A three-dimensional finite difference simulation of sonic logging

Abstract

A three-dimensional finite-difference ͑FD͒ method is used to simulate sonic wave propagation in a borehole with an inhomogeneous solid formation. The second-order FD scheme solves the first-order elastic wave equations with central differencing in both space and time via staggered grids. Liao's boundary condition is used to reduce artificial reflections from the finite computational domain. In the staggered grids, sources have to be implemented at the discrete center in order to maintain the appropriate symmetry in an axisymmetric borehole environment. The FD scheme is validated for multipole sources in three special media: ͑i͒ a homogeneous medium; ͑ii͒ a homogeneous formation with a fluid-filled borehole; and ͑iii͒ a horizontally layered formation. The staircase approximation of a circular borehole introduces little error in dipole wave fields, although it causes a noticeable phase velocity error in the monopole Stoneley wave. This error has been drastically reduced by using a material averaging scheme and finer grids. Numerical examples show that the FD scheme can accurately model 3-D elastic wave propagation in complex borehole environments.

Acoustic detection of buried objects in 3-D fluid saturated porous media: numerical modeling

“…The strain field is computed at and velocity field is computed at . This staggered grid is similar to that for elastic waves in a solid [4], [22].…”

“…Then the time-stepping equations can be written as (22) (23) (24) where , , and are right-hand sides of equations (19), (20), and (21), respectively. It should be noted that the material parameters in the above equations must be properly averaged in order to arrive at a higher accuracy [4]. In order to save computer storage, the computational domain is divided into a PML region and an interior region.…”

“…Conventional methods for acoustic modeling include the popular finite-difference time-domain (FDTD) solution of pure elastic media [3], [4]. Unfortunately, the pure elastic model cannot adequately incorporate the attenuation mechanism in its governing equations, although some approximate models are possible [5].…”

Finite-difference time-domain method for heterogeneous orthotropic media with damping

“…(6)- (29). Herein, all the material parameters are defined at the reference points of normal stress, and the mean values are used as reference points for velocity and shear stress [13][14][15][16][17]. Arithmetic averages are employed at the reference points of velocity, for example, &ði þ 0:5; j; kÞ ¼ &ði; j; kÞ þ &ði þ 1; j; kÞ…”

“…One of the two types of damping is proportional only to the velocity and the other is proportional to the secondorder space derivative of the velocity and the strain velocity. The vibroacoustic FDTD method assumes that both solids and fluids are governed by a unique set of motion equations and viscoelastic constitutive equations using averaged material parameters [13][14][15][16][17]. If the simultaneous existence of solids and fluids can be considered as heterogeneity in one medium, then a vibroacoustic problem can be treated as a type of heterogeneous problem by employing averaged material parameters all over the solid and fluid regions.…”