Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 272

m u' t = - p' x in Equation 5.40. This proves our earlier assertion that a constant
background speed is dynamically insignificant, since it can be removed by a
simple Galilean transformation.
Limit No. 4. When the background mean flow varies weakly with x and t,
momentum and energy transfer between wave and mean states in general occurs
that cannot be ignored over large scales. This transfer is related to “Reynolds
stress” effects in fluid mechanics, responsible for laminar flow instabilities;
analogous terms in wave propagation are termed “radiation stresses.” These
weak variations do not exist in uniform area pipes and boreholes. They do,
however, in the variable mean current ocean applications treated later.
Limit No. 5. As the mud pulser opens and closes in MWD telemetry
operations, oncoming fluid is brought to sudden starts and stops, in the process
creating well known “water hammer” noise. Even though valve speeds are
much less than typical speeds of sound, the interaction between mean hydraulic
and local compressible flow is strong. This effect is three-dimensional and
highly localized. In principle, it can be calculated, say, using computational
fluid-dynamics (CFD) approaches. In practice, however, simulations that
provide physical resolution at the smallest length scales imply extremely long
computing times and huge memory resources; numerical artifacts such as
“artificial viscosity” may introduce rheological effects that are unrealistic. Thus,
in Su et al (2011) and the topic Measurement While Drilling Signal Analysis,
Optimization and Design (Chin et al , 2014), a hybrid analysis approach is
adopted: analytical models are used for wave propagation in the farfield while
wind tunnel measurements (rescaled to mud conditions) are used to determine
source characteristics as they depend on fluid properties and valve geometry.
We now digress to discuss finite difference model of transient wave fields.
5.3 Transient Finite Difference Modeling
We now discuss finite difference modeling using our Lagrangian
formulation as the host discussion model. Since this involves the displacement
function u(x,t), the derivations proceed along the lines of Chapter 4 for axial
vibrations. We will replace Equation 4.2.25 by
2 u/ t 2 + u/ t - B 2 u/ x 2 = 0 (5.51)
so that Equation 5.51 extends Equations 5.1 and 5.2 to include a simple
dissipation model.

5.3.1 Basic difference model.
Following steps similar to those leading to Equations 4.2.88 and 4.2.89, we
derive the finite difference equation
U i-1,n - {2 + ( /B)( x/ t) 2 + ( /2B)(( x) 2 / t)} U i,n + U i+1,n

= - ( /B)( x/ t) 2 (2U i,n-1 - U i,n-2 )- ( /2B)(( x) 2 / t) U i,n-2 (5.52)