Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 282

Mud

X = X

s

Figure 5.4. Positive pressure poppet pulser.
Put differently, not all delta-p' s propagate as sound: only acoustic signals
propagate rapidly with any efficiency, and the greatest practical potential is
reaped through transmission via “AC” rather than hydraulic “DC” means (again,
refer to Chapter 1 for an explanation of our AC/DC nomenclature).
Accordingly, it is this acoustic delta-p or discontinuous jump in acoustic
pressure that should be created mechanically and modeled analytically. So far,
we have developed many wave equation models for propagation in one-
dimensional systems. Dependent variables have included pressure, velocity
potential, density, velocity and so on. All of these are viable candidates as
mathematical models in typical engineering problems where excitations occur at
the boundaries of the system. In MWD applications, the acoustic source is
found within the transmission channel, and the appropriate governing equation
must account for the physics of the source, that is, the symmetries or
antisymmetries in created velocity or pressure (see Chapter 1).
For poppet valves, the acoustic source is characterized by a jump in
pressure through the source point. In Chapter 1, we showed that the addition of
a delta function (x-x s ) to the right side of a wave equation in the dependent
variable, say, h(x,t), is responsible for a jump in the first spatial derivative h/ x
through the point x = x s . Now consider the Lagrangian fluid displacement
function, which measures displacements from (static or flowing) equilibrium.
Earlier we found that it satisfies 2 u/ t 2 - B 2 u/ x 2 = 0. If we add to it the
delta function excitation P s (t) (x-x s ), we obtain,