Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 385

10.3.2 Successes in eccentricity modeling.
Concentric formulations can be useful since their analytically-based
solutions often result shed considerable insight into the physical problem. An
excellent example is our use of Equation 10.1 together with a “kinematic wave
focus” allowing us to prove all of the Stoneley wave features uncovered only
numerically by previous investigators - and then, additionally provide useful
algorithms for permeability prediction. But analytical models in cylindrical
geometries, whether they appear in borehole acoustics or electromagnetics, are
often “married” to Bessel function solutions, and unfortunately, just as often to
complicated ones with complex arguments. Such approaches, which can never
be extended to truly eccentric applications, hold little promise in practical
application. As Cheng and Blanch (2008) correctly observe, in the context of
borehole acoustics, Cartesian or rectangular mesh systems offer some
improvement; however, straight lines will always provide poor first
approximations to curved ones found in real world applications. In this section,
we review recent advances in eccentric geometry modeling using the “boundary-
conforming, curvilinear grids” developed by the present author over the years.
Figure 10.6 (center) shows how circular problems are best hosted by
circular grids - the side illustrations demonstrate how rectangular grids could be
used, e.g., through rectangular or circular “grids within grids,” but their
coordinate lines will never truly conform to the highly curved boundaries that
need accurate characterization. It is possible, as described in Figure 10.7, to
increase rectangular grid density and approximate curves using “staircase grids,”
but then, questions related to density and aspect ratio arise. Such gridding
methods introduce numerical noise and uncertainty into computational schemes.
Figure 10.6. Embedded fine-grid systems using classical coordinates.
Figure 10.7. Meshes requiring interpolation and “staircase” grids.