Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 322

where > 0, a “small number,” is related to the viscosity of the fluid, and the
term 2 u/ x 2 as a whole represents “viscous diffusion.” The exact “Cole-Hopf
solution” is discussed extensively in Whitham (1974). It is possible to show that
physical models that satisfy Equation 6.88 can be modeled by Equation 6.81,
which is considerably simpler. In other words, shock solutions of Equation 6.81
are also obtained from the more detailed description offered by Equation 6.88.
Why is this not intuitively obvious? This issue is subtle because we have
not stated what is small with respect to; when spatial gradients, as obtained
from Equation 6.81 become large, the 2 u/ x 2 term in the more complete
model of Equation 6.88 may no longer be small by comparison to the left hand
side. Thus, Equation 6.81 may or may not apply near the shock, and direct
recourse to the detailed physical model must be made.
Now consider the “Korteweg deVries equation,” obtained in the study of
long, inviscid, water waves. Instead of Equation 6.88, we have
u/ t + u u/ x = 3 u/ x 3 (6.89)
where > 0 is also a “small number.” While Equations 6.88 and 6.89 differ
only in the order of the “small” right-side term, the solution structure governing
Equation 6.89 is completely different. An exact solution to the general initial
value problem, using methods in “inverse scattering theory,” is again available
and described in Whitham (1974). It turns out, interestingly, that Equation 6.89
does not admit any solutions with shocks. Thus, even if

is small, Equation

6.81 is never relevant as a simplified model.
Our point is this: the high-order terms in Equations 6.88 and 6.89 control
the dynamics of the solution at all scales. If shocks exist, their structure and
thickness are completely determined by the diffusive term 2 u/ x 2 . In
gasdynamics, this term is related to the viscous stress tensor; in reservoir
engineering, the analogous high-order term is dictated by the capillary pressure
function. If high-order dispersive terms such as that in Equation 6.89 exist, e.g.,
the terms related to the high order wavenumber derivatives of the real frequency
in Equations 6.62 and 6.63, shocks may never form. At any rate, the interplay
between high-order diffusion (as in Equation 6.88) and high-order dispersion is
very subtle. Low-order models by themselves are never enough: one must
always examine the next higher order structure for clues to solution structure.

6.6.4 Entropy conditions.
We have emphasized that once the high-order model is agreed upon, say
Equation 6.88 or 6.89, the complete physical description of the problem is self-
contained. That is, the “entropy conditions” one normally “pulls from hats” in
thermodynamics courses can be straightforwardly obtained through simple
integration by parts. To see that this is so, let us consider Equation 6.88. For