The Hydraulics of Open Channel Flow: An Introduction
Hubert Chanson

Major Assignments

Preface
The following problem covers lecture material relevant to Basic Principles (Part I). Based upon real historical facts, the geometry of the hydraulic structures has been simplified to facilitate the calculations.
 

I. Introduction

I.1 Presentation

In the history of dams and the story of civilisations, the first dams were built in Egypt and Iraq around 3000 BC where they controlled canals and irrigation works. Often the civilisations originated in areas where irrigation was a necessity. The history of dams followed closely the rise and fall of civilisations, especially where these depended on the development of the water resources.

A typical example is the Egyptian civilisation. For centuries, the prosperity of Egypt relied on the annual flood of the Nile river from July to September and the irrigation systems. One of the most enormous efforts of the Egyptian Kings was the creation of the Lake Moeris in the Fayum depression and the construction of a 16 km long canal connecting the Lake to the Nile. The Lake was used to regulate the Nile river and as a water reservoir for irrigation purposes.
 

I.2 The Moeris Lake

The location of the Lake was the Fayum (or Fayoum) depression, located 80 km South-West of the city of Cairo (fig. P2-1). The depression is a vast area of 1700 km² and the lowest point in the depression is 45 m below the sea level.

Egyptian engineers connected the Nile river and the Fayum depression to lead flood flows into the depression during high floods. The connection between the river and the depression was a natural cut in the mountains. It was in existence at the time of the King Menes, founder of the 1st Egyptian dynasty (2900 BC). At that time, the Fayum depression contained only a natural lake filled from the Nile during large floods. [There are still some arguments upon whether the lake Moeris existed or not. There are suggestions that there were in fact two lakes (e.g. SCHNITTER 1994, pp. 4-6).]

King Amenembat (2300 BC) widened and deepened the canal between the Nile and the Fayum depression. He converted the existing lake into an artificial reservoir (i.e. Lake Moeris) which controlled the highest flood of the Nile. The canal connecting the river to the reservoir was regulated by the Ha-Uar dam. The regulation system consisted of two earthen dams at both ends of the canal, with gates by means of which the architects regulated the rise and fall of the water.

The Lake Moeris had three main purposes:

1- the control of the highest floods of the Nile during the July-September periods,

2- the regulation of the Nile river during the dry season by releasing water from the Lake, and

3- the irrigation of a large surface area around the Lake.

From its size and depth, the Lake Moeris was capable of receiving the overflow of the Nile during its rising and preventing the flooding of downstream cities like Memphis (see fig. P2-1). When the river fell, the Lake waters discharged again via the canal connecting the river to the Lake, and these waters were available for irrigation.

The Ha-Uar dam had also a strategic interest. At the time when Egypt was divided in two kingdoms, the Lower Egypt (i.e. Northern Egypt) and the Upper Egypt (i.e. Southern Egypt), the frontier fortress of Lower Egypt was at the head of the Lake Moeris canal (HATHAWAY 1958). The capture of the dams controlling the canal, and the injudicious or malicious use of the reservoir could deprive a great part of the Lower Egypt (i.e. Northern Egypt) of any basin irrigation at all, for such irrigation utilised only the surface waters of the Nile flood. The importance of the fortress commanding the regulator of the canal ceased when the kingdoms were re-united.

The abandonment of the lake Moeris was primarily caused by the fact that the Lahoun branch of the Nile (i.e. the West branch of the river, fig. P2-2) dwindled in size and reduced the use of the reservoir. From BC 230, the canal was abandoned and the area inundated by Lake Moeris became the province of Fayoum as it is today. Nile shells can still be found in the Fayoum area near the limits of the ancient Lake Moeris (WILLCOCKS 1919).
 

Geometrical characteristics
The Lake Moeris was located in the Fayoum (or Fayum) depression. The geometry of the Lake was:
 

Climatic conditions

The evaporation from the Lake was about 2.5 metres per annum in depth. When the Lake was full, the evaporation was around 4.25 x 10⁹ m³ of water per year.

It must be noted that the artificial regime of the lake had effect on the climate. The regulation of the Nile by the Moeris reservoir prevented or reduced stagnant and dirty waters downstream, and hence suffocating air in the cities downstream of the Ha-Uar dam (BELYAKOV 1991).
 

I.3 The Ha-Uar Dam

The canal connecting the Nile river to Lake Moeris was controlled by two dams at each end of the canal (fig. P2-2). The two regulators were earthen dams.

As shown in figure P2-2, the Nile flowed in two channels opposite the head of the Lake Moeris canal. The upper regulator (i.e. Eastern dam) consisted of the existing Lahoun bank (i.e. West bank of the river), a broad spill channel, cut out of the rock to a suitable level for passing ordinary floods, and a massive earthen dam across a ravine, which was cut in dangerously high floods. [Lahoun is also spelled Lahun.] The other end of the canal (i.e the Western end) was a much simpler earth dam. During a flood, the dam was cut: the cutting was easy enough.

Fortresses and barracks were located on either sides of the two dams to protect them. Access to the Eastern dam and to the Eastern end of the canal was difficult: it was written that a fleet was essential to gain possession of the lower great dam (i.e. Western dam) (WILLCOCK 1919).

The passing of very large floods was possible by cutting the dams. But their reconstructions after the passing of a flood entailed an expense of labour which even an Egyptian Pharaoh considered excessive!

 
Canal connecting the Nile and the Lake Moeris

The canal connecting the river to the Fayum depression was initially a natural cut through the mountain bordering the Libyan desert (fig. P2-2). The natural canal was 16 km long and 1.5 km wide (GARBRECHT 1987b). This place is now called the Fayoum Bahr Yusuf Canal.

Around BC 2300, King Amenembat started the construction of an artificial canal along the natural valley. The artificial channel was 16 km long and 5 m deep. Its shape was trapezoidal with a 600 metres width at the bottom (fig. P2-3). The slope of the banks was 1V:10H to allow the use of non-cohesive rockfill and earth materials. The protection of the channel bottom and the bank slopes consisted of cut stones and cement joints. The covering blocks were placed on the bottom and on the slopes, and they were cemented together.

The average slope of the channel bed was 0.01 degrees. The canal was inclined towards the Fayum depression.

 
Eastern dam

The Eastern dam of the Ha-Uar dam structure blocked the valley connecting the Nile river to the Fayum depression. The axis of the valley was East-West. The dam stretched over 1550 m from the South to the North, and it consisted of three large parts:

1- On the North, the Lahoun bank blocked the natural valley over a 550 m length. The elevation of top of the bank was 10 m above the canal bottom. The bank consisted of non-erodable material.

2- Next to the Lahoun bank, there was a broad spill channel. The channel was an ungated broad-crested weir. The weir was 400 m wide with a rectangular cross-section. The crest elevation was 3 m above the canal bottom.

3- An earthen dam blocked the southern extremity of the natural valley. The top of the dam was 9 m above the channel bottom.

A hydraulic structure was located between the broad-crested weir and the earthen dam. The structure supported a sluice gate used to release waters to the river. The gate was 5 m high and 10 m wide. The bottom of the gate opening was at the same elevation as the canal bottom.

 
I.4 Historical events

History indicates that Joseph arrived in Egypt in the time of the Hyksos who ruled the Lower Egypt (i.e. Northern Egypt) while Theban dynasties ruled Upper Egypt. It was a time of unending wars between the two kingdoms.

One of the famines of long duration in Egypt occurred during Joseph's time and was described in the Book of Genesis. It was caused by the capture and the breaching of the Ha-Uar dam by the King of Upper Egypt (i.e. Southern Egypt). The famines were ended by the recapture and repair of the dam by the King of Lower Egypt. According to a tradition, Joseph (BC 1730) worked on the restoration of the canal and the dams. Later Jewish slaves were used for the maintenance of the works.

When the Jews fled from Egypt with Moses, stories relate that the Egyptian army was destroyed by the sea when it crossed the "Red Sea". In fact, the drowning of the army was probably caused by the breaching of the Ha-Uar dam.

 
I.5 Important dates
 

 
I.6 Discussion

Although there are a lot of stories about Lake Moeris, the earliest of these historians wrote his account of the lake in BC 430. There is no trace left of the Western dam and only remains of the Eastern structure may be seen (e.g. GARBRECHT 1987, SCHNITTER 1994).

In fact there are still doubts and arguments about the existence of Lake Moeris. SCHNITTER (1967) mentioned the enigmatic Lake Moeris. SMITH (1971) and BELYAKOV (1991) implied that Lake Moeris was a natural lake and not an artificial reservoir. GARBRECHT (1987b) suggested that the Fayum depression was transformed by large-scale reclamation works into a fertile province around BC 1700 and was never used as an artificial reservoir. Recent studies (SCHNITTER 1994, GARBRECHT 1996) suggested the existence of two lakes since around BC 300-250. Prior to BC 300 the water level in the Fayum depression was about 15-20 m above sea level. It dropped down to about 2 m below sea level (and later to about 36 m below sea level) as a result of land reclamation works associated with high evaporation rate. An artificial reservoir was created around BC 300-250 to irrigate the reclaimed land. The artifical lake was in use up to the 19th century.

In any case, WILLCOCKS (1919) and HATHAWAY (1958) provided a lot of evidences supporting the existence of Lake Moeris. Further the constructions of the pyramids, of large temples and of a large dam (i.e. the Sadd el-Kafara dam) indicate that Egyptian engineers had the expertise and the knowledge to build large-scale civil engineering works. There is no doubt that the Egyptians were able to divert the floods of the Nile River into Lake Moeris. Further they had the technology and the engineering skills to build canals and earthen dams.

Interestingly a new mini hydro scheme will be installed by the year 2000 (Hydropower & Dams 1997). The purpose of the project is the regulation of irrigation water in the depression as well as the generation of hydro-electricity using low-head Kaplan turbines.

 
I.7 Bibliography

Books

Articles

II. Hydraulics problem

Introductory note

We shall consider the Lake Moeris and the artificial canal connecting the Nile river to the reservoir (fig. P2-3). On figure P2-3, the flood plain on each side of the trapezoidal channel consisted of grass, bush and rocks. An equivalent roughness height of k_s = 100 mm could be considered, if necessary.
 

II.1 Study of the upper regulator

A- Draw a sketch of the Eastern dam and the canal cross-section, view from the West (i.e. view from the canal). Indicate the main dimensions on the sketch. Show the North and South directions.
 

B- During a flood, the discharge in the Nile River south of Ha-Uar is 8,000 m³/s. At the same time, the Chief Engineer records a 1.1 m flow depth above the sill of the broad-crested weir.

In this question, you will assume that the weir crest and the transition between the weir and the channel are smooth and horizontal. For each sub-question, students are asked to explain in words each formula and assumption that you use. Indicate clearly each fundamental principle(s).

B.1 What is the flow rate in the Nile river downstream of the Ha-Uar dam?

B.2 Sketch a cross-section profile and a top view of both the broad-crested weir and the start of the channel. On the cross-section view, plot the water surface profile.

B.3 What are the values of the specific energy: (a) upstream of the weir, (b) mid-sill, (c) downstream of the weir where the cross-section is rectangular (and 400 m wide) and (d) downstream of the weir at the start of the canal where the cross-section is trapezoidal (see fig. P2-3)? You will assume that the channel bed elevation is the same upstream and downstream of the weir.

B.4 Develop the dimensionless relationship between the specific energy and the flow depth (i.e. E/d_c as a function of d/d_c) for a channel of irregular cross-section. Explain clearly in words each step of your development. Deduce the expression of the dimensionless specific energy E/d_c for a rectangular channel.

B.5 Plot the dimensionless specific energy diagram (i.e. E/d_c versus d/d_c) on a graph paper. Indicate on the graph the points representing the flow conditions: (a) upstream of the weir, (b) mid-sill and (c) downstream of the weir where the cross-section is rectangular (and 400 m wide).

B.6 Downstream of the weir, is the flow subcritical or supercritical: (a) at the end of the weir where the cross-section is rectangular (400 m width) and (b) at the entrance of the canal where the cross-section is rectangular? Justify your answers in words.
 

C- The flow rate in the trapezoidal canal was initially 1500 m³/s. The initial flow depth in the canal was 2 m. A large flood arrives from the Nile river and the flow depth at the start of the canal becomes instantly 2.5 m. A surge develops and travels downstream in the canal towards Lake Moeris.

In this question, the channel will be assumed smooth and horizontal.

C.1 Draw the appropriate sketch(s) of the travelling surge. Indicate clearly the direction of the initial flow and the direction of the surge.

C.2 What type of surge is taling palce in the channel? Can you make any appropriate assumption(s) to compute the new flow conditions. Justify your answer. If you are not able to do any calculation, go to the next question. If you can do the calculations, continue this question.

C.3 Define a control volume across the surge front for the trapezoidal canal. Indicate the control volume on your sketch(s) (Q. C.1). Show on your sketch(s) (Q. C.1) the forces acting on the control volume. Show also your choice for the positive direction of distance and of force.

C.4 Write the continuity and momentum equation as applied to the control volume shown on your sketch.

C.5 Compute the surge velocity and the new flow rate.

C.6 Neglecting flow resistance, how long would it take for the surge to reach the downstream end of the canal?

C.7 A horseman would need 25 minutes to cover the distance between the two ends of the channel. Starting from the upstream end at the same time as the surge, will he reach the downstream end before the surge.
 

II.2 Study of the channel

A- The canal is discharging 3000 m³/s.

A.1 For a canal of irregular shape, how are defined the critical flow conditions? Explain clearly your answer. Use appropriate sketch(s) if necessary.

A.2 For the trapezoidal artificial canal, what is the critical depth for the above discharge? Is the critical depth a function of the channel roughness and/or slope?

 
B- Uniform equilibrium flow

B.1 What is the definition of an uniform flow? Give at least two practical examples of uniform flow situations.

B.2 Draw a sketch of an uniform flow situation. Choose an appropriate control volume. Show on your sketch the forces acting on the control volume. Show also your choice for the positive direction of distance and of force.

B.3 Write the momentum equation for an uniform flow in a channel of irregular cross-section.

B.4 How do you deduce all the uniform flow conditions for a channel of irregular cross-section? Explain your answer in words. No more than one equation is required.

 
C- The canal is discharging 3000 m³/s (as for Q. II.2.A).

C.1 What roughness height would you choose for the trapezoidal channel? Justify in words your answer.

C.2 What is the uniform flow depth for that discharge? Is the uniform flow depth a function of the channel roughness and/or slope?

C.3 What value of the Manning coefficient would you choose for the channel? Justify your choice. Compute the uniform flow depth using the Manning formula? Does your result differ from the result obtained in Q. C.2? Discuss the comparison (if any) between the results obtained in Q. C.2 and Q. C.3.

C.4 Assuming that uniform flow is obtained in the canal, where can you control the flow in the canal (e.g. upstream, downstream)? Justify and discuss your answer.

C.5 Give at least two examples of hydraulic controls that could be used to regulate the flow in the canal (Q. C.4). Discuss each example and explain clearly the difference(s) between each possibility. Sketch each example.

 
D- In BC 2251, a very large flood of the Nile river discharged into the canal to Lake Moeris. The level of water in the valley connecting the river and the reservoir was 2.1 m above the bed elevation of the valley (i.e. 7.1 m above the bed elevation of the canal).

Assuming that uniform flow conditions were reached in the long canal and the valley:

D.1 Deduce the water discharge into the reservoir.

D.2 Provide all the flow conditions in the artificial canal (i.e. water discharge, velocity of water, flow depth, hydraulic diameter, cross-section area, wetted perimeter, friction factor).

D.3 Provide all the flow conditions in the flood plain (i.e. water discharge, velocity of water, flow depth, hydraulic diameter, cross-section area, wetted perimeter, friction factor).

D.4 Is the flow supercritical? Explain your answer.

Students must detail, discuss and justify every step of your method: e.g. calculation of flow resistance, choice of roughness height(s). For the trapezoidal canal, students shall use the roughness height selected in Q. II.2.C1. The roughness height of the flood plain (i.e. on each side of the artificial canal) is given at the beginning of the assignment.

III. Hydrology of Egypt's Lake Moeris

Part A

(a) Using an atlas like "The Times Atlas of the World", find out which 2 months of the year normally yield the greatest rainfalls in the upper Nile catchment, near Khartoum (Sudan), and near Adis Abeba (Ethiopia). In this region, which direction are winds blowing from during those months? [This question prompts consideration of where is the likely evaporative source of moisture for this rainfall.]

(b) In the upper Nile catchment, why do winds tend to blow in that direction during that period? [This question prompts consideration of what is driving the airflow. If the wind direction doesn't conform to the general circulation described in class, then the driving force must be a very strong phenomenon.]

(c) Explain in a few sentences why the flood absorbtion capacity of Lake Moeris is significantly less if a major flood occurs during the previous year than if not, but greater still if preceded by 2 or more relatively dry years.

(d) What does the atlas indicate for the general order of magnitude of annual evaporation rates in the Nile region? Explain how this relates to (c) above. How can an (empty) depression form below sea-level, and remain empty?

(e) Do you think high flows in the Nile would have snow-melt as a significant contributor? Explain. In what way is soil at the bottom of Lake Moeris likely to be infertile?

 
Part B

This question addresses the potential for attenuating flood flows in the Nile river by means of the off-stream storage provided by Lake Moeris. [The previous question dealt with flood attenuation due to direct (on-stream) storage.] Just upstream of the junction between the Nile and the Lake's canal, the river flow typically varies as indicated in figure P2-4, reaching a peak rate of just over 700 GL/day (about 8.2 ML/s) in early September. [Very wet years probably yield flows about 30% greater than this.]

For the questions below, increase the ordinates of figure P2-4 to somewhere between the "typical" wet season and the "extreme" wet season, by adding about 30%.

Although Part II of this assignment asks you to find normal depth in the Lake's canal at a particular flowrate, for this question you will need the whole stage-discharge curve (Q versus H) up to about 10 ML/s, which may include a portion of over-bank flow.

From figure P2-4, note that the flowrate in June is typically steady (at about 50 GL/day). During this period, assume that the Lake's surface level is the same as the river surface, with no flow in the canal. As the wet season brings higher flows from up-river each week, assume that the river level at the junction must rise until the sum of the Lake's canal flow and the downstream river flow equals the given total flowrate from upstream each week. For this exercise, assume that the canal has no dams at either end, and perform all requested calculations manually (without computer assistance). [NOTE: The imprecise calculations requested below are intended merely to provide a rough estimate of the attenuating effect of flood diversion to the Lake. More accurate hydraulic calculations would be needed, particularly as the lake approaches "full".]

(a) Replot the total inflow hydrograph (appropriately rescaling the data from figure P2-4) dividing each month into 4 equal "weeks" of 7.6 days, and showing the hydrograph in stepped form (as if each "weekly" flow was constant).

(b) Integrate the total Nile flow volume, from July 1st to when the falling hydrograph passes half its peak value. Deduce what depth of Lake Moeris (assuming constant plan area) would be required to store all of this volume.

(c) Plot a stage-discharge curve for the river at (above and below) the canal junction:

Q(ML/s) = 0.12H²

[This means, for example, that when H = 8.2 m, then Q = 8.2 ML/s = 8200 m³/s, at which time the velocity is about 1.5 m/s, the sectional area is about 5500 m², and the river width is about 550 m.]

(d) On the plot of (c), superimpose a hypothetical [but incorrect] stage-discharge curve for the canal:

Q = 0.72H^1.4

[From your Part II results, plot points to show how incorrect this curve is.]

Set up a tabulation (with each row representing a "week", starting at the end of June), showing how each "week's" upstream inflow is divided into two components, one of which represents the diversion to Lake Moeris. A column should show the stage (H) at the canal junction. [The objective of this table is to identify how much attenuation of the upstream river flow is achieved by the diversion.] Superimpose the deduced downstream river flow on the plot of (a). Highlight the "attenuation".

(e) Sum the canal flow volumes to the lake each "week" from July 1st, and deduce the new surface level of the lake at the end of each week. Stop the calculations in (d) as soon as the lake surface level matches the stage-level at the canal-river junction (i.e. the canal flow is completely "drowned"). [From this point, continued falling river levels would result in outflows from Lake Moeris, unless these are intercepted by blocking the canal (to save the water for later irrigation purposes).]

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