Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 173

4.2.11.4 Basic mathematical approach.
The interaction between the rock bit and the formation was assumed in
Equations 4.2.22 and 4.2.45 to take the general form u x + u t + u = 0, where
u denotes the value u(0,t) at the bit. In reality, the actual condition may be
nonlinear, but this would preclude analytical solution; of course, complicated
boundary conditions can be used numerically. In our forward simulations, the
constants , and were assumed to be known from detailed laboratory studies
where various types of bits are drilled into different rocks using vibration
isolated apparatus; thus, u x + u t + u = 0 provides a characteristic signature
for the particular rock-bit pair. In formation imaging , the inverse problem
determining , and given measured values of u, u x and u t is considered. The
problem is difficult if it is tackled using Equations 4.2.25 to 4.2.30 in their
entirety. Even if a closed form solution were available, uncertainties will arise
because input parameters (e.g., our surface mass-spring-damper constants)
cannot be characterized accurately. From this viewpoint, an improved
understanding of the vibration process only leads to increased uncertainty in the
results. But without a useful host solution, seeing ahead of the bit is destined to
failure. The crucial idea is the use of a single limited and stable aspect of the
total problem that relies on a minimum of uncertain information.
To motivate the ideas, we consider a semi-infinite drillstring in order to
develop a simple inverse model. Recall that undamped axial vibrations satisfy
the classical wave equation for u(x,t), whose general solution takes the form u =
f(ct-x) + g(ct+x); here, g represents a downgoing incident wave and f represents
the reflected upgoing wave. Suppose that processed information from all
necessary sensors is available, and that in this first example, the data indicates a
reflected displacement wave equal in magnitude to the incident, but out-of-phase
by 180 o . Thus, we would choose f(ct-x) = -g(ct-x); from the wave solution
u(x,t) = -g(ct-x) + g(ct+x) that follows, we find that u(0,t) = 0. Thus, we infer
that the drillstring is in contact with rigid, unyielding, hard rock. If in our
second example, we detect a reflected displacement wave equal in magnitude
but perfectly in phase, we would select f(ct-x) = g(ct-x). In this case, the
solution u(x,t) = g(ct-x) + g(ct+x) similarly constructed would lead us, upon
differentiation with respect to x, to the conclusion that the strain satisfies u x (x,t)
= -g'(ct-x) + g'(ct+x) or u x (0,t) = 0. Thus, one would infer a free end, or soft,
unconsolidated formation.
We discussed our above input information in terms of displacements . We
could have posed the problem via stress . The corresponding interpretations
would be made from the rule derived in Chapter 1: stresses reflect with like
signs at rigid (hard) boundaries and opposite signs at free (soft) boundaries.
These simple but exact “thought experiments” drew upon several idealizations.
First, we used known rigid and fixed end results from classical theory to assist in