I want to find an ODE or PDE for modeling water height in manhole. In other words I want a formula for water velocity in manhole that consider density, diameter, friction and etc parameters.
I don't think that there is a 'formula'; instead,I think you need to look at solving the St Venant or related equations, as the upstream and downstream conditions will affect the water height.
Most people look to using packaged solutions - you could look through the SWMM5 code for inspiration.
try the following:
http://arxiv.org/pdf/1402.4254.pdf
http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/181516/1/a56b0p55.pdf
http://www.nws.noaa.gov/oh/hrl/hsmb/docs/hydraulics/papers_before_2009/hl_226.pdf
http://web.extension.illinois.edu/iwrc/pdf/149.pdf
Good luck!
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