Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 57

2.1.8 Ease of use is important to practical engineering.
In conventional wave studies, authors investigate the consequences of
partial differential equations and their auxiliary conditions; clearly, equation-
based methods can produce elegant closed form dispersion relations. But often
in research, the governing equations and boundary conditions behind a new
physical phenomenon may be unknown. Engineers focus on two objectives, the
first of which determines formulations that replicate observed results. For
example, it is easily demonstrated experimentally that gravity waves in deep
water satisfy r (k) = (gk) 1/2 , where g is gravitational acceleration (Lamb, 1945).
However, it would have been difficult to guess the formulation that produces
this result; in fact, it arises as the solution of (an elliptic, not wave) Laplace's
equation subject to free surface constraints (see Chapter 3). Thus, experiments
can motivate math models. The second objective focuses on applying measured
results immediately, since analytical models tend to be complicated and not
useful. This is true for seismic waves in oil bearing rocks; many theoretical
models have been proposed to describe rock matrix and pore fluid interaction
without definitive conclusions. KWT provides an alternative. All that is
required to use Equations 2.26 and 2.27 is a frequency function = (k,x,t).
The dependence of on k, x and t may be obtained analytically using
differential equations, numerically using computational methods, or empirically,
from field or laboratory observations. The latter objective opens up the greatest
application potential, but before discussing examples, we need to develop new
ideas on attenuation modeling.
2.2 Simple Attenuation Modeling
Wave motions in real media survive over many wavelengths or periods
before damping significantly. Thus, damping is sometimes ignored during the
early stages of modeling. Over larger space and time scales, damping must be
accounted for since it affects wave amplitude and kinematics. The terms
attenuation, damping, dissipation, internal friction and nonconservative effects
are used interchangeably, and we follow this custom. There are two approaches
that have been used to describe attenuation, and we present several unifying
ideas in this section.
2.2.1 The Q-model.
We will reconcile the “Q-model” popular in geophysics, with the complex
frequency approaches used in vibrations and hydrodynamic stability; then, we
integrate these ideas with recent developments in high-order KWT. First, let us
define the Q-model and relate it to simple propagating and standing waves
before adapting it for general use in kinematic wave theory. In geophysical
problems, a dimensionless measure of attenuation Q is commonly used. If a
volume of material is cycled in stress at a fixed frequency, Q can be defined as