Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 182

F'(t -x/c) = - {c u/ x - u/ t - u}e t /2 (4.2.159)
= - {c (u i+1,n - u i-1,n )/2 x - (u i,n - u i,n-1 )/ t - u i,n }/2 (4.2.160)
to leading order, where we have used a “finite difference representation” for the
right-side of Equation 4.2.159 (see Chapter 5), with x and t now interpreted
as “transducer separation distance” and “sampling time,” respectively.
The left side of Equation 4.2.160 depends only on present data and single
time step old information; thus, the scheme is robust because the effects of
history completely disappear. The scheme is particularly useful because the “c”
in Equation 4.2.160 represents the easily measured mud sound determined at the
surface. This is crucial because the actual sound speed of the mud varies along
the drillstring; mud properties are not constant over the wide range of pressures
and temperatures found throughout the flowing system. We stress that while we
have digressed into MWD telemetry, the ideas apply equally to drillstring axial
wave formation imaging; only the physical scales, and hence, transducer
separations and sampling requirements, change.
The appearance of the indexes i-1 and i+1 indicates that at least two
transducers are required; u values at “i” can be taken as arithmetic averages of
those at i-1 and i+1. On the other hand, the appearance of n and n-1 indicates
that two levels of time storage at least are required. Whether we deal with
MWD telemetry, or drillstring acoustic imaging, what is meant by a derivative is
crucial. Suppose, for the latter, that c = 15,000 ft/sec and bit excitations having
frequencies up to 15 Hz are considered. Then, the wavelength of 15,000
ft/sec/15 Hz or 1,000 ft suggests a transducer separation of 10% of wavelength
or 100 ft (a 10% of period rule might apply to the discretization t). More
accurate “difference molecules” for alternative x and t derivative
approximations can be used for increased accuracy; these will require additional
transducers in space, more time level memory capacity, and increased computer
resources, although the total required work is still modest even at data rates up
to a hypothetical 100 bits/sec.
Three-wave formulation. In some physical systems, more than two waves
may coexist simultaneously; for example, in borehole geophysics, upgoing and
downgoing waves having like speeds may work in the presence of surface
waves. The differential method is easily extended, but we need to evaluate
derivatives at the next higher order. For example, corresponding to
u(x,t) = F(t +x/c 1 ) + G( t +x / c 2 ) + H(t+x/c 3 ) (4.2.161)

are the second derivative formulas

F" + G" + H" = u tt (4.2.162a)

1/c 1 F" + 1 / c 2 G" + 1 / c 3 H" = u xt (4.2.162b)

1/c 1 2 F" + 1 / c 2 2 G" + 1 / c 3 2 H" = u xx (4.2.162c)