A 3D cylindrical PML/FDTD method for elastic waves in fluid‐filled pressurized boreholes in triaxially stressed formations

A new 3D cylindrical perfectly matched layer (PML) formulation is developed for elastic wave propagation in a pressurized borehole surrounded by a triaxially stressed solid formation. The linear elastic formation is altered by overburden and tectonic stresses that cause significant changes in the wave propagation characteristics in a borehole. The 3D cylindrical problem with both radial and azimuthal heterogeneities is suitable for numerical solutions of the wave equations by finite-difference time-domain (FDTD) and pseudospectral time-domain (PSTD) methods. Compared to the previous 2.5D formulation with other absorbing boundary conditions, this 3D cylindrical PML formulation allows modeling of a borehole-conformal, full 3D description of borehole elastic waves in a stress-induced heterogeneous formation. We have developed an FDTD method using this PML as an absorbing boundary condition. In addition to the ability to solve full 3D problems, this method is found to be advantageous over the previously reported 2.5D finite-difference formulation because a borehole can now be adequately simulated with fewer grid points. Results from the new FDTD technique confirm the principle of superposition of the influence of various stress components on both the borehole monopole and dipole dispersions. In addition, we confirm that the increase in shear-wave velocity caused by a uniaxial stress applied in the propagation direction is the same as that applied parallel to the radial polarization direction.

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