Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 258

can we use constructive interference to increase MWD signal strength? How are
multiple transducer methods designed to filter signal reflections and pump noise
from true information content? How are swab-surge pressures in tight holes
accurately modeled within a water hammer framework? In this chapter, we
extend our classical wave equation discussions to such problems and highlight
recent applications to swab-surge analysis and high-data-rate mud pulse
telemetry design. We emphasize math modeling techniques and also summarize
state-of-the-art results from the author' s recent topic Measurement While
Drilling Signal Analysis, Optimization and Design (Chin et al , 2014).
5.1 Governing Lagrangian Equations
In mud acoustics, two fluid-dynamical models can be formulated, each
with its own strengths and areas of application. These are the “Lagrangian”
model, in which momentum book-keeping follows a fixed element of mass, and
the “Eulerian” model, in which we focus our attention at specific points in
space. The Lagrangian derivation is unchanged from that of Chapter 4 for
drillstring axial vibrations, except to the extent that Young' s modulus E is
replaced by the “bulk modulus” B. Let us first repeat this derivation for a
uniform cross-sectional area, using terminology appropriate to the present
chapter. We emphasize that viscous shear stresses acting on a smaller hydraulic
scale can be ignored in the propagation formulation. Viscous flow details will
no doubt affect the attenuation model, but these effects can be introduced after-
the-fact through an imaginary frequency. In modeling long wave propagation,
only compressibility effects due to pressure need to be considered.
5.1.1 Hydraulic versus acoustic motion.
In will be convenient to distinguish between two fluid-dynamic limits
using the terms “hydraulic” and “acoustic.” By “hydraulic,” we refer to the
constant density, incompressible component of flow generally associated with
the inviscid Bernoulli equation, or alternatively, with viscous pipe flows, or with
both. These flows may be steady or unsteady, as in a uniform wind or one that
suddenly changes direction. By “acoustic,” we refer to transient, compressible
flow effects only. To explain this simply, imagine a row of infinitely rigid
billiard balls in contact with each other. Any displacement at one end is
instantly felt at the opposite end: incompressibility is responsible for the infinite
speed with which information travels. If these balls were made of rubber, with
each ball compressing and expanding before passing on the motion, the effect at
the opposite end would require a time delay - this acoustic effect is due to
compressibility. From a layman's perspective, air is certainly compressible and
water may not appear to be; however, water and drilling mud certainly are
compressible, a fact not realized by all engineers.