Siren Design and Evaluation in Mud Flow Loops and Wind Tunnels - Measurement While Drilling Signal Analysis, Optimization and Design - page 258

Piston Acoustic signal Semi-infinite pipe

V(t)

c,U

P(t)

Figure 9.11. Three pressure disturbance modes.
In summary, the acoustic pressure developed by a piston that fully
occupies the cross-section of the pipe is P(0,t) = UcV(t). The middle drawing in
Figure 9.11 shows a piston with nonzero wall clearance, which allows flow
leakage as the piston moves. One therefore expects that the pressure predicted
by the foregoing formula cannot be realized fully, and it is reasonable to modify
that relationship in the form P(0,t) = UcV(t)G where G < 1 is a dimensionless
efficiency factor associated with piston geometry. Thus, we write P/(UcV) = G,
where G can be determined from wind tunnel or mud flow loop analysis.
Because the dimensionless G represents a unique descriptor for the piston, then,
as in our torque analysis, we can infer that P air /(U air c air V air ) = P mud /(U mud c mud V mud )
or P mud /P air = (U mud /U air )(c mud /c air ) (V mud /V air ). For example, a heavy mud might
have U mud /U air = 1,500 and c mud /c air = 3. If the wind tunnel test is carried out at
300 gpm and the desired downhole volume flow rate is 1,200 gpm, then P mud /P air
= 1,500 u 3 u 4 = 18,000. Thus, a 'p of 0.01 psi measured in the wind tunnel
translates to 180 psi under the downhole conditions assumed.
Once G is determined from wind tunnel analysis, one can use the model
P air = U air c air V air G for air flow and P mud = U mud c mud V mud G for mud. We caution
that our discussion so far assumes that the only physically important
phenomenon is acoustical. In the bottom sketch of Figure 9.11, we do not have
a piston; instead, any analogous piston action is deemed to be so slow that the
left end acts as an orifice. When this is the case, the sound speed c is no longer
important, and UU 2 is the only quantity with dimensions of pressure. Thus, the
ratio P/(UU 2 ) must depend on the geometrical details of the orifice only,
characterized by a dimensionless constant H, also measured in the wind tunnel.
In fact, we can write P air /(U air U 2 ) = P mud /(U mud U mud 2 ) = H, from which it follows
that P air = U air U 2 H and P mud = U mud U mud 2 H. Note that in the purely acoustic
model, the pressure depends linearly on V(t), while in the constant density
hydraulic model, it depends quadratically on V(t).