obtained for D=1.60 m, ksat=5·10-5 m/s and h0=-16 m (dotted line). Such high value of h0 is due to the great depth of groundwater table in the considered case.
The (D, hp) curve crosses the (D*, hp*) curve already for T=2; therefore, all of the considered return periods give rise to failures. Thus, the critical event, according to the above given definition, is associated with T=2 years and, consequently, extremely high failure risk and local reliability index Ri for the considered channel smaller than 0.5. This result has been sadly confirmed by tragic events happened in 1997, when a large cavity developed up to the road level from the channel buried at about 25 m below the ground level.
A new methodology for the assessment of risk of
excavations, caused by pressurised flow, around a sewer channel, buried into
unsaturated pyroclastic soil, is presented. Since excavations are due to rapid
decay of mechanical properties of wet soil, a hydrograph is defined as critical,
when leading to a height D/2 of wet
soil over the top of a channel of diameter D. By means of hydrological study and simple hydraulic
schematisation, duration of pressurised flow and mean pressure head inside the
channel can be defined, corresponding to any given rainfall duration. Time
dependent flow through surrounding unsaturated soil, due to leaching from
pressurised channel, is numerically solved, leading to dimensionless abacus
useful for general identification of critical
events. An application to a real case is then briefly described, confirming
the validity of the followed approach. Furthermore, it is shown how the outlined
methodology leads to the definition of local and overall reliability indices of
sewer networks with respect to the considered phenomenon.
The authors would like to acknowledge the support of Prof. Giacomo Rasulo, Director of the Hydraulic and Environmental Engineering Department of the University of Naples “Federico II” and Head of the Neapolitan Hydraulic Division of “Large risks forecast and prevention Center” (C.U.G.RI.).
References
Cabrera, E. and A.F. Vela (Editors), 1995, Improving efficiency and reliability in water distribution systems, Ed. Kluwer Academic Publishers, Dordrecht.
Del Giudice G. (2000): La pluviometria della città di Napoli (in Italian). From “ Il sistema fognario della città di Napoli alle soglie del 2000”. Edited by Rasulo G. Published by the City Council of Naples (Italy). CUEN, Napoli (Italy), October 2000.
Gisonni C.: I moderni criteri di calcolo e la verifica della rete (in Italian). From “Il sistema fognario della città di Napoli alle soglie del 2000”. Edited by Rasulo G. Published by the City Council of Naples (Italy). CUEN, Napoli (Italy), October 2000.
Mays L. W. (Editor), 1989, Reliability analysis of water distribution system, Ed. American Society of Civil Engineers, New York.
Philip, J. R., 1968, Steady Infiltration from Buried Point Sources and Spherical Cavities, Water Resources Research, 4: 1039-1047.
Philip, J. R. and J. H. Knight, 1997, Steady infiltration flows with sloping boundaries, Water Resources Research, 33: 1833-1841.
Scotto di Santolo, A., M. V. Nicotera and M. Ramondini, 2000, Analysis of instability phenomena affecting a cut slope in unsaturated pyroclastic soils, Proc. of 8th International Symposium on Landslides, Cardiff.