If the Chinese engineers are involved the Rapidan dam can be rebuilt in a year. Just based on my observations on how China does its hydro it should be possible (1) Starting from the shore dump rock and granular fill with controlled cohesive soil content on the damaged section upstream and downstream of the dam. This is to create a temporary bund wall, reasonably watertight with an area to work in the dry while diverting the water back to the dam via its spillways. The rubbish in front of the spillways needs to be cleared too. (2) install two parallel sheet pile wall, most likely the bottom width of the new dam say 8 to 10m wide to a suitable depth beneath the riverbed, say 5 to 6 m depth. (3) Dredge the interior of the sheet-piled section and back filled with lean concrete by tremie technique if the base is cover with water. This volume can be filled with sound rock as filler. It only needs to be reasonably watertight and have sufficient dead weigh for stability. By working section by section the foundation of the new dam is form, say to riverbed level. The sheet-piled foundation needs to be key into existing soil not yet scoured. (4) Build the dam with a modern design hydro electric plant, from the river bed level to the top crest. The intake can be arranged on the side parallel to the stream flow, possibly with large diameter penstock pipe exposed or buried inside the dam body. Dowstream the tailrace is also built in the dry. (5) reinstate the bank and add shore protection work. (6) When the half of dam has been rebuilt on the eroded side the temporary bund walls upstream and downstream can be dredged away allowing the new part to discharge water as well as generating electricty. (7) Using identical method to seal off the undamaged section of the dam, with bund wall upstream and downstream. This create the undamaged section to be worked in the dry. (8) I would break up the concrete work of the old dam and replicate the construction of the first half as the old dam may be uneconomical for modification. (9) The the second section has been rebuilt the bund walls can be removed to allow the new dam fully operational. (10) The above method allow a more flexible approach to install a new hydroelectric plant plus a new set of spillways suitably sized for say 1 in 500 years flood. (11) The river isn't large so the scope of the power plant is limited in capacity. Once the key parameters have been known the preliminary design can go ahead for the foundation while the more demanding design of the superstructure can take place while the foundation is installed. This will save time and cost. The river isn't wide and the flow is modest so a rebuild and design should be executed with robustness in mind with simple construction and generous dimension. This will save time which is money in construction. Remember the undamaged section of the dam didn't wash away because it has adequate self weight or/and suitably key into the soil or rock. If the riverbed has rock then the foundation can be built directly on sound rock, with unsuitable portions removed. Adequate rock anchors can be installed to secure the dam as the second line of defence. The first line of defence should always be the self weight plus any horizontal friction against the moving water. China took 77 hours to seal the breached levee which has been widened to 226m with basically just rock and granular fill plus fines. The progress was slow because of almost no turning for the lorries which had to wait 7 to 8 hours in a queue several kilometer long. Chinese engineers always use this method to create a dry working condition in hydro dam construction so watch a few youtube videos to get some idea of the construction sequence.
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