Siren Design and Evaluation in Mud Flow Loops and Wind Tunnels - Measurement While Drilling Signal Analysis, Optimization and Design - page 257

positive pulsers since surface values range up to 100 psi. If this acoustic model
is valid, then p s is linearly proportional to Q (and also to U and c) as opposed to
Q 2 . But because siren signal generation is both acoustic and hydraulic (due to
leakage through the gap), the truth, most likely, lies somewhere in between, with
U depending on a combination of Q and rotation rate Z.
9.3.5 Predicting mud flow ' p's from wind tunnel data.
How to analyze and extend wind tunnel pressure results to actual mud
flowrates and densities raises questions best answered through discussion. We
motivate the mathematical ideas using our wind tunnel methods for turbine stall
torque analysis (see Chapter 8 for a complete discussion). In any physical
problem, it is important to identify the key parameters. Since torque is primarily
determined by the inviscid character of the fluid, viscosity and rheology are
unimportant to leading order (these parameters are, of course, important in
assessing viscous losses). This being the case, the primary variables needed to
characterize the torque T are the oncoming speed U, the density U, the surface
area A and the mean moment arm R. This observation, while simple, is key.
The “dimensional analysis” methods used in engineering argue that natural
phenomena can and should be expressible in terms of fundamental
dimensionless relationships. The quantity UU 2 AR is the only product that can be
formed with dimensions of torque. Thus, the ratio T/(UU 2 AR) can only depend
on the remaining dimensionless parameter, namely, the shape of the turbine.
This idea is expressed by the equation T/(UU 2 AR) = C T where C T is a
dimensionless torque coefficient associated with the geometry being considered.
Now suppose that measurements are made in both air and mud. Since C T is
invariant, one must have T air /(U air U air 2 A air R air ) = T mud /(U mud U mud 2 A mud R mud ).
For full-scale testing, we have A air = A mud and R air = R mud . Hence, we find that
the equation T mud = (U mud /U air )(U mud /U air ) 2 T air can be used to determine mud
torque at any flow rate and density from any set of wind tunnel data.
Now, we turn to MWD signal generation and attempt to understand the
underlying physics using similar techniques. Consider the semi-infinite pipe at
the top in Figure 9.11 with a piston at the left end. If this piston is struck, for
instance, by a hammer, an acoustic “water hammer” pressure wave is created
which propagates to the right. Fortunately, in this example, an exact, analytical
solution to the governing one-dimensional wave equation can be developed.
Again, we consider the fluid displacement u(x,t), which satisfies u tt - c 2 u xx = 0
where c is the sound speed. The general solution takes the form u = f(t - x/c)
which represents a right-going wave. At the piston face where x = 0, this
reduces to the boundary condition u(0,t) = f(t) for piston position. Recall that
the acoustic pressure P(x,t) satisfies P(x,t) = - B u x (x,t) where B is the bulk
modulus. Then, P(x,t) = (B/c) f ' (t - x/c) or, since c 2 = B/U where U is fluid
density, a simple P(0,t) = Ucf '(t) = UcV(t) holds, where V(t) is piston speed.