Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 391

Specifying the problem domain. The governing equations that result are
transient with second-order time derivatives. Three solution approaches are
possible, as discussed in Chapter 1. First, a direct finite-difference time-
marching scheme, e.g., second-order accurate in time and fourth-order in space,
can be used. Second, Equations 10.27a,b,c can be Fourier-transformed in time
and the resulting complex system of equations integrated with elliptic equation
solvers. The solution process must be repeated for a large number of
frequencies, although the work load can be reduced significantly if field
solutions for one frequency are used to initialize the solution process for a
subsequent frequency. This process is repeated for a wide range of frequencies.
With the Fourier transform defined for all space and frequency parameters, time-
based solutions can be easily constructed. And finally, Laplace transform
methods are possible following a procedure identical to that for Fourier
transformations. The well-known Stehfest algorithm, for instance, can be used
to support the inversion process. We have not approached the overall problem
using “staggered grid” velocity-stress formulations containing only first-order
time derivatives; second-order wave-equation type approaches seem more
natural given the wider body of literature available. We note, in closing, that
characterizing the monopole, dipole or quadrupole nature of the source proved
to be the greatest challenge given demanding space and time requirements on
physical resolution. The use of delta functions and couples, as described in
Chapter 4, provides cost-effective means to represent disturbance symmetries
and antisymmetries associated with these signal generators.
10.4 References
Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions ,
Dover, 1964.
Aki, K. and Richards, P.G., Quantitative Seismology: Theory and Methods , W.
H. Freeman and Co., 1980.
Biot, M.A., “Theory of Propagation of Elastic Waves in a Fluid-Saturated
Porous Rock: I. Low-Frequency Range,” Journal of the Acoustical Society of
Ame r i c a , Vol. 28, 1956a, pp. 168-178.
Biot, M.A., “Theory of Propagation of Elastic Waves in a Fluid-Saturated
Porous Rock: II. Higher-Frequency Range,” Journal of the Acoustical Society of
Ame r i c a , Vol. 28, 1956b, pp. 179-191.
Biot, M.A., and Willis, D.G., “The Elastic Coefficient of the Theory of
Consolidation,” Journal of Applied Mechanics , Vol. 24, 1957, pp. 595-601.
Bouchon, M. and Aki, K., “Discrete Wavenumber Representation of Seismic-
Source Wavefields,” SSA Bu l l e ti n , 67, 1977, pp. 259-277.