Some Aspects of the

Hydraulic Design of Roman Aqueducts

Hubert Chanson © 2004

http://www.uq.edu.au/~e2hchans/
TRAIANVS © 2004

Published in La Houille Blanche, 2002, No. 6/7, pp. 43-57 (in French)

Abstract :

The Roman aqueducts were large water supply systems delivering water for public health usage. However little is known on the hydraulic design of the aqueducts. The Roman engineers, contemporary of Hero of Alexandria, had a strong expertise and experience in hydraulic engineering. Several regulations basins were found along the aqueducts. It is proposed that the operation was based upon a dynamic regulation principle. Associated issues are discussed. two types of major hydraulic structures were also built : dropshaft cascades and culverts. The hydraulic engineers who designed these structures has a sound knowledge and understanding of basic hydraulics. Who were they ? Although we do not know, the writer is impressed by their technical expertise. They knew more than modern hydraulic engineers !

Introduction

The hydraulic expertise of the Romans contributed significantly to the advance of science and engineering in Antiquity. Aqueducts were built primarily for public health and sanitary needs: i.e., public baths, thermes, toilets (HODGE 1992, FABRE et al. 1992,2000). Many were used for centuries; some are still in use, for example at Carthage. Magnificent aqueduct remains at Rome, in France, Spain and North Africa, for example, are still standing (e.g. Carthage, CLAMAGIRAND et al. 1990; Mons à Fréjus, VALENTI 1995a,b) (Fig. 1). Aqueduct construction was an enormous task often performed by the army and the design was undertaken by experienced army hydraulicians. The construction cost was gigantic considering the small flow rates (less than 0.5 m³/s) and was around one to three millions sesterces per kilometre on average (e.g. FEVRIER 1979, LEVEAU 1991)[1].

  Figure 1

Although magnificient remains are still standing (e.g. ASHBY 1925, RAKOB 1974, BURDY 1996), little is known on the hydraulic engineering of Roman aqueducts, the design procedure and the design engineers. Some suggested that Roman engineers did not know the principle of conservation of mass (GARBRECHT 1987, HODGE 1992). However, the major aqueducts in Gaul and North-Africa were built at the time of Hero of Alexandria. Hero understood the concepts of continuity and momentum, and he built the first steam turbine. His influence might have affect Roman hydraulic engineers. Herein the writer demonstrates the soundness of design of Roman aqueducts, a good common sense of the engineers, and even some innovatgive design techniques, such as regulation basins, dropshaft cascades and culverts.

Hydrology and operation of two aqueducts

Hydrology

Despite their fame, little is known on the water supply and flow rates of the aqueducts. Some answers may derive from the studies of the Roman aqueduct catchment and its hydrology. A comparison between the Gorze and Nîmes aqueducts is pertinent (Table 1). Both water supply systems were equipped with wide channels and equipped with a monumental aqueduct bridge [2]. The aqueducts were supplied by natural springs with similar catchment area. Details on the Gorze aqueduct are given in Table 1. FABRE et al. (1991, 2000) presented a comprehensive study of the Nîmes aqueduct.

The aqueduct springs are still used today. Figure 2 presents the daily streamflows of the Gorze aqueduct source, the data being recorded in 1997 and 1998 [3]. It shows the monthly average, minimum and maximum daily flow rates. For the study period, the average flow rate was 8,050 m³/day (93 L/s). Figure 2 suggests that a modern aqueduct could not operate at maximum flow rates for more than few months per year in average : i.e., 3 to 5 months at Gorze. During dry periods, the water supply was reduced by the source input. At Gorze, the minimum daily flow rate (over the two-year period) was less than 10% of the maximum (Fig. 2, October month).

Figure 2

FABRE et al. (1991, 2000) (see also BOSSY et al. 2000) reviewed a number of hydrological studies of the Eure spring [4] at Uzès (Nîmes aqueduct). The average streamflow was 29,600 m³/day (343 L/s). The results showed an important  variability of the spring output (at Uzès). Over the study period, the minimum flow rate was 10,800 m³/day (125 L/s) in Sept. 1976 and the maximum discharge was 143,400 m³/day (1660 L/s) in Oct. 1976. The difference between the maximum and minimum daily flow rates was about 13.3 to 1.

VALENTI (1995a) detailed 13 years of spring outflows for the source of La Siagnole at Mons (period 1981-1993). The daily average flow rate was 1,125 L/s. The results demonstrated however a high variability of daily flow rates. The lowest daily flow was zero (Aug. 1986), and the highest record was 17,900 L/s (April 1993).

Ancient flow rates are unknown and there is little information on the ancient climate. Dr P. LEVEAU argued that the climate in Southern France, at the beginning of our era, differed from present days. Nonetheless, the knowledge of present discharges may provide some information on the water supply variability. In the 20th century, the minimum daily flow rate of the aqueducts was less than 10% of the maximum daily flow rate. In a given month, variations of the daily discharge were in average about +/-35% at Gorze, with extremes ranging from 40% up to 200% of the mean in October. That is, from 1,100 m³/day to 8,300 m³/day in October at Gorze. The writer believes that such discharge variations were likely to occur also in Roman times, and these large daily variations of the water supply must have had implications on the water distribution. That is, reservoirs and cisterns had to be used to regulate the water distribution in town. For example, baths were equipped with cisterns and reservoirs : e.g., Grands Thermes of Cuicul (ALLAIS 1933); at Autun, a cistern was found Place St Louis, at the upper end of the city where the castellum was supposedly located, at the arrival the Montjeu and Mondru aqueducts. Large reservoirs were found at the intakes of the Gier and Anio Vetus aqueducts, and in Carthage.

In terms of maximum flow rates, no definite conclusion is achievable. One would note, however, that the maximum flow rate observation of about 10,000 to 15,000 m³/day at Gorze, and the average streamflow of 30,000 m³/day at Uzès are close to the maximum aqueduct flow rates suggested by FABRE et al. (1992), LEFEBVRE (1996) and CHANSON (1998).

Regulation basins

Despite arguments suggesting that the aqueduct flow was not regulated, the present study demonstrates the existence of in-stream regulation systems. These were large regulation basins equipped with sluice gates. Their operation is discussed based upon two cases : the Nîmes and Gorze aqueducts in France. The writer will show that regulators operated between the source and the city. Based upon basic engineering considerations, new ideas will be proposed.

In pipes, valves and taps are the most common types of regulatory devices. In open channels, gates (or sluices) are more appropriate. The two basic types of sluice gate are the undershoot and overshoot gates. With an undershoot, or underflow gate, the outflow is delivered underneath the gate edge. The outflow may be a 'free' jet or a 'drowned' flow. In the former case, the jet flow is supercritical (torrential) while the flow upstream of the gate is tranquil (subcritical). In the latter case, the flow is subcritical downstream of the gate. At an overshoot, or overflow gate, the water discharges over the upper edge of the sluice. For low tailwater levels, the overflow forms a free-falling nappe. For high downstream water levels, the jet is drowned.

There is a major difference between the hydraulic operation of undershoot and overshoot sluices. With an overshoot gate, a small variation of the upstream water level induces a large change in discharge. At an undershoot gate, a change in upstream water level is associated with a smaller variation in flow rate. For underflow and overflow gates, the relationship between the flow rate and the upstream water level satisfies respectively:

	Undershoot sluice
	Overshoot sluice

where d₁ is the upstream water depth and H is the weir height. Practically the overflow gate is commonly used in spillway design and overflow system. A small increase in upstream water level induces a large increase in flow rate. An underflow sluice is better used to control the downstream flow rate : i.e., as a regulation gate.

Aqueduct regulation systems

Although VITRIVIUS recommended to install regulatory devices at regular intervals (e.g. HODGE 1992, p. 165), little information is known on the practical details. Few remains of regulators were found or studied so far (Table 2 , Fig. 3) [5]. Why were regulation devices required ? In the writer's professional opinion, the regulation of the flow was a necessity : (1) to prevent overflows and unsatisfactory aqueduct operations during wet seasons, and (2) providing optimum flow conditions (minimum energy losses and maximum flow rates) to supply satisfactorily the city during low-flow seasons. During wet periods, large flow rates could overflow the sidewalls, or exert pressures on the rood of covered sections. Overflowing waters could wash away foundations in soft soils. Altogether the control of the aqueduct flow was required to prevent damage to the aqueduct and disruption of the water supply.

CHANSON (1998,2000) highlighted the existence of steep sections. The operation of these chutes was characterised by supercritical (torrential) flow motion. Both upstream and downstream control structures had to be installed. That is, upstream of the steep chute and downstream of the dissipation structure. Supercritical flows can only be controlled from upstream while subcritical flows are best controlled from downstream (e.g. HENDERSON 1966, CHANSON 1999,2004a). Control structures were a necessity to prevent improper flow operation that could result in major scour, damage and destruction.

Sluices and control structures had to be a design feature of the aqueduct. Optimum locations [6] included at the source of the aqueduct, upstream of steep sections, upstream of bridges and tunnels, downstream of flat sections [7]. In term of gate design, the writer believes that in-stream gates, installed in the main aqueduct channel, were undershoot sluices to deliver reasonably constant flow rates. The gates of overflow systems were most likely an overshoot gate type (Fig. 3).

Figure 3

Figure 4

Two case studies: Gorze and Nîmes

Both the Gorze and Nîmes aqueducts were equipped with regulation basins (Table 2). The Gorze aqueduct crossed the Moselle river between Ars-sur-Moselle and Jouy-aux-Arches. The 1-km long aqueduct bridge was equipped with an unusual canal geometry: i.e., two 0.85-m wide parallel channels with a 0.35-m wide dividing wall (LEFEBVRE 1996) [8]. At the upstream end, the canal was regulated by a basin (Ars-sur-Moselle) (Fig. 2A) while a stilling basin was located at the downstream end (Jouy-aux-arches). The Nîmes aqueduct crossed the Gardon river at the Pont-du-Gard. FABRE et al. (1991) found a regulation basin at the upstream end of the bridge canal while BOSSY et al. (2000) reported two further regulation systems.

The two regulation basins located upstream of the aqueduct-bridges were equipped both with an overflow channel (on the left side of the basin) and an outlet feeding the bridge canal. Slots for sluice gates were found at the start of the outlet(s) and at the overflow (FABRE et al. 1991, LEFEBVRE 1996). The writer believes that the bridge canal intakes were equipped with an undershoot gate system while the overflow systems were equipped with overshoot gates (CHANSON 2002a).

For each aqueduct, the sluice gate operation at the canal intake was investigated for flow rates up to the maximum flow rate. Typical results are shown in Figure 3 showing the discharge in the bridge canal, in horizontal axis, as a function of the water level in the regulation basin, in vertical axis. The calculations were performed for several gate openings and flow rates. The results show that an undershoot sluice gate could regulate the flow only for gate openings less than 0.07 to 0.1 m at Gorze and less than 0.1 to 0.12 m at Nîmes. For larger openings, the flow was not be affected by the presence of gate. Practically, the regulation of the aqueduct required relatively small gate openings. Fine gate adjustment devices (e.g. +/- 1 cm) had to be used if an accurate flow control was required.

Bridge-canal hydraulics and backwater effects

The backwater profile (i.e. free-surface profile) was investigated in the downstream canal for several flow rates, gate openings and for each aqueduct bridge [9]. At Gorze, the results highlight the unfavourable presence of a hydraulic jump for flow rates ranging from 700 to 15,000 m³/day. That is, an undular hydraulic jump. The jump location was a function of the gate opening : the distance gate-to-jump toe increases with decreasing gate opening heights. At Pont-du-Gard, the calculations show a drowned gate operation for most flow conditions, but very-small gate openings. That is, drowned gate operation occurred for h > 0.01 to 0.015 m for flow rates between 10,00 and 35,000 m³/day, where h is the gate opening. The downstream flows were subcritical (fluvial or tranquil regime).

Discussion : what type of regulation ?

At Pont-du-Gard, it believed that the regulation system was built to prevent spillage over the bridge canal sidewalls and in the downstream section. FABRE et al. (1991,1992) highlighted indeed that the average gradient between Pont-du-Gard and St-Bonnet (further downstream) was very small (S_o = sin = 0.007%). The very flat geometry created a backwater effect causing higher water levels in the channel for identical flow rates. Indeed the water depth is inversely proportional to the cube of the bed slope ( where d is the flow depth and  is the angle between the invert and the horizontal) [10].

The Ars-sur-Moselle basin at Gorze was designed to regulate the bridge-canal operation, and possibly to interrupt the flow. The overflow canal was large and could pass the maximum flow rate [11]. Overall the basin was a better design in the writer's opinion. The lower basin invert and the larger cross-section area contributed to better flow conditions in the basin [12], leading to better gate operation. Indeed sluice gate operation may be affected by improper inflow conditions leading to a discharge reduction (for a given gate opening) and gate vibrations.

There is still some uncertainty on the operation of the gates. Were the gates operated only for On/Off operation (i.e. 100% flow rate or zero flow rate) ? Were fine adjustments of the gate position performed regularly (e.g. daily) ? Based upon present experience in irrigation water systems, the writer believes that the gates were designed : (a) to limit the downstream discharge during wet seasons (the overflow discharge would spill over the regulation basin sidewall gate), and possibly (b) to control the downstream flow rate in response to the city water needs during the rest of the year.

The first type operation requires only one gate opening. Regular inspection and maintenance check are needed only for the overflow channel. In addition, the aqueduct channel had to be cleaned regularly for water sanitation. The simplest procedure was to perform the cleaning when the aqueduct was empty, but flushing some water and the debris up to a pit and to clean the pit. (This technique is still used today to clean hydraulic laboratories worldwide.) The second type of operation implies a regular transfer of information from the city water engineer to the gate operators, and the suitable adjustment of the gate opening by the caretaker(s). It is conceivable that this operation was done on a daily basis (i.e. daily gate position adjustment).

At Nimes (bassin de la Source de l'Eure, and regulator upstream of the Pont du Gard) and at Gorze (upstream of Pont), the overflows and the outflow channels were equipped with a double groove system, suggesting two vertical gates at each location. In modern dam outlets, it is common to design two gates. The upstream gate operates fully-open or fully-closed. Its main purposea are to dewater the downstream outlet, allowing maintenance, and to act as a safety gate if the second gate fails. The downstream gate is a regulation gate used to control the downstream water release. The writer believes that a similar double-gate system was installed at the Nîmes and Gorze aqueducts.

FRONTINUS indicated the need for gangs of maintenance workmen (FRONTINUS, 117 [13]). Outside of the city of Rome, repairs demanded "prompt attention". He further classified repair works as either conducted "without stopping the flow of water, or such as cannot be made without diverting the flow" (FRONTINUS, 121). FRONTINUS' statements indicate clearly that the aqueduct flow could be stopped completely, although he added that the aqueduct flow should "be out of commission as few day as as possible" (FRONTINUS, 122). BOSSY et al. (2000) showed the existence of several regulation basins along the Nîmes aqueduct suggesting the possible interruption of the flow in several locations. They hypothesised further a daily gate operation for a dynamic regulation of the aqueduct flow with water storage at night.

Discussion

There is a definite relationship between water supply and water usage. Up to date, little if none information is available to estimate the daily water consumption in Roman times. HODGE (1992, p. 464) suggested a water consumption around 200L/day/person although he mentioned also the work of ESCHEBACH at Pompeii leading to an individual water consumption of about 500L/day (HODGE 1992, p. 305). A comparison with modern times is interesting. For the period 1997-98, the water consumption per capita in Brisbane (Australia) was 240 L/day (Reference: Brisbane City Council). By comparison, the household of the writer (3 people) used about 400 L/day (Ref. : Water rate, Brisbane City Council) for the period Jan.-Mar. 2000. The latter figure does not include industrial usage, but it applies to an efficient water pipe systems with modern valves (i.e. no running water).

Dropshaft cascades

Most aqueducts consisted of long, flat sections with sometimes shorter steep drops (Table 3). Despite arguments suggesting that Roman aqueducts maintained a fluvial flow regime, it was shown that steep drops produced supercritical flows requiring an energy dissipation system to ensure normal downstream flow operation (CHANSON 2000a,2002b,2004b). Despite unusual hydraulic features, Roman dropshafts (Fig. 5) were not well documented and their role was not clearly understood. CHANSON (1998, 2000a) presented probably the first well-documented survey of dropshafts, installed along Roman aqueducts. He further studied the hydraulic performances of the dropshaft design (CHANSON 1998,1999b). The survey showed the existence of single dropshaft construction as well as complex cascades (or series) of dropshafts. The study is incomplete however.

Figure 5

Although over 30 dropshaft structures were listed by CHANSON (2000a), some information were found subsequently to be inaccurate or misleading. Up to date, the writer has obtained detailed information and drawings of only a small number of dropshaft sites. That is, the following ten dropshafts :

- Cherchell (1 dropshaft),

- Valdepuentes, Cordoba, Spain (2 dropshafts) after VILLANUEVA (1993,1996) [14],

- Recret, Yzeron (2 dropshafts),

- Vaugneray, Yzeron (1 dropshaft),

- Gunudu, Tunisia (1 dropshaft),

- Cologne, Germany (1 dropshaft),

- Montjeu, Autun (1 dropshaft) [15],

- Beaulieu, Aix-en-Provence (1 dropshaft) [16].

Most studies on the Montjeu aqueduct, Autun, relied on the original work of ROIDOT-DELEAGE (1879?) : e.g., COQUET (1966), PINETTE and REBOURG (1986), CHANSON (1998). Jean ROIDOT-DELEAGE (1794-1878) was an engineer at the Ponts-et-Chaussées from 1820 to 1833. He became the architect of the city of Autun in 1859 and he studied the Montdru and Montjeu aqueducts until his death. His work was published posthumously and the only remains are drawings kept at the Société Eduenne, Musée Rolin, Autun (ROIDOT-DELEAGE 1879?) [17]. His drawings of the Montjeu aqueduct show 24 "puits" (or shafts), a term used for both dropshafts ("puits de rupture") and inspection shafts ("regards"). The writer inspected parts of the aqueduct in September 2000. The path of the aqueduct is relatively flat but for two short steep sections in the Forêt de Brisecou and at Pierre de Couhard. It is the writer's opinion that the only dropshafts were the "puits" No. 18, 19, 20, 21, 22, and 24 (at Pierre de Couhard), and possibly the "puits" No. 10 and 23. The "puit" No. 10 was probably a small dropshaft. Further the dimensions of the shafts were not identical as suggested by PINETTE and REBOURG and by CHANSON.

The studies of the Valdepuentes aqueduct at Cordoba [18] showed the existence of at least three major dropshaft cascades : i.e., at Cerro de los Pinos upstream of Valdepuentes bridge (Fig. 5B), at Madinat-al-Zhara, and at Cortijo los N., downstream of junction with Veneros branch junction. The total drop in elevation at Cerro de los Pinos and Madinat-al-Zhara dropshaft cascades was respectively 120 m and 200 m. The Cerro de los Pinos dropshaft cascade was very steep (total drop of 120 m over a 400 m long distance) and an unusual spiramina design was used. It consisted of three 90-degree dropshafts. This is the only documented dropshaft design of this type. Five 90-degree shafts were possibly parts of the Montjeu aqueduct although it is uncertain whether these were dropshafts.

Further observations of Roman shafts were reported but there are doubts whether these were wells, cisterns, inspection shafts or dropshafts (CHANSON 1998, 2002ba). At Cherchell, LEVEAU and PAILLET (1976) described several very-steep sections : Zabrir Clift (H ~ 20 m) [19], Bouchaoun gorge (H ~ 10 m) [20] and Oued Bellah (H = 37m) [21]. LEVEAU and PAILLET hypothesised the existence of series of dropshafts at these three sites. As no dropshaft trace was found, it is uncertain whether the final design was dropshafts, cascades, steep chutes or a combination of different types as at Chabet Ilelouine. Near the source of the Aïn Nadour branch of the Hippo Zarite aqueduct (Tunisia), GAUCKLER (1902, pp. 126-127) observed the presence of seven circular shafts (1-m diameter, 2.5-m deep). The main channel specus was 0.2-m wide and 0.3-m high. GAUCKLER [22] described four shafts located on a steep slope [23]. It is likely that these four were circular dropshafts.

Six kilometres West of Zaghouan (Tunisia), GAUCKLER (1902, p. 146) described a Roman circular well (3-m diameter, over 7-m deep), feeding a 500-m long subterranean rectangular conduit. The geometry of the shaft is close to that of Chabet Ilelouine, Cherchell, suggesting possibly a dropshaft design.

Discussion

The design of an aqueduct was a difficult task (GREWE 1992, HODGE 1992). In particular the hydraulic design of dropshaft was not and is still not today a simple job (CHANSON 2002a,b). It was a highly specialised task and the advice of an experienced engineer was required. GREWE (1992) and HODGE (1992) elaborated on the difficulties to design an aqueduct.

Dropshaft hydraulic calculations are among the most difficult hydraulic engineering calculations. Even research on dropshaft hydraulics is limited : i.e., there are only 5 international refereed journal articles on dropshaft design listed in Science Citation Index, the Web of Science® for the period 1985-2000.

The writer believes that dropshaft expertise in Roman times was restricted to a handful of engineers. Who were they ? Although there is no written proof that the engineers understood the basic concepts of continuity and energy, as used in modern hydraulics, they were contemporaries of Hero of Alexandria who understood the principle of continuity, probably those of momentum and energy [24]. It is likely that he also influenced the Roman hydraulicians of the 1st, 2nd and 3rd centuries AD, and possibly, the designers of the Montjeu and Valdepuentes aqueducts.

In any case, the aqueduct engineers designed very reliable dropshaft structures, used for centuries [25]. The sound design implied a great deal of engineering experience.

Oversizing

CHANSON (2000) argued that the dropshafts of the Yzeron (Vaugneray branch), Cherchell and Montjeu aqueducts were oversized. Dr P. LEVEAU commented that oversizing a structure ("surdimensionnement") was not a problem : i.e., neither economical cost nor aesthetics were an issue. Mr STRASBERG similarly mentioned that many structures at Autun  [26] were unusually oversized, possibly to display a show of power and wealth.

Although the writer may accept the ideas of oversized visible structures (e.g. roads, buildings, theatres), he is sceptical of the reasons to oversize (intentionally) underground structures, like dropshafts.

Dropshaft cascade hydraulics

CHANSON (1998, 2002a, 2004c) analysed the hydraulic performances of the dropshaft. The study was however limited to a single dropshaft operation. It was shown that series of dropshaft were built systematically for large drop in specus invert elevation : e.g., at Recret, Valdepuentes, Montjeu (Table 3). The hydraulic performances of a dropshaft cascade differ from those of a single dropshaft.

The operation of the cascade [27] is characterised by hydraulic interference between adjacent dropshafts. In particular the geometry and slope of the connecting channels (in-between dropshafts) may affect significantly the operation of the dropshaft cascade. For flat connecting channels (e.g. Vaugneray), a hydraulic jump takes place downstream of each dropshaft. The hydraulic jump is a very energetic process associated with scour beneath the roller and possibly wave propagation further downstream (e.g. CHANSON 1999a, pp. 56-67). With steep connecting channel (e.g. Cerro de los Pinos, Valdepuentes), the flow in between dropshafts is supercritical or torrential. Lesser energy dissipation takes place at each dropshaft. The topic requires more engineering investigations.

Culverts

A culvert is a covered channel of relatively short length designed to pass water through an embankment : e.g., underneath a highway or a railroad. The Romans utilised the culvert design (BALLANCE 1951). Table 4 shows examples of Roman road culverts. O'CONNOR (1993) indicated that a significant number of culverts were built for small water crossings while a bridge construction was preferred for longer crossings. The most common culvert designs were the arched culvert and the rectangular (or box) culvert (Table 4).

Along the Nîmes aqueduct, a large box culvert was recently excavated (FABRE et al. 1992, CHANSON 2002c) (Table 4). The culvert was designed to allow storm water passage beneath the aqueduct (Fig. 6). It is located between the Combe de la Sartanette and Combe Joseph, downstream of Pont du Gard. While the aqueduct crossings of the Combe de Sartanette and Combe Joseph were bridge constructions (one arch in each case), the culvert was a multi-cell structure equipped with 3 rectangular parallel cells. The cells were made of series of large limestone blocks placed on supporting pillars (or dividing walls) founded on worked bedrock (FABRE et al. 1992). The upstream end of each dividing wall was cut in a pointed shape to form cut-waters. Note that the Bornègre bridge, located between Uzès and Pont du Gard, was composed of three arches with two central piers equipped with upstream cut-waters (FABRE et al. 1992). The writer visited both the culvert and Bornègre bridge sites. It is this opinion that the cut-waters of the multi-cell culvert were better shaped. The cut-waters of the Bornègre bridge are more sturdy and less profiled that those of the multi-cell culvert (60º convergence angle at Bornègre, 45º for the culvert).

Figure 6

It must be recognised that culverts were seldom used beneath aqueducts. Few examples were found and documented (Table 2). In the writer's opinion, the culvert downstream of Pont-du-Gard is an unique example of Roman aqueduct structure. Unusual features include :

- a box culvert design of large dimensions (Table 4),

- a multi-cell structure, and

- a modern and sound design from a hydraulic perspective (see next paragraph).

Hydraulic design of culverts : a modern approach

A culvert is designed to pass water through an embankment. A modern design requires a careful analysis of the upstream catchment to estimate the maximum (design) discharge and the risks of exceptional (emergency) floods. The dimensions of the culvert are based on hydraulic, structural and geotechnical considerations.

A culvert consists of three parts : the intake (also called inlet or fan), the barrel (or throat) and the diffuser (also called outlet or expansion fan ). The cross-sectional shape of the barrel may be circular, rectangular (i.e box culvert), or multi-cell. The bottom of the barrel is called the invert while the barrel roof is called the soffit or obvert. The training walls of the inlet and outlet are called wing walls.

Hydraulic design

The hydraulic characteristics of a culvert are its design discharge Q_des, the upstream water level and the maximum (acceptable) head loss H. Head losses must be minimised to reduce upstream flooding, especially in urban environments. In modern times, the primary design constraints of a culvert are :

[1] the cost must be (always) minimum,

[2] the afflux [28] must be small and preferably minimum,

[3] eventually the embankment height may be given or may be part of the design, and

[4] a scour protection may be considered, particularly if a hydraulic jump might take place near the culvert outlet.

The hydraulic design is basically an optimum compromise between discharge capacity and head loss (CHANSON 1999a, pp. 365-397). From a hydraulic aspect, a dominant feature of a culvert is whether it runs full or not. In practice short culverts are designed for free-surface flow with critical flow conditions [29] in the throat.

When free-surface flow takes place in the barrel (i.e. inlet control), the discharge is fixed only by the entry conditions. The discharge Q is typically estimated as :

	Free-surface inlet flow
	Submerged entrance (2)

where B is the barrel (internal) width, D is the barrel height, g is the gravity constant, H₁ is the upstream total head and z_inlet is the inlet invert elevation (HENDERSON 1966, CHANSON 2004a). C_D equals 1 for rounded vertical inlet edges and 0.9 for square-edged inlet. C equals 0.6 for square-edged soffit and 0.8 for rounded soffit.

For drowned culverts (i.e. outlet control), the discharge is determined by the culvert resistance (i.e. primary and secondary losses) (e.g. US Department of the Interior 1987, Concrete Pipe Association of Australasia 1991, CHANSON 1999a, 2004a).

Hydraulic performances of the multi-cell culvert of the Nîmes aqueduct

The writer inspected the culvert site in September 2000. He noted that the culvert barrel was well located, at the trough of the valley and aligned with the Combe axis. The cells were of similar dimensions as modern precast concrete box culverts. In the writer's opinion, the culvert was a good hydraulic design [30].

Detailed hydraulic calculations were conducted for the multi-cell culvert on the Nîmes aqueduct [31]. Assuming a barrel internal height of 0.65 m, the culvert operated with free-surface inlet flow conditions for flow rates up to 2 m³/s corresponding to an upstream water depth of 0.78 m. For greater upstream flow depths, the barrel inlet was submerged (Fig. 7B). Calculations are summarised in Figure 7A, showing the relationship between the discharge Q in the barrel and the upstream water depth d₁. Figure 7B shows a typical free-surface pattern for large flow rates.

Figure 7

The results indicate that the discharge capacity of the culvert was large. Considering an acceptable maximum upstream water depth of 2 m [32], the culvert could pass up to 4.2 m³/s : i.e., 363,000 m³/day or more than 12 times the aqueduct maximum flow rate. Further the barrel operated with relatively high flow velocities at large flow rates. For example, for a 3 m³/s flow rate, the mean barrel velocity was in excess of 2.5 m/s. For comparison, the larger Bornègre (located upstream Pont du Gard) bridge may experience discharges over 5 m³/s in modern times (FABRE et al. 2000).

While many studies have highlighted the hydraulic expertise of the Romans for small to medium discharges, the writer believes that the sound hydraulic design of the multi-cell culvert, including its large discharge capacity and modern design, demonstrates some hydraulic experience, if not knowledge, in dealing with large stormwater runoff and its conveyance beneath a major structure (CHANSON 2002c).

Summary and Conclusion

The Roman engineers who designed and built the major aqueducts in Roman Gaul and Germany (e.g. Lyon, Metz, Nîmes) were contemporary to Hero of Alexandria who understood the fundamental principles of conservation of mass and of momentum  in fluid mechanics. Despite the lack of written evidence, indicating that Roamn engineers knew Hero's works, the soundness of design and successful operation of their aqueudcts for centuries demonstated their "savoir-faire" and technical expertise. Even today, in the 21st century, the design of a water supply aqueduct is a major technical challenge.

In conclusion, the writer is very impressed by the hydraulic knowledge, experience and expertise of the Roman engineers who designed the regulation basins, dropshaft cascades and culverts. They knew much more that most modern hydraulic engineers ! Yet we know so little of their background.

Acknowledgements

The writer thanks all the people who provided him with relevant infromation, including : Professor C.J. APELT, University of Queensland, Australia; Mr G. BERGE Jussy, France; Dr D. BLACKMAN, Monash University, Australia; Dr J. BURDY, Lyon, France; Mme P. CHARDON-PICAULT, Autun, France; Mme CHOU Y.H., Brisbane, Australia; Dr J.L. FICHES, France; Dr A.T. HODGE, Carleton University, Canada; Mr G. ILLIDGE, The University of Queensland, Australia; Mr C. LEFEBVRE, Châtel-St-Germain, France; Dr P. LEVEAU, Université d'Aix-en-Provence, France; Mr J.C. LITAUDON, Saint-Etienne, France; Mr D. MURPHY, Houston, USA; Professeur N. Rajaratnam, University of Alberta, Canada; Société Mosellane des Eaux, France; Mr A. STRASBERG, Musée Rolin, Autun, France; Mr V. VALENTI, Fréjus, France; Dr A.V. VILLANUEVA, University of Cordoba, Spain.

Appendix I - Hydraulic calculations for a modern culvert

A culvert is designed to carry floodwaters beneath an embankment. Basic hydraulic charactersitics include the design discharge Q_max, the downstream water level and the maximum acceptable afflux. In practice, the design is a balance between the minimum culvert cross-section area and the maximum head loss. For short barrels, the optimum design is the smallest cross-section for which inlet control operation occurs, with critical flow conditions in the barrel. With free-surface flow in the barrel, the discharge equals (for a rectangular cell) :

	Free-surface inlet operation  (1)
	Submerged inlet  (2)

where H is the upstream specific energy, B is the internal width, D is the internal barrel height and g is the gravity acceleration (HENDERSON 1966, CHANSON 1999,2004a). The coefficient C_D equals 1 for rounded edges and 0.9 for sharp edges, while C equals 0.6 for sharp edges and 0.8 for rounded edges.

If the culvert barrel is drowned (i.e. outlet control operation), the discharge can be deduced from design charts (e.g. US Department of the Interior 1987, Concrete Pipe Association of Australasia 1991, CHANSON 1999, 2004a).

Bibliography

ALLAIS, Y. (1933). "L'Alimentation en Eau d'une Ville Romaine d'Afrique : Cuicul (Djemila)," 5th Congrès International d'Archéologie, Alger, pp. 93-117.

ASHBY, T. (1935). "The Aqueducts of Ancient Rome." Clarendon Press, Oxford, UK, edited by I.A. RICHMOND, 342 pages.

BAILHACHE, M. (1977). "Etude de l'Evolution du Débit des Aqueducs Gallo-Romains." ('Study of the Discharge Evolution of Gallo-Roman Aqueducts.') Journées d'Etude sur les Aqueducs Romains, Lyon, France, 26-28 Mai (also Les Belles Lettres Publ., Paris, France, 1983, pp. 19-49) (in French).

BALLANCE, M.H. (1951). "The Roman Bridges of the Via Flaminia." Papers of the British School at Rome, Vol. 19, pp. 78-117 & plates xiv to xix.

BOSSY, G., FABRE, G., GLARD, Y., and JOSEPH, C. (2000). "Sur le Fonctionnement d'un Ouvrage de Grande Hydraulique Antique, l'Aqueduc de Nîmes et le Pont du Gard (Languedoc, France)." Comptes Rendus de l'Académie des Sciences de Paris, Sciences de la Terre et des Planètes, Vol. 330, pp. 769-775.

BOUILLOT, R. (1976). "Hydraulique Agricole et Urbaine." Polycopié de cours, INPG, Grenoble, France, Tome 1.

BURDY, J. (1996). "Préinventaire des Monuments et Richesses Artistiques. IV Lyon. L'Aqueduc Romain du Gier." Bosc Frères Publ., Lyon, France, Conseil Général du Rhône, 407 pages & 1 Map.

BURDY, J. (2002). "Les Aqueducs Romains de Lyon." Presses Universitaires de Lyon, Lyon, France, 204 pages.

CAQUOT, A. (1941). "Sur la Quantité d'Eau Pluviales à Ecouler dans les Agglomérations Urbaines Modernes." Comptes Rendus des Séances de l'Académie des Sciences, Vol. 213, 2nd sem., pp. 509-514.

CHANSON, H. (1998). "The Hydraulics of Roman Aqueducts : Steep Chutes, Cascades and Dropshafts." Research Report No. CE156, Dept. of Civil Engineering, University of Queensland, Australia, 97 pages.

CHANSON, H. (1999). "The Hydraulics of Open Channel Flows : An Introduction." Butterworth-Heinemann, Oxford, UK, 1st edition, 512 pages. {http://www.uq.edu.au/~e2hchans/reprints/errata.htm}

CHANSON, H. (2000a). "Hydraulics of Roman Aqueducts : Steep Chutes, Cascades and Dropshafts." American Jl of Archaeology, Vol. 104, No. 1, Jan., pp. 47-72 {http://www.uq.edu.au/~e2hchans/reprints/aja_200.pdf }.

CHANSON, H. (2000b). "Notes on the Hydraulic Engineering of Roman Aqueducts." Report, Dept. of Civil Engrg., Univ. of Queensland, Brisbane, Australia, Dec., 48 pages.

CHANSON, H. (2000-2001). "Historical Development of Stepped Cascades for the Dissipation of Hydraulic Energy." Trans. Newcomen Society, Vol. 71, No. 2, pp. 295-318 {http://www.uq.edu.au/~e2hchans/reprints/newcom01.pdf}.

CHANSON, H. (2002a). "Certains Aspects de la Conception hydrauliques des Aqueducs Romains." ('Some Aspect on the Hydraulic Design of Roman Aqueducts.') Jl La Houille Blanche, No. 6/7, pp. 43-57 (in French) {http://www.uq.edu.au/~e2hchans/reprints/hb2002.pdf}.

CHANSON, H. (2002b). "An Experimental Study of Roman Dropshaft Hydraulics." Jl of Hyd. Res., IAHR, Vol. 40, No. 1, pp. 3-12 {http://www.uq.edu.au/~e2hchans/reprints/jhr02_11.pdf}.

CHANSON, H. (2002c). "Hydraulics of a Large Culvert beneath the Roman Aqueduct of Nîmes." Jl of Irrigation and Drainage Engrg., ASCE, Vol. 128, No. 5, pp. 326-330 {http://www.uq.edu.au/~e2hchans/reprints/jide02_2.pdf}.

CHANSON, H., and JAMES, D.P. (2002). "Historical Development of Arch Dams : from Cut-Stone Arches to Modern Concrete Designs." Aust. Civil Engrg. Trans., IEAust, Vol. CE43, pp. 39-56 & front cover {http://www.uq.edu.au/~e2hchans/reprints/acet_02.pdf}.

CHANSON, H. (2004a). "The Hydraulics of Open Channel Flows : An Introduction." Butterworth-Heinemann, Oxford, UK, 2nd edition, 630 pages {http://www.uq.edu.au/~e2hchans/reprints/book3_2.htm}.

CHANSON, H. (2004b). "Hydraulics of Rectangular Dropshafts." Jl of Irrigation and Drainage Engrg., ASCE, Vol. 130, No. 6 (in Print) (ISSN 0733-9437).

CHATZIS, K., and DUPUY, G. (1997). "Coment se Passer des Observations Locales : la Formule "Caquot" et l'Assainissement en France de l'Après-Guerre." Annales des Ponts et Chaussées, No. 82, pp. 67-75.

CLAMAGIRAND, E., RAIS, S., CHAHED, J., GUEFREJ, R., and SMAOUI, L. (1990). "L'Aqueduc de Carthage." Jl La Houille Blanche, No. 6, pp. 423-431.

Concrete Pipe Association of Australasia (1991). "Hydraulics of Precast Concrete Conduits." Jenkin Buxton Printers, Australia, 3rd edition, 72 pages.

Conseil Général du Rhône (1993). "Préinventaire des Monuments et Richesses Artistiques. III L'Aqueduc Romain de la Brévenne." Bosc Frères Publ., Lyon, France, Henri HOURS Editor, 230 pages & 1 Map.

COQUET, M. (1966). "Les Aqueducs Romains de Beaulieu et les Puits de Rupture de pente." Cahiers Ligures de Préhistoire et d'Archéologie, Vol. 15, pp. 283-295.

FABRE, G., FICHES, J.L., and PAILLET, J.L. (1991). "Interdisciplinary Research on the Aqueduct of Nîmes and the Pont du Gard." Jl of Roman Archaeology, Vol. 4, pp. 63-88.

FABRE, G., FICHES, J.L., LEVEAU, P., and PAILLET, J.L. (1992). "The Pont du Gard. Water and the Roman Town." Presses du CNRS, Caisse Nationale des Monuments Historiques et des Sites, Collection Patrimoine au Présent, Paris, France, 127 pages.

FABRE, G., FICHES, J.L., and PAILLET, J.L. (2000). "L'Aqueduc de Nîmes et le Pont du Gard. Archéologie, Géosystème, Histoire." CNRS Editions, CRA Monographies Hors Série, Paris, France, 483 pages & 16 plates.

FEVRIER, P.A. (1979). "L'Armée Romaine et la Construction des Aqueducs." Dossiers de l'Archéologie, Séries Les Aqueducs Romains, Vol. 38, Oct./Nov., pp. 88-93.

FRONTINUS, Sextus Julius. "De Aquis Urbis Romae." ('Des Aqueducs de la Ville de Rome"') Latin text par LOEB (1925). Traduction française par C. BAILLY.

GARBRECHT, G. (1987). "Hydraulics and Hydraulic Research : a Historical Review." Balkema Publ., Rotterdam, The Netherlands.

GERMAIN DE MONTAUZAN, C. (1907). "Rapport sur une Mission Scientifique en Italie et en Tunisie." Nouvelles Archives des Missions Scientifiques, Tome XV, Fasc. II, pp. 71-123.

GERMAIN DE MONTAUZAN, C. (1908). "Aqueducs Antiques de Lyon. Etude Comparée d'Archéologie Romaine." Ernest Leroux, Paris, France.

GREWE, K. (1986). "Atlas der Römischen Wasserleitungen nach Köln." ('Atlas of the Roman Hydraulic Works near Köln.') Rheinland Verlag, Köln, Germany, 289 pages (in German).

GREWE, K. (1992). "Plannung und Trassierung römisher Wasserleitungen." ('Planning and Surveying of Roman Water Supplies.') Verlag Chmielorz GmbH, Wiesbaden, Germany, Schriftenreihe der Frontinus Gesellschaft, Suppl. II, 108 pages (in German).

HAMM, J.A., and LITAUDON, J.C. (2000). "L'Aqueduc Romain du Gier ou du Pilat." Fédération de la Recherche Archéologique de la Loire, Graphi-Print printer, La Fouillouse, France, 41 pages.

HAUCK, G.W., and NOVAK, R.A. (1987). "Interaction of Flow and Incrustation in the Roman Aqueduct of Nîmes." Jl of Hyd. Engrg., ASCE, Vol. 113, No. 2, pp. 141-157.

HENDERSON, F.M. (1966). "Open Channel Flow." MacMillan Company, New York, USA.

HODGE, A.T. (1992). "Roman Aqueducts & Water Supply." Duckworth, London, UK, 504 pages.

LEFEBVRE, C. (1996). "L'aqueduc Antique de Gorze à Metz." ('The Antic Aqueduct of Gorze at Metz.') Itinéraire du Patrimoine No. 119, Direction Régionale des Affaires Culturelles de la Lorraine, Nancy (in French).

LEVEAU, P. (1991). "Research on Roman Aqueducts in the past Ten Years." Future Currents in Aqueduct Studies, Leeds, UK, T. HODGE ed., pp. 149-162.

LEVEAU, P. (1996). "The Barbegal Water Mill in its Environment: Archaeology and the Economic and Social History of Antiquity." JRA, Vol. 9, pp. 137-153.

LOPEZ-CUERVO, S. (1985). "Medina Az-Zahra Ingeniera y Formas." Publicaciones del Ministerio de Obras Publicas y Urbanismo, Madrid, Spain169 pages (in Spanish).

O'CONNOR, C. (1993). "Roman Bridges." Cambridge University Press, Cambridge, UK, 235 pages.

PINETTE, M., and REBOURG, A. (1986). "Autun (Saône-et-Loire) Ville Gallo-Romaine." Guides Archéologiques de la France, Imprimerie Nationale, Ministère de la Culture et de la Communication, France (In French).

RAKOB, F. (1974). "Das Quellenheigtum in Zaghouan und die Römische Wasserleitung nach Karthago." Mitt. des Deutschen Archaeologischen Instituts Roemische Abteilung, Vol. 81, pp. 41-89, Plates 21-76 & Maps (in German).

REBOURG, A. (1999). "L'Urbanisme d'Augustodunum (Autun, Saône-et-Loire)." Gallia, Presses du CNRS, Paris, France, Vol. 55, pp. 141-236.

ROIDOT-DELEAGE, J. (1879?). "Autun Ancient et Moderne." Société Eduenne, Autun, France, 2 volumes.

US Department of the Interior (1987). "Design of Small Dams." Bureau of Reclamation, Denver CO, USA, 3rd edition.

VALENTI, V. (1995a). "Aqueduc Romain de Mons à Fréjus. 1. Etude Descriptive et Technique. Son Tracé, son Profil, son Assise, sa Source ..." Research Report, Fréjus, France, 97 pages.

VALENTI, V. (1995b). "Aqueduc Romain de Mons à Fréjus. 2. Etude Hydraulique. Son Débit ... de sa Mise en Service à son Déclin." Research Report, Fréjus, France, 123 pages.

VILLANUEVA, A.V. (1993). "El Abastecimiento de Agua a la Cordoba Romana. I : El Acueducto de Valdepuentes." ('The Water Supply of the Roman Cordoba. I : Aqueduct of Valdepuentes.') Monografias No. 197, Universidad de Cordoba, Servicio de Publicaciones, Cordoba, Spain, 172 pages (in Spanish).

VILLANUEVA, A.V. (1996). "El Abastecimiento de Agua a la Cordoba Romana. II : Acueductos, Ciclo de Distribución y Urbanismo." ('The Water Supply of the Roman Cordoba. II : Aqueduct, Distribution System and Urbanism.') Monografias No. 251, Universidad de Cordoba, Servicio de Publicaciones, Cordoba, Spain, 222 pages (in Spanish).

WILSON, A. (1998). "Water Supply in Ancient Carthage." Jl Roman Archaelogy, Suppl. series No. 28, pp. 65-103.

Websites

CHANSON, H. (2002). "Some Hydraulics of Roman Aqueducts. Myths, Fables, Realities. A Hydraulician's Perspective." Internet resource {http://www.uq.edu.au/~e2hchans/rom_aq.html}.

CHANSON, H., and JAMES, D.P. (2000). "Historical Development of Arch Dams. From Cut-Stone Arches to Modern Concrete Designs." Internet resource {http://www.uq.edu.au/~e2hchans/arch_dam.html}.

Translation:

Table 1 - Comparison between the Roman aqueducts of Nîmes, Gorze (Metz), Mons (Fréjus) and Mont d'Or (Lyon)

References : FABRE et al. (1991,1992,2000), VALENTI (1995a,b), LEFEBVRE (1996), BURDY (2002), CHANSON (2002a)

Table 2 - Distribution basins installed along Roman aqueducts

Notes: Left = left when looking downstream; Right = right when looking downstream.

Table 3 - Well-documented steep chutes along Roman aqueducts les aqueducs romains

Notes : L : chute length; Q_max : maximum discharge; H : total head loss; (--) : no available information.

Table 4 - Culverts and small bridges under Roman aqueducts

Notes : (a) terminology after O'CONNOR (1993); (b) : after second bridge refurbishment (Stage 2); (--) : no available information.

[1] During the Augustan period (BC 33 to AD 14), one sesterce weighted about 1/336 of a pound of silver which would bring the cost of one kilometre of aqueduct to about US$ 23 to 69 millions  (based on US$485.5 per ounce of silver on 25 November 1998) ! By comparison the pipeline for the Tarong power station (70-km long, 0.9 m³/s) in Queensland costed AUD$ 0.2 millions per km (Courier Mail 3 Dec. 1994, p.13).

[2]In each case, a regulation system was found upstream of the bridge-canal suggesting needs for operation control.

[3]The "Source des Bouillons" is used today to supply water to the city of Metz (catchment area: 58 km²). Streamflows were measured at the downstream end of the water supply system. The data did not account for any overflow. It is believed that some overflows along the water supply occurred for Q 10,000 m³/day.

[4]Flow rates measured at the source. The catchment area is about 45 to 50 km². The study period covered July 1967 to May 1968 and Jan. 1976 to Dec. 1978.

[5]Table 2 does not include the castella near the cities. The castellum divisorium is a distribution structure, often located at the arrival into a city: e.g., Nîmes, Pompeii, Fréjus, Arles. At Nîmes, it is argued that the castellum was equipped with a sluice at its upstream end. At Tebourda (Thuburbo Minus, Tun.), the castellum was a rectangular basin, 2.8-m long, 1.5- wide and 1.5-m deep, distributing the water into three separate directions (GERMAIN DE MONTAUZAN 1907, pp. 108-109).

[6]That is, in terms of hydraulic control as well as maintenance.

[7]That is, sections characterised by a mild slope and subcritical tranquil flow (e.g. HENDERSON 1966, CHANSON 1999).

[8]The channels are sketched in Fig. 3.

[9]Backwater calculation based upon a standard step method, distance calculated from depth (e.g. HENDERSON 1966, pp. 126-130; CHANSON 1999, pp. 112-113 & 289-294). [CHANSON (1998) used the same method.] Assuming a long prismatic downstream conduit, the flow depth far downstream, or tailwater depth, is the uniform equilibrium flow depth or normal depth in the downstream canal.

[10]In uniform equilibrium flows, the relationship between the flow rate Q and flow depth d yields : (e.g. HENDERSON 1966, CHANSON 1999).

[11]With the bridge canal flow stopped, the overflow could pass up to 165,000 m³/day (gate fully-opened).

[12]That is, slower flow velocities and a longer retention time.

[13]The number is the paragraph number in the Latin version (Monte Cassino manuscript).

[14]Not listed in CHANSON (2000a).

[15]The original drawing by ROIDOT-DELEAGE (1879?), re-used by PINETTE and REBOURG (1986), is inaccurate. For example, the drawing of the "puit" (shaft) No. 10 is incorrect. The writer is definite after studying the original manuscript in Autun.

[16]Dr P. LEVEAU questioned whether the site was a well or a dropshaft.

[17]No text accompanies the sketches and gravures.

[18]Also called Aqua Vetus. The aqueduct was used for several centuries, by the Romans and later by the Muslims. The Valdepuentes aqueduct was thoroughly studied by the the engineer S. LOPEZ-CUERVO (1985) and in the Ph.D. thesis of Dr VILLANUEVA (1993,1996) who went both on site several times.

[19]LEVEAU and PAILLET (1976), pp. 46-47 including Fig. 24.

[20]LEVEAU and PAILLET (1976), pp. 56-57.

[21]LEVEAU and PAILLET (1976), pp. 104-107.

[22]P. GAUCKLER was the son of the famous French hydraulic engineer Philippe Gaspard GAUCKLER (1826-1905) (e.g. CHANSON 1999).

[23]GAUCKLER (1902), Fig. 45.

[24]HERO designed the first steam turbine and he impressed Italian scientists for many centuries including Galileo (LEVI 1995).

[25]For example, the dropshaft cascades of the Valdepuentes aqueduct were later re-used by the Muslims (VILLANUEVA 1993,1996).

[26]For example, the theatre of 148-m diameter (the largest in Gaul); the amphitheatre measuring 154-m by 130-m (agin the largest in Gaul); the roads (no less than 14 roads converged to Autun); the streets; the aqueducts (REBOURG 1999).

[27]A series of dropshafts is also called a dropshaft cascade by CHANSON.

[28]The afflux is the rise of water level above normal free-surface level upstream of the culvert. It is a measure of the upstream flooding caused by the culvert design.

[29]In open channel flows, the flow conditions such as the specific energy is minimum are called the critical flow conditions (e.g. HENDERSON 1966, CHANSON 1999a).

[30]The writer is strongly involved in professional consultancy and in the teaching of culvert hydraulics at both undergraduate and postgraduate levels (CHANSON 1999a).

[31]Calculations were conducted assuming inlet control operation because the upstream and downstream bed slope are relatively steep (writer's own survey).

[32]For larger upstream water levels, the reservoir formed upstream upstream of the aqueduct would induce a very-large pressure force on the aqueduct structure with high risks of overturning and sliding (in the writer's opinion). FABRE et al. (2000, pp. 419-420) reported that the culvert cells were blocked progressively during the aqueduct operation. Near the end the aqueduct acted as a dam wall.