Situation: A very large reservoir of water exists raised up above a large expanse of almost flat land. From the bottom of the reservoir the land falls away at a very gentle gradient of ~1 in 10,000. The output of the water in the high reservoir can be controlled via a series of sluice gates to deliver anywhere between 0 and 1000 cubic meters water flow per second down onto the flat land below.

The people want to build a canal across the flat land to deliver water from the bottom of the reservoir into a river 400km away. Which of the flowing would be the easiest way to do this and what are the likely difficulties?

  1. They must dig the whole 400km out before letting the water in.
  2. They can just let the water out and it will find its own way to the river and dig its own channel
  3. They will need to dig a wide shallow guide channel and then let the flowing water expand that
  4. They will need to dig a narrow deep guide channel and then let the flow erode the sides

Would it be better to use a steady or pulsed flow rate?#

edit for clarity

The ground is generally loose material such as sand, loam or clay and is mostly homogeneous. The surface is dry but becomes increasingly damp as you dig through it down to the water table. If it makes a big difference describe why it does.

Assume the reservoir is large enough to provide a constant flow of water indefinitely at the flow rate selected at the sluice gates. The water can flow at any desired rate and can be channeled into a deep torrent or a wide shallow stream when it passes through the sluice gates. The foundation of the reservoir will not be undermined (out of scope).

Another way to think of this is given the starting situation can the water dig it's own channel? If so how best to organise the flow to encourage it.

Example of situation

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    $\begingroup$ What is the material making up the plane? $\endgroup$
    – L.Dutch
    Nov 17, 2020 at 16:22
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    $\begingroup$ "A very large reservoir of water exists raised up above a large expanse of almost flat land. From the bottom of the reservoir the land falls away at a very gentle gradient of ~1 in 10,000." Those two sentences do not add up to any kind of sensible picture. Maybe draw a diagram? And of the four alternatives presented, only alternative (1) represents (potentially) a canal; the other three represent uncontrolled torrents. (Ah, and remember that when water flows without interruptions such as dams or weirs, it does not flow at constant speed over constant inclination.) $\endgroup$
    – AlexP
    Nov 17, 2020 at 16:57
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    $\begingroup$ Several additional facts need to be known. First, what is the composition of the material on the flat - is it homogenous? Sand-like? Clay-like? Rocky? Solid rock? Water absorption characteristics? Second, how high is the reservoir? (potential energy/kinetic energy) Third, how will the water be constrained within the channel? Artificial walls, natural banks, rock? If you go for natural errosion, how do you STOP or CONTROL it? Fourth, technology level? Powered machines, huge excavators, or pick and shovel? You have imagined your world, we can not read minds. Share the details with us. $\endgroup$ Nov 17, 2020 at 17:08
  • $\begingroup$ By your figures, the water drops 40 meters, maybe 24 stories, ( in 400 km., right? $\endgroup$ Nov 17, 2020 at 17:15
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    $\begingroup$ 1000 cubic meters of water per second is roughly half of the water flowing through the Sir Adam Beck power generating station in Niagara Falls. A picture of one of the channels that deliver this water flow wiki2.org/en/Sir_Adam_Beck_Hydroelectric_Generating_Stations#/… $\endgroup$ Nov 17, 2020 at 18:02

3 Answers 3

  1. "A very large reservoir of water exists raised up above a large expanse of almost flat land. From the bottom of the reservoir the land falls away at a very gentle gradient of ~1 in 10,000. [...] The people want to build a canal across the flat land to deliver water from the bottom of the reservoir into a river 400km away."

    First of all, they need to dig a large compensation reservoir in order to be able to control the hidraulic head of the water flowing into the canal. You absolutely don't want the water source to be hundreds of meters above your canal, with the water being delivered directly under high pressure by a penstock. What they want is something like this:

     \ - - Upper - - - /  control valve       ___ constant level of water
      \ - reservoir - /   ||                  |
       \ - - - - - - /____||___\--------------v--/--||-----------------------
        \ - - - - - __penstock___ compensation  ____||_________canal_________
         \---------/      ||     \  reservoir  /    ||
                                  \-----------/     outflow sluice
  2. 1 in 10,000 is a small gradient; water will flow slowly: they will need a very large canal in order to deliver a debit of 1,000 m³/sec.

    For comparison, the average slope of the lower course of the Nile, from Aswan to the sea, is 1 in 13,300; with an average discarge of about 2800 m³/sec, the river is 2.8 km (1.7 miles) wide and about 10 meters deep. Their canal would be about 1 km (0.6 miles) wide. That is a very wide canal.

  3. Rivers flowing over small gradients tend to shift their courses unpredictably unless controlled. The people will need to expend a significant budget of resources and workforce in maintaining the canal.

  4. A large river is a powerful erosion force. They won't be able to convince it to flow freely over a constant gradient from the compensation reservoir to the sea. This is a consequence of the principle of least action; if they let the water flow freely, it will tend to dig down a deep valley at the source, and a very wide valley towards its mouth. They don't want this, and therefore they will need to engineer the course of the canal into sections separated by dams or weirs.

  5. Now coming to the four options presented by the question:

    • Option 1, dig the canal before letting the water in: this may result in a stable canal, provided they know what they are doing. They don't need to dig all of it before letting the water in: they can dig a section, let water flow into it, then dig another section and so on.

    • Options 2, 3, and 4 all come down to letting the river cut its course any way it sees fit: this will result in a natural river, not a canal. They will have to adopt option 1 for the initial part of the river at the exit from the compensation reservoir, or else the river will erode their work. Those options are great if the plain is not populated when they let the water flow -- they can let the river stabilize and then bring in the inhabitants. But if they want to be able to predict the course of the river, or if they don't want to allow it to adopt the variable slope profile of a natural river, then these options are not recommended.

  • $\begingroup$ The channel can not be perhaps as deep as you imagine. The total difference in drop across the 400 km, is only 40 meters, the height of a 13 or 14 story building. This certainly limits how deep the channel can be. The Nile depth, according to your figures, is one quarter of the total drop. $\endgroup$ Nov 17, 2020 at 18:13
  • $\begingroup$ @JustinThymetheSecond: A forty meters deep valley is not conducive to extensive irrigation works... I don't mean the level of the bottom of the river, I mean the level of the surface of the water. That's how natural rivers do it, because that's how physics requires it. $\endgroup$
    – AlexP
    Nov 17, 2020 at 18:24
  • $\begingroup$ 40 meters is ALL you have, from start to finish, in that 400 km. long channel - one meter in 10 km. drop. The surface of the water in the channel at the start is only 40 meters above the surface of the water at outflow. $\endgroup$ Nov 17, 2020 at 18:30
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    $\begingroup$ @JustinThymetheSecond: Yes, I realised that. A natural river will drop 30 meters over the first 50 km or so (going 25 meters below the plain) and then idle very slowly over the last 350 km, meandering wildly over the last 200. $\endgroup$
    – AlexP
    Nov 17, 2020 at 18:33
  • $\begingroup$ According to the OP, this plain does not drop then rise back up again, the slope is an even, continuous gradient. Thus, it has to be the channel itself that is dug out to conform to that 'paraboloc' shape. The Nile works because of periodic fooding. $\endgroup$ Nov 17, 2020 at 19:09

@AlexP's answer was fantastic. I upvoted it. So should you. But there might be an alternative....

Use Hydraulic Mining Techniques

Hydraulic mining is a form of mining that uses high-pressure jets of water to dislodge rock material or move sediment. (Source, also image below)

enter image description here

Set up pipes that bring the water to the termination (the sea) such that there's a lot of pressure, then carve out the canal back-to-front, washing (slowly) the sediment out to sea. Depending on the mineral deposits in the area, this method can be combined with placer mining techniques (the use of sluices to extract heavier minerals) to extract those minerals and (from a suspension-of-disbelief perspective) pay for the project.

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    $\begingroup$ Water pressure maintained through a 400 km. long hose? Methinks pumping stations along the route are required. A 40 meter drop is not going to do much to retain pressure against friction. It's like puting a water tank on a 14 story building and hoping there is enough pressure in a hose stretched from Toronto to Montreal. $\endgroup$ Nov 17, 2020 at 18:22
  • $\begingroup$ @JustinThymetheSecond Oh, I ignored that fact completely - principally because I live in an area that has gullies devastated by hydraulic mining in the late 1800s. They weren't using pumping stations, just staged reservoirs (and pretty small ones, too). It's not really the pressure that you need - it's volume (ok, and some pressure, but not as much as you're envisioning). $\endgroup$
    – JBH
    Nov 17, 2020 at 18:25
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    $\begingroup$ @JustinThymetheSecond: The pressure is provided by the upper reservoir. I shudder to think of a pre-industrial society trying to build a 400 km long pipe containing water at 10 atm pressure. $\endgroup$
    – AlexP
    Nov 17, 2020 at 18:38
  • $\begingroup$ It's that 400 km. long thing. The water pressure at the first floor of that 14 story building would be enormous. Any apartment building higher that say 8 stories does not use city water pressure, Normal city water pressure will not get water that high, The building has pumbs that push the water up to tanks in the roof, and the water flows down from these tanks. But 400 km.? They would have to use high-pressure pumps to get the water into reservoirs close to the hydraulic mining hoses, since they do not yet have the supply channel built. $\endgroup$ Nov 17, 2020 at 18:40


It just might turn out, given the fact that there is only a 40 meter drop from one end to the other, that the ground is relatively flat and unobstructed, and that you are after a flow of 1000 cubic meters per second, your best option just might be to forget about erroding the channel, forget about digging a pilot channel, in fact forget about a dug channel complterely, and just build a twin 400 km. long 5 m. high dike to channel the water.

With a wide channel, you just might end up moving far less material - the dike walls are no where near the dimensions of the channel.

Water flows because of differences in energy. Potential energy is converted to kinetic energy. The higher the potential energy, the greater the kinetic energy and the resultant flow. Ten tons of water flows through a channel because there was, first and foremost, enough energy applied to that ten tons of water to get it moving (accelerate it). To keep that ten tons of water moving through the friction of 400 km. of channel, you need a combination of a lot of inertia to begin with, and enough energy to overcome friction. A drop of 40 m. in 400 km. is not supplying a lot of potential energy to move ten tons of water through rough, rocky, meandering waterways at 1000 cubic meters per second. The water will just overflow the banks at the head end, and flood out over the plain, until it dissipates into the ground, evaporates, or stagnates in pools.

The smoother, straighter, more frictionless the banks are, the more water can flow in the same dimension of channel.

Roman aquaducts were successful because they were sealed pipes, and thus the water channel could be pressurized. The water could be 'pushed' through the pipes at high source pressure, so the pressure energy could be contained and dissipated much further along the course than just at the source. Try that with a river, and the water just expands under the pressure at the source, dissipating it quickly through flooding.

Errosion only happens when the water has enough energy to actually carry the erroded material some distance away. If the water does not have enough energy to overcome the inertia of the sand (or whatever) particle, the sand particle does not go anywhere. To errode an entire channel, one needs enough energy to overcome the inertia of all of the material in that channel that you have to move - the entire mass. You are not just moving ten tons of water, you are moving ten tons of water AND ten tons of earth (numbers are for illustration only).

To get your water across that 400 km. distance, you need to build up a lot of energy at the source, enough to get it the entire distance. Now, to get it to errode, you need to supply enough energy at the source to move the water AND the earth that same 400 km. Rivers do this by using a parabolic shape. Steep at the beginning, shallower at the end. Like a roller coaster track, or a slide. Build up speed coming down the steep slope, build up the energy, then coast. That ten tons of water in a steep drop has now built up enough energy to move both itself AND the ten tons of earth across 400 km. of surface, hopefully. There is going to be very little additional energy added along the way.

You want to START the ten tons of water moving at the TOP of the reservoir, letting it fall down the slope, building up speed. But you do not want it simply crashing into the ground. That is like a one-way trip off a cliff. A big splat on the ground, a big hole, but not much lateral movement. You want a parabolic channel at the bottom. Something that will change the vector. Vertical motion into horizontal motion. You don't want the energy dissipated in one big bang, you want it to carry on downstream.

So you want a lot of engineering at the source. A channel built down the slope of the reservoir, in a parabolic shape. A design that can not be left to chance. Nature seems to prefer the 'splat' option, the waterfalls. Straight down.

Once you get the engineering at the head end, and the engineering to direct the flow at the base in a single direction instead of fanning out, then you can decide if that ten tons of water has enough energy to carry the ten tons of soil, or whatever the mass actually is. If it is more than ten tons, far more energy is needed. If it ls less than ten tons, not as much energy is needed. That is an engineering calculation - how much material needs to be moved, how far it has to be moved, and the energy needed to first accelerate it (inertia) then to overcome friction. We can't know that until the actual material is known, how high the head end is (potetial energy due to gravity), and how much water is moving.

One thing is for sure - in order for the water to reach the river, it needs to be directed. Is there a natural path for the water to follow, or is the land almost flat? The flatter the land, the more the water just wants to spread out, dissipating its energy over a very wide area. In a fan, the area squares as the distance from the source, so the water very quickly loses the ability to carry any material as it moves further out from the source.

If there is no natural straight course, then the water must be 'directed'. Yes, you can do that by digging a pilot channel, but you can also do that by building a dike. Whichever you chose depends on the material you are digging through vs the avaiiability of suitable material for building the dike. If the ground is flat, solid, strong contiguous bedrock, but there is an abundant supply of more granular fill closeby, then building a dike is probably the best option. The land is flat, and so the walls of the dike will be parallel to the ground. Alternately, if the ground is easy to dig into, or you have a prolific source of explosives or power beams, then excavating the channel is probably preferred. It is a lot easier to move a big rock than it is to chip that same rock out of solid rock and then move it.

Now, you have to consider the end. What is this channel flowing into? A river, pool, lake, or ocean? Remember, the surface of this end point is only 40 meters lower than the head end. Tidal forces? Wind-driven waves? Flooding? is it allowable for the water to be pushed back up the channel? How deep is the water flow at the end? If it is a river that is only ten meters deep, obviously the channel can not be more than ten meters deep, or you get the potential for back flow. If it has a shallow shore then the dug channel has to be dug far out from the shore - the deeper the channel, the further out it has to be dug. Is the water flow capable of accepting 1000 cubic meters of water per second, PLUS all of the sediment? Do you need to worry about all of the sediment settling out of the water? Will you end up with one collossal river delta? Or is there enough water movement in the destination to also carry away the sediment?

If you use a dike, instead of digging the channel, the bottom of the water flow (ground levvel) is still above the surface level of the river or such. The channel water flows on TOP of the existing water flow, not INTO it. It does not matter how shallow the shoreline is, the additional water flow can still be added to it.

  • $\begingroup$ Seems the river at the far end will become overloaded with sediment. I think they might have to dig the main channel out manually or mostly manually to avoid that. It sounds like it could benefit from a solid rock foundation perhaps a few metres down to prevent or at least greatly slow erosion of the base. And maybe some "guidance" from the surrounding terrain directing the flow a little bit. $\endgroup$
    – Slarty
    Nov 18, 2020 at 9:36
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    $\begingroup$ No matter what you do, with the slope you have, sediment is always going to be a problem. You just are not going to get enough energy in the water at the end to move the mass of the sediment, let alone the mass of the water, unless you pressurize the water in an enclosed pipe. With the soil conditions you specify, better to pave the ground and then build the dike around it. Maybe even move the dike every so often to a new location, or build two dike systems, and rotate closing one down for maintenance. $\endgroup$ Nov 18, 2020 at 16:54

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