Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 384

10.3 Effects of Borehole Eccentricity
10.3.1 Industry formulations, solutions and approaches.
Borehole acoustics (or geophysics) is a specialty which, like annular flow,
electromagnetic logging or reservoir engineering, requires exhaustive study, in
each case focusing on differential equation formulations, custom solution
techniques and complementary experimental validation. Despite their
differences, each of these disciplines has been approached from a common
perspective in its scientific evolution - concentric problems were tackled first
with eccentric modifications handled later. Borehole acoustics is no exception.
The Earth Resources Laboratory at the Massachusetts Institute of Technology,
through its well known Full Waveform Acoustic Logging Consortium , has
developed over the years an impressive array of modeling tools essentially for
concentric problems having radial symmetry, and detailed (client proprietary)
annual reports are available which document the range of problems solved.
While concentric solutions have been useful in the past, particularly in the
drilling of vertical wells, they are less useful in deviated and horizontal well
applications where drillstrings rest near the bottom of the hole and render the
cross-sectional geometry highly eccentric. In fact, one can emphatically state
that, in modern well logging tool design and data interpretation, eccentricity
effects now predominate and make concentric models less and less useful. The
review article of Cheng and Blanch (2008), entitled “Numerical Modeling of
Elastic Wave Propagation in a Fluid-Filled Borehole,” discusses recent methods
for simulating elastic wave propagation in a borehole. It considers two different
approaches: a quasi-analytic approach using the Discrete Wavenumber
Summation Method and the purely numerical Finite Difference Method .
The article explains strengths of these models as well as numerical
difficulties that make the approaches less useful in simulating reality. For
instance, the discrete wavenumber approach is quasi-analytic and limited to
situations with radial symmetry, and in particular, formations homogeneous in
the vertical direction (that is, classical concentric vertical well problems). The
finite difference approach described by the authors offers a solution with radial
and vertical heterogeneities, but is presumably limited by numerical instabilities.
This, perhaps, is not too much of a practical impediment - researchers usually
solve such problems in due course. The problem, however, is the coordinate
system used to host the differencing - it is “r, and z” based, therefore
implicitly assuming near-concentric applications. It is noted, however, that “the
most efficient method is to do the simulation in three dimensions and in
Cartesian coordinates,” but that the most obvious limitation is “the
approximation of the curved surface of the borehole” in rectangular coordinates.
And rightly so - for this reason, a large body of work has been developed in
recent years in several petroleum disciplines by the present author utilizing
general “curvilinear grid generation” to remedy these deficiencies.