Drilling mud circulation—M . Prove that the equivalent or hydraulic mean diameter for flow in the annular space between two concentric cylinders of diameters d1 and d2 is given by De = d1.
Fig. P3.12 illustrates the mud-circulation system on an oil-well drilling rig. Drilling mud from a mixing tank T flows to the inlet of the pump P, which discharges through BD to the inside of the drill pipe DE. During drilling, the mud flow is to be steady at Q = 100 gpm. The mud is a Newtonian liquid with μ = 5.0 cP at the average flowing temperature of 70 ◦F, and its density, due to weighing agents and other additives, is ρ = 67 lbm/ft3. The drill pipe DE, of depth 10,000 ft, is surrounded by the casing C. At the bottom, the mud jets out through the drill bit and recirculates back through the annular space to F, where it is piped back to the tank T. The surface piping has a total equivalent length (including all valves, elbows, etc.) of 1,000 ft. The mild steel piping has a roughness ε = 0.00015 ft. Other properties of the piping are given in Table P3.12.
Calculate:
(a)The flow rate Q in ft3/s throughout the system. (b) The mean velocities (ft/s) in the surface piping, the drill pipe, and the annular space between the casing and the tubing.
Then, assuming for the moment that all friction factors are the same, show that the frictional dissipation F for the annular space is likely to contribute only on the order of 1% to F for the surface and drill pipe, and may therefore be reasonably neglected. Finally, if the pump is running at 79% overall efficiency, compute the required pumping horsepower, within 2%.