Wavespeed for grouted riser

[stanier]

Where a mine dewatering pump discharge is grouted in how does one determine the modulus from which one can determine the wavespeed (celerity). The application is a vertical mine dewatering pump discharge line running vertically through rock and grouted in position. it is presumed that the modulus will be much higher than that for the pipe material as it is basically reinforced by the rock. Is there a maximum that should be used for wavespeed?

 

[sms]

The effective acoustic velocity for fluid in a pipe is as follows:

C= CL * [ (D*El/Ep*t) +1]^ -1/2 ref.

Where

C is the acoustic velocity in the pipe
CL is the theoretical acoustic velocity in the liquid as in a unbounded medium
D is the pipe diameter
El is the bulk modulus of the fluid (lb/ft^2)
Ep is the modulus of the pipe, including the effects of the grout and rock
‘t is the effective thickness of the pipe.

From this equation, as the pipe thickness, or the pipe modulus gets very large the acoustic velocity in the liquid approaches that of an unbounded medium. The more compliant the pipe wall is, the lower the acoustic velocity. In your case I suspect that the pipe is not very compliant, so the acoustic velocity should be close to theoretical for an unbounded medium.

Ref: This equation is from the Southwest Research Institute course on controlling the effects of pulsation and fluid transients in piping systems.

 

[butelja]

I don't have the formula, but it seems that the rock should be modeled as an infinite cross sectional area with a hole in it undergoing plain strain. Given the rock modulus, you should, with the appropriate formula, be able to work out a radial stiffness. Combine this with the pipe radial stiffness to obtain the stiffness of the rock/grout/pipe system.

Try posting this to the Structural Engineers forum, as there are people very knowledgeable in structural mechanics in that forum.

As a reality check, the wave speed will fall somewhere between the wave speed of water in an unreinforced pipe and water in an infinitely rigid pipe. I'm guessing that it will fall closer to the infinitely rigid case.