I believe it's safe to say that analytical solutions to fluid flow problems are generally only available for laminar flow. For turbulent flows the fundamental equations of change need to be supplemented by experiments or by modelling conjectures (boundary layer theory, for example). You might find the attached manual useful.
The formulae for laminar, isothermal flow through pipes (tubes) are well-known. See, for example,
https://www.uio.no/studier/emner/matnat/math/MEK4450/h11/undervisningsmateriale/modul-5/Pipeflow_intro.pdf
Mehdi, the method used by some of the most common commercial packages such as Wellflow and Prosper are based on traditional hydraulics. However, the presence of gas and different density fluids make the process a combination of a simple formulae and a number of correlations to account in the change of the friction factor as a result of the presence of gas and water along with the hydrocarbons (on a foot by foot basis). Three main laws run the calculation, gravity, friction and acceleration, perhaps you have access to books such as the Whiston or similar on Well performance where these calculations are well explained in detail. I hope it helps
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