Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 71

2.5.1 Detailed formal analysis.
We introduced waves in nondissipative, nonuniform media earlier in
Equations 2.21 to 2.27. We now consider dissipative, nonuniform media,
restricting ourselves to low-order effects, in much greater detail; quantities
similar to the high-order terms in Equations 2.94 and 2.95 are not considered.
Again, Equation 2.100 can be simplified and solved in characteristic form.
Using the chain rule of calculus, we have
k/ t + r k (k,x,t) k/ x = - r x (k,x,t) (2.101)
The total derivative of k(x,t) is just dk = k/ t dt + k/ x dx. Dividing through
by dt, we obtain dk/dt = k/ t + k/ x dx/dt. On comparison with Equation
2.101, it is clear that
dk/dt = -

r x (k,x,t)

(2.102)

along the ray path
dx/dt = r k (k,x,t) (2.103)
In Equations 2.102 and 2.103, the total derivative d/dt denotes changes
following the wave (a.k.a., the convective, material, or substantive derivative).
The speed of interest is the group velocity C g = r k (k,x,t) obtained as the partial
derivative of r k (k,x,t) with respect to k, holding x and t fixed. Since the
(k,x,t) is available, from analytical modeling or empirical measurement, these
first-order ordinary differential equations in time are easily integrated to give x
and k along a wave trajectory; Equations 2.26 and 2.27 apply exactly as before.
When the dispersion function does not depend explicitly on x and t, the
result dk/dt = 0 indicates that k is a constant equal to its initial value; the
corresponding equation dx/dt = r k (k) = constant shows that the characteristic
must be a straight line. On the other hand, when a dependence on x is allowed,
k will vary along the ray, and the ray will no longer be straight. In a drilling
application studied later, we show that ray convergence due to nonuniformities
is responsible for wave trapping and high-amplitude lateral vibrations in drill
collars. The (mathematical) inhomogeneities, as far as the beam equation is
concerned, arise from axial stress variations along the drillstring. The results
given here will be significant in downhole vibration modeling.

2.5.2 Wave energy and momentum.
In our earlier work dealing with waves in uniform media, we found that
a 2 / t + { r k (k)a 2 }/ x = 2 i a 2 describes amplitude (see Equation 2.55).
But when heterogeneities are present, work interaction with the underlying
medium in general exists. In mechanics topics, concepts such as kinetic,
potential strain, or potential gravitational energy are developed from differential
equation models for specific distributed systems, but the subject can be treated