How are underwater tunnels constructed?

You can actually dig a tunnel under a river, without disturbing the river bed.

The way you do this today is very similar to the original (Canadian! [1]) engineering contribution to tunneling, where a circular tunnel is cut using a boring machine:

Tunnel Construction:

Tunnels are dug in types of materials varying from soft clay to hard rock. The method of Tunnel construction depends on such factors as the ground conditions, the ground water conditions, the length and diameter of the tunnel drive, the depth of the tunnel, the logistics of supporting the tunnel excavation, the final use and shape of the tunnel and appropriate risk management.

How Tunnel Works:

Lutech Engineering construct Tunnel through soft rock and tunneling underground require different approaches. Blasting in soft, firm rock such as shale or limestone is difficult to control.

Instead, engineers use tunnel boring machine, or moles, to create the tunnel. TBMs are enormous, multimillion-dollar pieces of equipment with a circular plate on one end.

The circular plate is covered with disk cutters --chisel-shaped cutting teeth, steel disks or a combination of the two. As the circular plate slowly rotates, the disk cutters slice into the rock, which falls through spaces in the cutting head onto a conveyor system.

The conveyor system carries the muck to the rear of the machine. Hydraulic cylinders attached to the spine of the TBM propel it forward a few feet at a time.

TBMs don't just bore the tunnels -- they also provide support. As the machine excavates, two drills just behind the cutters bore into the rock.

Then workers pump grout into the holes and attach bolts to hold everything in place until the permanent lining can be installed.

The TBM accomplishes this with a massive erector arm that raises segments of the tunnel lining into place.

There are many ways in which you can construct underwater tunnels:
One way of building them is to floats the tubes into position using boats, sink them and then get divers to seal the pieces together. After this pump the water out of the tubes and you have a tunnel.

Another way is to dig down so deep you reach below the sea and potentially dig a tunnel through the rock how you would a mine. There shouldn't be any affect from water as there will be a thick layer of stone above the tunnel keeping you away from water. If the rock / soil is not able to keep water out you can pump compressed air into the tunnel so the pressure inside is greater than the outside and then proceed to seal the inside with concrete and steel lining.
Here are a couple construction methods for underwater tunnels:

The BART Transbay tube was generally constructed by:
- preparing a pad on the floor of the San Francisco Bay,
- placing and connecting tunnel segments in the wet,
- placing ballast on the tunnel, and
- pumping water out of the tunnel.

The Channel Tunnel between England and France was constructed with the use of tunnel boring machines (also known as TBMs).  TBMs drill through soil or rock while simultaneously placing structural segments behind the drilled cavity. 

Some links:

Channel Tunnel
A History of BART: The Project Begins
Tunnel boring machine

You can precast the tunnel on a site near the location which is little bit lower in elevation , seal the structure , fill it with water so that it can float and transport it using a tugboat to the specific location . Then submerge it and back filling it with selected materials so that it becom stable. Finally after connecting all the tunnel remove the seal.

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